solution stringlengths 1 23 | id int64 1 15k | problem stringlengths 19 3.19k | question stringlengths 19 3.19k | answer stringlengths 1 23 | source stringclasses 1
value |
|---|---|---|---|---|---|
3 | 14,901 | 函数 $f(x)$ 对于任意的实数 $x$ 满足 $f(x+3)=-\frac{1}{f(x)}$。若 $f(0)=2$，求 $f(2013)$ 的值。答案的形式为 $-\frac{k}{m}$，请给出 $k + m$ 的值。 | 函数 $f(x)$ 对于任意的实数 $x$ 满足 $f(x+3)=-\frac{1}{f(x)}$。若 $f(0)=2$，求 $f(2013)$ 的值。答案的形式为 $-\frac{k}{m}$，请给出 $k + m$ 的值。 | 3 | dapo_math |
0 | 14,902 | 已知 $x, y \in \mathbf{Z}$ ，若 $\left(x^{2}+x+1\right)^{2}+\left(y^{2}+y+1\right)^{2}$ 为完全平方数，则数对 $(x, y)$ 的对数是多少？ | 已知 $x, y \in \mathbf{Z}$ ，若 $\left(x^{2}+x+1\right)^{2}+\left(y^{2}+y+1\right)^{2}$ 为完全平方数，则数对 $(x, y)$ 的对数是多少？ | 0 | dapo_math |
36 | 14,903 | 12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of the others. We need to choose 5 knights to save the princess such that no two knights in the group are enemies. How many ways can this be done? | 12 knights are sitting at a round table. Each knight is an enemy with the two knights adjacent to them, but not with any of the others. We need to choose 5 knights to save the princess such that no two knights in the group are enemies. How many ways can this be done? | 36 | dapo_math |
78 | 14,904 | In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$. | In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$. | 78 | dapo_math |
40 | 14,905 | Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they ... | Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they ... | 40 | dapo_math |
4 | 14,906 | Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the cont... | Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the cont... | 4 | dapo_math |
3 | 14,907 | All solutions of the equation $\cos 4x = -\frac{1}{2}$ can be expressed in the form $\frac{(kn \pm 1) \pi}{6},$ where $n$ is an integer. Find the positive value of $k.$ | All solutions of the equation $\cos 4x = -\frac{1}{2}$ can be expressed in the form $\frac{(kn \pm 1) \pi}{6},$ where $n$ is an integer. Find the positive value of $k.$ | 3 | dapo_math |
100 | 14,908 | A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$. | A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$. | 100 | dapo_math |
35 | 14,909 | Let $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ . | Let $a_n=6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ . | 35 | dapo_math |
2014 | 14,910 | Let $a_1 < a_2 < a_3 < \ldots < a_n < \ldots$ be positive integers such that, for $n = 1, 2, 3, \ldots,$
\[ a_{2n} = a_n + n. \]
Given that if $a_n$ is prime, then $n$ is also, find $a_{2014}$. | Let $a_1 < a_2 < a_3 < \ldots < a_n < \ldots$ be positive integers such that, for $n = 1, 2, 3, \ldots,$
\[ a_{2n} = a_n + n. \]
Given that if $a_n$ is prime, then $n$ is also, find $a_{2014}$. | 2014 | dapo_math |
6 | 14,911 | If \(10^{2y} = 25\), then find the value of \(10^{-y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | If \(10^{2y} = 25\), then find the value of \(10^{-y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 6 | dapo_math |
34 | 14,912 | A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops, what is the probability that the cricket has returned to the leaf where it started? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the... | A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops, what is the probability that the cricket has returned to the leaf where it started? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the... | 34 | dapo_math |
3 | 14,913 | How many real pairs $(x, y)$ are there such that
\[
x^2 + 2y = 2xy \\
x^3 + x^2y = y^2
\]
Find the number of real pairs $(x, y)$ that satisfy the equations. | How many real pairs $(x, y)$ are there such that
\[
x^2 + 2y = 2xy \\
x^3 + x^2y = y^2
\]
Find the number of real pairs $(x, y)$ that satisfy the equations. | 3 | dapo_math |
751 | 14,914 | A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | 751 | dapo_math |
47 | 14,915 | The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. What is a number that could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? | The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. What is a number that could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? | 47 | dapo_math |
