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
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75 | 201 | Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$... | Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$... | 75 | dapo_math |
1940 | 202 | Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\). | Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\). | 1940 | dapo_math |
6 | 203 | 复数 $z^{11}+z=1$，求 $z$ 的一个正确的幂次形式。答案应为 $z^k$ 的形式，请给出 $k$ 的值。 | 复数 $z^{11}+z=1$，求 $z$ 的一个正确的幂次形式。答案应为 $z^k$ 的形式，请给出 $k$ 的值。 | 6 | dapo_math |
4018 | 204 | Let $\omega$ be a nonreal root of $x^3 = 1,$ and let
\[\mathbf{M} = \begin{pmatrix} -\omega^2 & - \omega \\ 1 & 0 \end{pmatrix}.\]Find the sum of the entries of $\mathbf{M} + \mathbf{M}^2 + \mathbf{M}^3 + \dots + \mathbf{M}^{2009}.$ | Let $\omega$ be a nonreal root of $x^3 = 1,$ and let
\[\mathbf{M} = \begin{pmatrix} -\omega^2 & - \omega \\ 1 & 0 \end{pmatrix}.\]Find the sum of the entries of $\mathbf{M} + \mathbf{M}^2 + \mathbf{M}^3 + \dots + \mathbf{M}^{2009}.$ | 4018 | dapo_math |
109 | 205 | Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value o... | Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value o... | 109 | dapo_math |
48 | 206 | Equilateral triangle \(ABC\) has been creased and folded so that vertex \(A\) now rests at \(A'\) on \(\overline{BC}\) as shown. If \(BA' = 1\) and \(A'C = 2\), then find the length of the crease \(\overline{PQ}\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n. | Equilateral triangle \(ABC\) has been creased and folded so that vertex \(A\) now rests at \(A'\) on \(\overline{BC}\) as shown. If \(BA' = 1\) and \(A'C = 2\), then find the length of the crease \(\overline{PQ}\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n. | 48 | dapo_math |
561 | 207 | What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$? | What are the last three digits of $49^{303} \cdot 3993^{202} \cdot 39^{606}$? | 561 | dapo_math |
0 | 208 | Find the imaginary part of \[(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)^6.\] | Find the imaginary part of \[(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)^6.\] | 0 | dapo_math |
25 | 209 | Let u, f, and g be functions, defined for all real numbers x, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine u(x) in terms of f and g. Please provide the value of u(x) when x = 1. The answer should be in the format $u(x)=g(a)-f(b)+f(c)-f(d)+f(e)$, directly give the valu... | Let u, f, and g be functions, defined for all real numbers x, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text { and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine u(x) in terms of f and g. Please provide the value of u(x) when x = 1. The answer should be in the format $u(x)=g(a)-f(b)+f(c)-f(d)+f(e)$, directly give the valu... | 25 | dapo_math |
5 | 210 | Let $ABC$ be an acute triangle with side lengths $AB = 7$, $BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting the midpoin... | Let $ABC$ be an acute triangle with side lengths $AB = 7$, $BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting the midpoin... | 5 | dapo_math |
6 | 211 | How many positive cubes divide $3!\cdot 5!\cdot 7!\,$? | How many positive cubes divide $3!\cdot 5!\cdot 7!\,$? | 6 | dapo_math |
20 | 212 | What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$? Provide your answer as an integer. | What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$? Provide your answer as an integer. | 20 | dapo_math |
231 | 213 | 定义 $x * y=\frac{x+y}{1+x y}$, 则 $(\cdots((2 * 3) * 4) \cdots) * 21$结果的格式为$\frac{m}{n}$，给出$m + n$的值。 | 定义 $x * y=\frac{x+y}{1+x y}$, 则 $(\cdots((2 * 3) * 4) \cdots) * 21$结果的格式为$\frac{m}{n}$，给出$m + n$的值。 | 231 | dapo_math |
26 | 214 | Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | 26 | dapo_math |
10 | 215 | Kelly drove north for 9 miles and then east for 12 miles at an average rate of 42 miles per hour to arrive at the town of Prime. Brenda left from the same location, at the same time, and drove along a straight road to Prime at an average rate of 45 miles per hour. How many minutes earlier than Kelly did Brenda arrive? | Kelly drove north for 9 miles and then east for 12 miles at an average rate of 42 miles per hour to arrive at the town of Prime. Brenda left from the same location, at the same time, and drove along a straight road to Prime at an average rate of 45 miles per hour. How many minutes earlier than Kelly did Brenda arrive? | 10 | dapo_math |
457 | 216 | The number $0.428125$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$? | The number $0.428125$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$? | 457 | dapo_math |
165 | 217 | Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isosceles triangle with $a$, $b$, and $c$ as the lengths of sides. Determine how many possible three-digit numbers $N$ there are.
($N=\overline{abc}$ means that $a$, $b$, and $c$ are digits of $N$, and not $N=a \times b \times c$.) | Let $N=\overline{abc}$ be a three-digit number. It is known that we can construct an isosceles triangle with $a$, $b$, and $c$ as the lengths of sides. Determine how many possible three-digit numbers $N$ there are.