7 | 14,916 | Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. The original answer is in \(... | Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. The original answer is in \(... | 7 | dapo_math |
243 | 14,917 | 设集合 $M=\{1,2, 3, 4,5,6,7,8,9,10\}$ ，
$A=\{(x, y, z) \mid x, y 、 z \in M$, 且 $\left.9 \mid\left(x^{3}+y^{3}+z^{3}\right)\right\}$ 。
则集合 $A$ 中元素的个数为 $\qquad$. | 设集合 $M=\{1,2, 3, 4,5,6,7,8,9,10\}$ ，
$A=\{(x, y, z) \mid x, y 、 z \in M$, 且 $\left.9 \mid\left(x^{3}+y^{3}+z^{3}\right)\right\}$ 。
则集合 $A$ 中元素的个数为 $\qquad$. | 243 | dapo_math |
651222 | 14,918 | Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 2... | Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 2... | 651222 | dapo_math |
31 | 14,919 | 已知正四棱雉 $\Gamma$ 的高为 3, 侧面与底面所成角为 $\frac{\pi}{3}$ 。先在 $\Gamma$ 内放入一个内切球 $O_{1}$, 然后依次放入球 $O_{2}, O_{3}, O_{4}, \cdots$, 使得后放入的各球与前一个球及 $\Gamma$ 的四个侧面均相切，则放入的所有球的体积之和的原始答案为$\frac{m\pi}{n}$，请给出$m + n$的值。 | 已知正四棱雉 $\Gamma$ 的高为 3, 侧面与底面所成角为 $\frac{\pi}{3}$ 。先在 $\Gamma$ 内放入一个内切球 $O_{1}$, 然后依次放入球 $O_{2}, O_{3}, O_{4}, \cdots$, 使得后放入的各球与前一个球及 $\Gamma$ 的四个侧面均相切，则放入的所有球的体积之和的原始答案为$\frac{m\pi}{n}$，请给出$m + n$的值。 | 31 | dapo_math |
18 | 14,920 | Dilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A}$, and $f_{C,B}$. Each function $f_{X,Y}$ takes an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his 6 functions, without repeats, in such a way that the resulti... | Dilhan has objects of 3 types: $A$, $B$, and $C$. He also has 6 functions: $f_{A,B}$, $f_{A,C}$, $f_{B,A}$, $f_{B,C}$, $f_{C,A}$, and $f_{C,B}$. Each function $f_{X,Y}$ takes an object of type $X$ and outputs an object of type $Y$. Dilhan wants to compose his 6 functions, without repeats, in such a way that the resulti... | 18 | dapo_math |
397 | 14,921 | Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | 397 | dapo_math |
763 | 14,922 | Let $P_0(x) = x^3 + 313x^2 - 77x - 8\,$. For integers $n \ge 1\,$, define $P_n(x) = P_{n - 1}(x - n)\,$. What is the coefficient of $x\,$ in $P_{20}(x)\,$? | Let $P_0(x) = x^3 + 313x^2 - 77x - 8\,$. For integers $n \ge 1\,$, define $P_n(x) = P_{n - 1}(x - n)\,$. What is the coefficient of $x\,$ in $P_{20}(x)\,$? | 763 | dapo_math |
70 | 14,923 | In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$, and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\an... | In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$, and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\an... | 70 | dapo_math |
6 | 14,924 | If there is only \(1\) complex solution to the equation \(8x^3 + 12x^2 + kx + 1 = 0\), what is \(k\)? | If there is only \(1\) complex solution to the equation \(8x^3 + 12x^2 + kx + 1 = 0\), what is \(k\)? | 6 | dapo_math |
75 | 14,925 | Suppose that $x,$ $y,$ and $z$ are complex numbers such that \[\begin{aligned} xy &= -80 - 320i, \\ yz &=60, \\ zx &= -96 + 24i, \end{aligned}\]where $i^2 = -1.$ Compute $|x+y+z|.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | Suppose that $x,$ $y,$ and $z$ are complex numbers such that \[\begin{aligned} xy &= -80 - 320i, \\ yz &=60, \\ zx &= -96 + 24i, \end{aligned}\]where $i^2 = -1.$ Compute $|x+y+z|.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 75 | dapo_math |
781 | 14,926 | How many positive integer solutions are there to \( w + x + y + z = 20 \) where \( w + x \ge 5 \) and \( y + z \ge 5 \)? | How many positive integer solutions are there to \( w + x + y + z = 20 \) where \( w + x \ge 5 \) and \( y + z \ge 5 \)? | 781 | dapo_math |
12 | 14,927 | At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred? | At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred? | 12 | dapo_math |
2014 | 14,928 | Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compu... | Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compu... | 2014 | dapo_math |
-4 | 14,929 | If $(0.2)^x = 2$ and $\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$? | If $(0.2)^x = 2$ and $\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$? | -4 | dapo_math |
0 | 14,930 | Let $p(x)$ be a polynomial of degree 4 such that $p(55) = p(83) = p(204) = p(232) = 8$ and $p(103) = 13.$ Find
\[p(1) - p(2) + p(3) - p(4) + \dots + p(285) - p(286).\] | Let $p(x)$ be a polynomial of degree 4 such that $p(55) = p(83) = p(204) = p(232) = 8$ and $p(103) = 13.$ Find
\[p(1) - p(2) + p(3) - p(4) + \dots + p(285) - p(286).\] | 0 | dapo_math |
8024 | 14,931 | The expression $16^n + 4^n + 1$ is equivalent to the expression \( \frac{2^{p(n)} - 1}{2^{q(n)} - 1} \) for all positive integers \( n > 1 \), where \( p(n) \) and \( q(n) \) are functions and \( \frac{p(n)}{q(n)} \) is constant. Find \( p(2006) - q(2006) \). | The expression $16^n + 4^n + 1$ is equivalent to the expression \( \frac{2^{p(n)} - 1}{2^{q(n)} - 1} \) for all positive integers \( n > 1 \), where \( p(n) \) and \( q(n) \) are functions and \( \frac{p(n)}{q(n)} \) is constant. Find \( p(2006) - q(2006) \). | 8024 | dapo_math |
16 | 14,932 | In a right prism with triangular bases, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base) is 24, find the maximum volume of the prism.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F;
A = (0,0);
B = (3,-1);
C = (-1,-2);
D = A + (0,-4);
E = B + (0,-4);
F = C + (0,-4)... | In a right prism with triangular bases, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base) is 24, find the maximum volume of the prism.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F;
A = (0,0);
B = (3,-1);
C = (-1,-2);
D = A + (0,-4);
E = B + (0,-4);
F = C + (0,-4)... | 16 | dapo_math |
100 | 14,933 | 设 $S=\{1,2,3, \cdots, 100\}$. 求最大的整数 $k$ ，使得 $S$ 有 $k$ 个互不相同的非空子集，具有性质：对这 $k$ 个子集中任意两个不同子集，若它们的交集非空，则它们交集中的最小元素与这两个子集中的最大元素均不相同。已知k形如m^{n}+k，求m+n+k的值 | 设 $S=\{1,2,3, \cdots, 100\}$. 求最大的整数 $k$ ，使得 $S$ 有 $k$ 个互不相同的非空子集，具有性质：对这 $k$ 个子集中任意两个不同子集，若它们的交集非空，则它们交集中的最小元素与这两个子集中的最大元素均不相同。已知k形如m^{n}+k，求m+n+k的值 | 100 | dapo_math |
14 | 14,934 | A positive integer $m$ has the property that $m^2$ can be expressed in the form $4n^2 - 5n + 16$, where $n$ is an integer (of any sign). Find the maximum value of $|m - n|$. | A positive integer $m$ has the property that $m^2$ can be expressed in the form $4n^2 - 5n + 16$, where $n$ is an integer (of any sign). Find the maximum value of $|m - n|$. | 14 | dapo_math |
321 | 14,935 | 已知 $M=\{1,2, \cdots, 8\}, A 、 B$ 是集合 $M$的两个不同子集，满足：
（1） $A$ 的元素个数比 $B$ 的元素个数少；
（2） $A$ 中的最小元素比 $B$ 中的最大元素大。
则所有满足条件的有序集合对 $(A, B)$ 的个数为$\qquad.$ | 已知 $M=\{1,2, \cdots, 8\}, A 、 B$ 是集合 $M$的两个不同子集，满足：
（1） $A$ 的元素个数比 $B$ 的元素个数少；
（2） $A$ 中的最小元素比 $B$ 中的最大元素大。
则所有满足条件的有序集合对 $(A, B)$ 的个数为$\qquad.$ | 321 | dapo_math |
14 | 14,936 | In triangle $ABC$, $AB = 9$, $BC = 12$, $AC = 15$, and $CD$ is the angle bisector. Find the length of $CD$.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | In triangle $ABC$, $AB = 9$, $BC = 12$, $AC = 15$, and $CD$ is the angle bisector. Find the length of $CD$.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 14 | dapo_math |
6 | 14,937 | Simplify
\[\frac{\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \sin 40^\circ + \sin 50^\circ + \sin 60^\circ + \sin 70^\circ + \sin 80^\circ}{\cos 5^\circ \cos 10^\circ \cos 20^\circ}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | Simplify
\[\frac{\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \sin 40^\circ + \sin 50^\circ + \sin 60^\circ + \sin 70^\circ + \sin 80^\circ}{\cos 5^\circ \cos 10^\circ \cos 20^\circ}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 6 | dapo_math |
16 | 14,938 | In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\tan \angle APD = 3,$ then find $AB.$ | In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\tan \angle APD = 3,$ then find $AB.$ | 16 | dapo_math |
5 | 14,939 | Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-dollar bill. What percentage of the $20.00 will she receive in change? Provide your answer as an integer percentage. | Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-dollar bill. What percentage of the $20.00 will she receive in change? Provide your answer as an integer percentage. | 5 | dapo_math |
3 | 14,940 | A square with side length 1 is rotated about one vertex by an angle of $\alpha,$ where $0^\circ < \alpha < 90^\circ$ and $\cos \alpha = \frac{4}{5}.$ Find the area of the shaded region that is common to both squares.
[asy]
unitsize(3 cm);
pair A, B, C, D, Bp, Cp, Dp, P;
A = (0,0);
B = (-1,0);
C = (-1,-1);
D = (0,-1... | A square with side length 1 is rotated about one vertex by an angle of $\alpha,$ where $0^\circ < \alpha < 90^\circ$ and $\cos \alpha = \frac{4}{5}.$ Find the area of the shaded region that is common to both squares.
[asy]
unitsize(3 cm);
pair A, B, C, D, Bp, Cp, Dp, P;
A = (0,0);
B = (-1,0);
C = (-1,-1);
D = (0,-1... | 3 | dapo_math |
33 | 14,941 | 定义在 \(\mathbf{R}\) 上的函数 \(f(x)\) 满足 \(f(0)=0\), \(f(x)+f(1-x)=1\), \(f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)\)，且当 \(0 \leqslant x_{1}<x_{2} \leqslant 1\) 时, \(f\left(x_{1}\right) \leqslant f\left(x_{2}\right)\)。求 \(f\left(\frac{1}{2007}\right)\) 的值。答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。 | 定义在 \(\mathbf{R}\) 上的函数 \(f(x)\) 满足 \(f(0)=0\), \(f(x)+f(1-x)=1\), \(f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)\)，且当 \(0 \leqslant x_{1}<x_{2} \leqslant 1\) 时, \(f\left(x_{1}\right) \leqslant f\left(x_{2}\right)\)。求 \(f\left(\frac{1}{2007}\right)\) 的值。答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。 | 33 | dapo_math |
200 | 14,942 | Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted. | Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted. | 200 | dapo_math |
5 | 14,943 | Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 | dapo_math |
3 | 14,944 | 一项赛事共有 100 位选手参加. 对于任意两位选手 x,y，他们之间恰比赛一次且分出胜负，用 x\rightarrow y 表示 x 战胜 y. 如果对任意两位选手 x,y，均能找到某个选手序列 u_1, u_2, \cdots, u_k (k\geq2)，使得 x=u_1 \rightarrow u_2 \rightarrow \cdots \rightarrow u_k=y，那么称该赛事结果是"友好"的.(1) 求证：对任意一个友好的赛事结果，存在正整数 m 满足如下条件：对任意两位选手 x,y，均能找到某个长度为 m 的选手序列 z_1,z_2,\cdots,z_m（这里 z_1, z_2, \cdots, z_m 可以有重... | 一项赛事共有 100 位选手参加. 对于任意两位选手 x,y，他们之间恰比赛一次且分出胜负，用 x\rightarrow y 表示 x 战胜 y. 如果对任意两位选手 x,y，均能找到某个选手序列 u_1, u_2, \cdots, u_k (k\geq2)，使得 x=u_1 \rightarrow u_2 \rightarrow \cdots \rightarrow u_k=y，那么称该赛事结果是"友好"的.(1) 求证：对任意一个友好的赛事结果，存在正整数 m 满足如下条件：对任意两位选手 x,y，均能找到某个长度为 m 的选手序列 z_1,z_2,\cdots,z_m（这里 z_1, z_2, \cdots, z_m 可以有重... | 3 | dapo_math |
0 | 14,945 | Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$ | Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$ | 0 | dapo_math |
5050 | 14,946 | Let $f(x) = 1 + x + x^2 + \cdots + x^{100}$. Find $f'(1)$. | Let $f(x) = 1 + x + x^2 + \cdots + x^{100}$. Find $f'(1)$. | 5050 | dapo_math |
13 | 14,947 | Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the n... | Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the n... | 13 | dapo_math |
51 | 14,948 | Consider the following six statements:
1. $x < x^2 < x^3$
2. $x < x^3 < x^2$
4. $x^2 < x < x^3$
8. $x^2 < x^3 < x$
16. $x^3 < x < x^2$
32. $x^3 < x^2 < x$
Enter the sum of the labels of statement that hold for some value of $x.$ For example, if you think the statements with labels 2 and 8 hold for some value of $x,... | Consider the following six statements:
1. $x < x^2 < x^3$
2. $x < x^3 < x^2$
4. $x^2 < x < x^3$
8. $x^2 < x^3 < x$
16. $x^3 < x < x^2$
32. $x^3 < x^2 < x$
Enter the sum of the labels of statement that hold for some value of $x.$ For example, if you think the statements with labels 2 and 8 hold for some value of $x,... | 51 | dapo_math |
4356 | 14,949 | Twelve friends participate in a tennis tournament, where each friend plays exactly one game against each of the other eleven friends. The winner of each game receives one point, while the loser receives zero points. There are no draws. The final points for the participants are denoted as $B_1, B_2, \ldots, B_{12}$. Det... | Twelve friends participate in a tennis tournament, where each friend plays exactly one game against each of the other eleven friends. The winner of each game receives one point, while the loser receives zero points. There are no draws. The final points for the participants are denoted as $B_1, B_2, \ldots, B_{12}$. Det... | 4356 | dapo_math |
3 | 14,950 | 平面上五点 $A, B, C, D, E$ 满足 $\overrightarrow{A B}=\overrightarrow{B C}=\overrightarrow{C D}, \overrightarrow{E A} \cdot \overrightarrow{E B}=4, \overrightarrow{E B} \cdot \overrightarrow{E C}=5$, $\overrightarrow{E C} \cdot \overrightarrow{E D}=8$, 则 $\overrightarrow{E A} \cdot \overrightarrow{E D}$ 的值为 $\qquad$. | 平面上五点 $A, B, C, D, E$ 满足 $\overrightarrow{A B}=\overrightarrow{B C}=\overrightarrow{C D}, \overrightarrow{E A} \cdot \overrightarrow{E B}=4, \overrightarrow{E B} \cdot \overrightarrow{E C}=5$, $\overrightarrow{E C} \cdot \overrightarrow{E D}=8$, 则 $\overrightarrow{E A} \cdot \overrightarrow{E D}$ 的值为 $\qquad$. | 3 | dapo_math |
580 | 14,951 | During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^{2}/2$ miles on the $n^{th}$ day of ... | During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^{2}/2$ miles on the $n^{th}$ day of ... | 580 | dapo_math |
93 | 14,952 | Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$. | Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$. | 93 | dapo_math |
1750 | 14,953 | An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{... | An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{... | 1750 | dapo_math |
13 | 14,954 | 设 $a_{1}, a_{2}, a_{3}, a_{4}$ 为四个有理数, 使得: $\left\{a_{i} a_{j} \mid 1 \leq i<j \leq 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}$, 求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的值。原始答案是 \frac{m}{n} 的形式，请给出 m + n 的值。 | 设 $a_{1}, a_{2}, a_{3}, a_{4}$ 为四个有理数, 使得: $\left\{a_{i} a_{j} \mid 1 \leq i<j \leq 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}$, 求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的值。原始答案是 \frac{m}{n} 的形式，请给出 m + n 的值。 | 13 | dapo_math |
39 | 14,955 | At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability. Then, Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probabi... | At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability. Then, Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probabi... | 39 | dapo_math |
8 | 14,956 | All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have? | All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have? | 8 | dapo_math |
4 | 14,957 | Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?
[asy]
size(4cm,4cm);
path... | Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?
[asy]
size(4cm,4cm);
path... | 4 | dapo_math |
3 | 14,958 | If \[f(x) =
\begin{cases}
2x-5 &\quad \text{if } x \ge 3, \\
-x + 5 &\quad \text{if } x < 3,
\end{cases}
\]then for how many values of $x$ is $f(f(x)) = 3$? | If \[f(x) =
\begin{cases}
2x-5 &\quad \text{if } x \ge 3, \\
-x + 5 &\quad \text{if } x < 3,
\end{cases}
\]then for how many values of $x$ is $f(f(x)) = 3$? | 3 | dapo_math |
210 | 14,959 | Triangle $PQR$ is isosceles and the measure of angle $R$ is $40^\circ$. The possible measures of angle $P$ are $x,y,z$. What is the value of the sum $x + y + z$? | Triangle $PQR$ is isosceles and the measure of angle $R$ is $40^\circ$. The possible measures of angle $P$ are $x,y,z$. What is the value of the sum $x + y + z$? | 210 | dapo_math |
4 | 14,960 | 椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{3}=\lambda, F$ 为左焦点, $A, B$ 为椭圆上两点且 $|F A|=5,|F B|=8$, 求直线 $A B$ 的斜率 $k$ 的范围。请给出该范围上下限绝对值之和的近似整数值 | 椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{3}=\lambda, F$ 为左焦点, $A, B$ 为椭圆上两点且 $|F A|=5,|F B|=8$, 求直线 $A B$ 的斜率 $k$ 的范围。请给出该范围上下限绝对值之和的近似整数值 | 4 | dapo_math |
6 | 14,961 | The sequence $6075, 2025, 675 \ldots$, is made by repeatedly dividing by 3. How many integers are in this sequence? | The sequence $6075, 2025, 675 \ldots$, is made by repeatedly dividing by 3. How many integers are in this sequence? | 6 | dapo_math |
694 | 14,962 | We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of resi... | We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2 \equiv a \pmod{n}$, $x^3 \equiv a \pmod{n}$, or $x^5 \equiv a \pmod{n}$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of resi... | 694 | dapo_math |
19 | 14,963 | A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$? | A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$? | 19 | dapo_math |
5 | 14,964 | Find the sum of all prime numbers that can be expressed both as a sum of two prime numbers and as a difference of two prime numbers. | Find the sum of all prime numbers that can be expressed both as a sum of two prime numbers and as a difference of two prime numbers. | 5 | dapo_math |
-89 | 14,965 | Triangle $ABC$ is a right triangle. If the measure of angle $PAB$ is $x^\circ$ and the measure of angle $ACB$ is expressed in the form $(Mx+N)^\circ$ with $M=1$, what is the value of $M+N$?
[asy]
draw((-10,0)--(20,0),linewidth(1),Arrows);
draw((0,0)--(10,10/sqrt(3))--(10+10/3,0),linewidth(1));
draw((10,10/sqrt(3))+di... | Triangle $ABC$ is a right triangle. If the measure of angle $PAB$ is $x^\circ$ and the measure of angle $ACB$ is expressed in the form $(Mx+N)^\circ$ with $M=1$, what is the value of $M+N$?
[asy]
draw((-10,0)--(20,0),linewidth(1),Arrows);
draw((0,0)--(10,10/sqrt(3))--(10+10/3,0),linewidth(1));
draw((10,10/sqrt(3))+di... | -89 | dapo_math |
89 | 14,966 | 给定凸 20 边形 $P$ .用 $P$ 的 17 条在内部不相交的对角线将 $P$ 分割成 18 个三角形.所得图形称为 $P$ 的一个三角剖分图.对 $P$ 的任意一个三角剖分图 $T, P$ 的 20 条边以及添加的 17 条对角线均称为 $T$ 的边. $T$ 的任意 10 条两两无公共端点的边的集合称为 $T$的一个完美匹配.当 $T$ 取遍 $P$ 的所有三角剖分图时, 求 $T$ 的完美匹配个数的最大值. | 给定凸 20 边形 $P$ .用 $P$ 的 17 条在内部不相交的对角线将 $P$ 分割成 18 个三角形.所得图形称为 $P$ 的一个三角剖分图.对 $P$ 的任意一个三角剖分图 $T, P$ 的 20 条边以及添加的 17 条对角线均称为 $T$ 的边. $T$ 的任意 10 条两两无公共端点的边的集合称为 $T$的一个完美匹配.当 $T$ 取遍 $P$ 的所有三角剖分图时, 求 $T$ 的完美匹配个数的最大值. | 89 | dapo_math |
1 | 14,967 | What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$? | What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$? | 1 | dapo_math |
19 | 14,968 | How many ordered triples $(a, b, c)$ of positive integers satisfy $a \le b \le c$ and $a \cdot b\cdot c = 1000$? | How many ordered triples $(a, b, c)$ of positive integers satisfy $a \le b \le c$ and $a \cdot b\cdot c = 1000$? | 19 | dapo_math |
16 | 14,969 | 已知自然数 $n \geqslant 3$, 实数 $x_{1}$, $x_{2}, \cdots, x_{n}$ 满足:
$x_{1}+x_{2}+\cdots+x_{n}=n, x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n^{2}$.
若$n=4$,求 $S=x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}$ 的最小值. | 已知自然数 $n \geqslant 3$, 实数 $x_{1}$, $x_{2}, \cdots, x_{n}$ 满足:
$x_{1}+x_{2}+\cdots+x_{n}=n, x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=n^{2}$.
若$n=4$,求 $S=x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}$ 的最小值. | 16 | dapo_math |
643210 | 14,970 | What is the largest number, with its digits all different, whose digits add up to 16? | What is the largest number, with its digits all different, whose digits add up to 16? | 643210 | dapo_math |
14 | 14,971 | A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to i... | A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to i... | 14 | dapo_math |
5 | 14,972 | A rhombus with side length $1$ has an inscribed circle with radius $\frac{1}{3}$. If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive integers $a$ and $b$, evaluate $a+b$. | A rhombus with side length $1$ has an inscribed circle with radius $\frac{1}{3}$. If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive integers $a$ and $b$, evaluate $a+b$. | 5 | dapo_math |
2022 | 14,973 | 求最小的正实数 $c$ ，使得存在二阶连续可微的函数 $f: \mathbb{R} \rightarrow \mathbb{R}$ ，满足 $f(0)=f(c)=0$ ，且对任意 $0<x<c$ ，有 $f({x})>0$ 和 $f^{\prime \prime}(x)+2021 f(x) \ge 0$. 原始答案为 \frac{m\pi}{\sqrt{n}}，请给出 m+n的值。 | 求最小的正实数 $c$ ，使得存在二阶连续可微的函数 $f: \mathbb{R} \rightarrow \mathbb{R}$ ，满足 $f(0)=f(c)=0$ ，且对任意 $0<x<c$ ，有 $f({x})>0$ 和 $f^{\prime \prime}(x)+2021 f(x) \ge 0$. 原始答案为 \frac{m\pi}{\sqrt{n}}，请给出 m+n的值。 | 2022 | dapo_math |
53 | 14,974 | The area of the quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and $(-4,2,5)$ is the number $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | The area of the quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and $(-4,2,5)$ is the number $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 53 | dapo_math |
9 | 14,975 | Two fair eight-sided dice have their faces numbered from 1 to 8. What is the expected value of the sum of the rolls of both dice? | Two fair eight-sided dice have their faces numbered from 1 to 8. What is the expected value of the sum of the rolls of both dice? | 9 | dapo_math |
7 | 14,976 | How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$? Find the number of such four-digit numbers. | How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$? Find the number of such four-digit numbers. | 7 | dapo_math |
1011100 | 14,977 | Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$? | Let $S$ be the set of all positive integers that have four digits in base $2$. What is the sum of all of the elements in $S$, when expressed in base $2$? | 1011100 | dapo_math |
31 | 14,978 | A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces? | A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces? | 31 | dapo_math |
9 | 14,979 | If $f(c)=\frac{3}{2c-3}$, find $\frac{kn^2}{lm}$ when $f^{-1}(c)\times c \times f(c)$ equals the simplified fraction$\frac{kc+l}{mc+n}$, where $k,l,m,\text{ and }n$ are integers. | If $f(c)=\frac{3}{2c-3}$, find $\frac{kn^2}{lm}$ when $f^{-1}(c)\times c \times f(c)$ equals the simplified fraction$\frac{kc+l}{mc+n}$, where $k,l,m,\text{ and }n$ are integers. | 9 | dapo_math |
6 | 14,980 | The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$? | The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$? | 6 | dapo_math |
3 | 14,981 | Find the sum of the indexes of the singular points other than zero of the vector field:
\[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \] | Find the sum of the indexes of the singular points other than zero of the vector field:
\[ z\overline{z}^2 + z^4 + 2\overline{z}^4 \] | 3 | dapo_math |
25 | 14,982 | 对每个正整数 $n$ ，定义集合 $P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。 对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数 （不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数。 | 对每个正整数 $n$ ，定义集合 $P_{n}=\left\{n^{k} \mid k=0,1, \cdots\right\}$ 。 对于正整数 $a 、 b 、 c$, 若存在某个正整数 $m$ ，使得 $a-1 、 a b-12 、 a b c-2015$ 这三个数 （不必两两不等）均属于集合 $P_{m}$ ，则称正整数组 $(a, b, c)$ 为"幸运的". 求所有幸运的正整数组的个数。 | 25 | dapo_math |
12 | 14,983 | 已知 $\alpha, \beta \geqslant 0, \alpha+\beta \leqslant 2 \pi$, 则 $\sin \alpha+2 \cos \beta$ 的最小值为。原始答案为$-\frac{m \sqrt{n}}{q}$的形式，请给出m+n+q的值。 | 已知 $\alpha, \beta \geqslant 0, \alpha+\beta \leqslant 2 \pi$, 则 $\sin \alpha+2 \cos \beta$ 的最小值为。原始答案为$-\frac{m \sqrt{n}}{q}$的形式，请给出m+n+q的值。 | 12 | dapo_math |
5 | 14,984 | Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He kee... | Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He kee... | 5 | dapo_math |
240 | 14,985 | If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list. | If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list. | 240 | dapo_math |
86 | 14,986 | A square has sides of length 2. Set $\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | A square has sides of length 2. Set $\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | 86 | dapo_math |
400 | 14,987 | Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$. | Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is $60$, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$. | 400 | dapo_math |
3 | 14,988 | How many positive integers less than 500 have exactly 15 positive integer factors? | How many positive integers less than 500 have exactly 15 positive integer factors? | 3 | dapo_math |
4 | 14,989 | The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter use... | The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter use... | 4 | dapo_math |
90 | 14,990 | The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.
\[\text{[asy]}
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{pair A = (0,10), B=origin, C=(12,0), D=(24,0... | The diagram below shows rectangle $ABDE$ where $C$ is the midpoint of side $\overline{BD}$, and $F$ is the midpoint of side $\overline{AE}$. If $AB=10$ and $BD=24$, find the area of the shaded region.
\[\text{[asy]}
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{pair A = (0,10), B=origin, C=(12,0), D=(24,0... | 90 | dapo_math |
17 | 14,991 | 已知复数 z 的模为 1, 则 $|z-4|^{2}+|z+3 i|^{2}$ 的最小值为 $\qquad$. | 已知复数 z 的模为 1, 则 $|z-4|^{2}+|z+3 i|^{2}$ 的最小值为 $\qquad$. | 17 | dapo_math |
92 | 14,992 | Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 17,$ $p(2) = 34,$ and $p(3) = 51.$ Find $p(0) + p(4).$ | Let $p(x)$ be a monic polynomial of degree 4, such that $p(1) = 17,$ $p(2) = 34,$ and $p(3) = 51.$ Find $p(0) + p(4).$ | 92 | dapo_math |
50 | 14,993 | If the degree measure of an arc of a circle is increased by $20\%$ and the radius of the circle is increased by $25\%$, by what percent does the length of the arc increase? | If the degree measure of an arc of a circle is increased by $20\%$ and the radius of the circle is increased by $25\%$, by what percent does the length of the arc increase? | 50 | dapo_math |
12 | 14,994 | Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$. Point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $... | Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$. Point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $... | 12 | dapo_math |
3 | 14,995 | Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\f... | Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\f... | 3 | dapo_math |
17 | 14,996 | 在等比数列 $\left\{a_{n}\right\}$ 中, $a_{2}=\sqrt{2}, a_{3}=\sqrt[3]{3}$, 则 $\frac{a_{1}+a_{2011}}{a_{7}+a_{2017}}$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 在等比数列 $\left\{a_{n}\right\}$ 中, $a_{2}=\sqrt{2}, a_{3}=\sqrt[3]{3}$, 则 $\frac{a_{1}+a_{2011}}{a_{7}+a_{2017}}$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 17 | dapo_math |
2035 | 14,997 | 数列 \{a_n\} 满足 a_1=1,\df{a_{n+1}-a_n}{a_n}=\df{a_{n+2}-a_{n+1}}{a_{n+2}} (n\in\bN^\ast). 若 a_1a_2+a_2a_3+\cdots+a_6a_7=3，则 a_{2024}=__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 数列 \{a_n\} 满足 a_1=1,\df{a_{n+1}-a_n}{a_n}=\df{a_{n+2}-a_{n+1}}{a_{n+2}} (n\in\bN^\ast). 若 a_1a_2+a_2a_3+\cdots+a_6a_7=3，则 a_{2024}=__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 2035 | dapo_math |
101 | 14,998 | Find all prime numbers of the form $\frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}$, where $n$ is a natural number. | Find all prime numbers of the form $\frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}$, where $n$ is a natural number. | 101 | dapo_math |
5 | 14,999 | The infinite sequence $T=\{t_0,t_1,t_2,\ldots\}$ is defined as $t_0=0,$ $t_1=1,$ and $t_n=t_{n-2}+t_{n-1}$ for all integers $n>1.$ If $a,$ $b,$ $c$ are fixed non-negative integers such that \begin{align*}
a&\equiv 5\pmod {16}\\
b&\equiv 10\pmod {16}\\
c&\equiv 15\pmod {16},
\end{align*}then what is the remainder when $... | The infinite sequence $T=\{t_0,t_1,t_2,\ldots\}$ is defined as $t_0=0,$ $t_1=1,$ and $t_n=t_{n-2}+t_{n-1}$ for all integers $n>1.$ If $a,$ $b,$ $c$ are fixed non-negative integers such that \begin{align*}
a&\equiv 5\pmod {16}\\
b&\equiv 10\pmod {16}\\
c&\equiv 15\pmod {16},
\end{align*}then what is the remainder when $... | 5 | dapo_math |
83 | 15,000 | 若锐角 A,B,C 满足 \sin^2A+\sin^2B+\sin^2C=2，则 \df{1}{\sin^2A\cos^4B}+\df{1}{\sin^2B\cos^4C}+\df{1}{\sin^2C\cos^4A} 的最小值是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 若锐角 A,B,C 满足 \sin^2A+\sin^2B+\sin^2C=2，则 \df{1}{\sin^2A\cos^4B}+\df{1}{\sin^2B\cos^4C}+\df{1}{\sin^2C\cos^4A} 的最小值是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值 | 83 | dapo_math |
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