($N=\overline{abc}$ means that $a$, $b$, and $c$ are digits of $N$, and not $N=a \times b \times c$.) | 165 | dapo_math |
17 | 218 | 称一个九位数是平衡的，如果数字 1\sim9 都出现. 将所有平衡的数按递增顺序依次写下，设 S 是所得的数字序列. 求最小的正整数 k，使得 S 的任意两个由 k 个连续数字构成的子列互不相同. | 称一个九位数是平衡的，如果数字 1\sim9 都出现. 将所有平衡的数按递增顺序依次写下，设 S 是所得的数字序列. 求最小的正整数 k，使得 S 的任意两个由 k 个连续数字构成的子列互不相同. | 17 | dapo_math |
12 | 219 | For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005$? | For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005$? | 12 | dapo_math |
13 | 220 | Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$,... | Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$,... | 13 | dapo_math |
286 | 221 | 对矩阵 P，定义 e^P=I+\ds{m=1}{\infty}\df{1}{m!}P^m. 设 20 阶实矩阵 A 满足 e^A=I_{20}，且特征值的模长不超过 20，则这样的互不复相似的 A 有__________个. | 对矩阵 P，定义 e^P=I+\ds{m=1}{\infty}\df{1}{m!}P^m. 设 20 阶实矩阵 A 满足 e^A=I_{20}，且特征值的模长不超过 20，则这样的互不复相似的 A 有__________个. | 286 | dapo_math |
41 | 222 | Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$. | Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$. | 41 | dapo_math |
33 | 223 | 设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$, 对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$ ，请给出 $a_{1}+b_{1}$ 的所有可能值的和。 | 设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$, 对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$ ，请给出 $a_{1}+b_{1}$ 的所有可能值的和。 | 33 | dapo_math |
102 | 224 | Find the smallest positive three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$. | Find the smallest positive three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$. | 102 | dapo_math |
14 | 225 | The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is the value of $A+M+C$? | The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is the value of $A+M+C$? | 14 | dapo_math |
3 | 226 | What is the maximum possible value of $5 - |6x - 80|$ over all integers $x$? | What is the maximum possible value of $5 - |6x - 80|$ over all integers $x$? | 3 | dapo_math |
220 | 227 | Darwin takes an $11 \times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10 \times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew? | Darwin takes an $11 \times 11$ grid of lattice points and connects every pair of points that are 1 unit apart, creating a $10 \times 10$ grid of unit squares. If he never retraced any segment, what is the total length of all segments that he drew? | 220 | dapo_math |
8 | 228 | Let $\mathbb{N}$ denote the set of all positive integers. Find a specific function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for $x = 1$ and $y = 1$, the expression $x^{2}-y^{2}+2 y(f(x)+f(y))$ becomes a square of an integer. Provide the value of $f(8)$. | Let $\mathbb{N}$ denote the set of all positive integers. Find a specific function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for $x = 1$ and $y = 1$, the expression $x^{2}-y^{2}+2 y(f(x)+f(y))$ becomes a square of an integer. Provide the value of $f(8)$. | 8 | dapo_math |
22 | 229 | A particular number written in base 3 requires three digits (${\_ \_ \_}_3$). When the number is written in base 3 and 4, the digits are the reverse of each other. What is this number expressed in base 10? | A particular number written in base 3 requires three digits (${\_ \_ \_}_3$). When the number is written in base 3 and 4, the digits are the reverse of each other. What is this number expressed in base 10? | 22 | dapo_math |
16 | 230 | Jack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the integers 1 through 6 (fair meaning the numbers appear with equal probability). Jack wins if a prime number appears when he rolls, while Jill wins if a number greater than 1 appears when she rolls. The game... | Jack and Jill are playing a chance game. They take turns alternately rolling a fair six-sided die labeled with the integers 1 through 6 (fair meaning the numbers appear with equal probability). Jack wins if a prime number appears when he rolls, while Jill wins if a number greater than 1 appears when she rolls. The game... | 16 | dapo_math |
30 | 231 | Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$. | Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$. | 30 | dapo_math |
132 | 232 | For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radiu... | For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a disk of radiu... | 132 | dapo_math |
810 | 233 | On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$. | On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$. | 810 | dapo_math |
64 | 234 | Let $p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$
and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$. | Let $p(x) = x^{2008} + x^{2007} + x^{2006} + \cdots + x + 1,$
and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$. | 64 | dapo_math |
720 | 235 | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. | A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form
\[a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$. | 720 | dapo_math |
19990002000 | 236 | Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$ | Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$ | 19990002000 | dapo_math |
167 | 237 | Let $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$? | Let $f : \mathbb{R} \to \mathbb{R}$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$? | 167 | dapo_math |
7 | 238 | 若一个三角形的各边长均为整数且其面积为有理数，则该三角形某一边的长可以是以下哪些选项？计算这些选项的和
A. 1
B. 2
C. 3
D. 4 | 若一个三角形的各边长均为整数且其面积为有理数，则该三角形某一边的长可以是以下哪些选项？计算这些选项的和
A. 1
B. 2
C. 3
D. 4 | 7 | dapo_math |
36 | 239 | There are positive integers that have these properties:
$\bullet$ I. The sum of the squares of their digits is $50,$ and
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of the largest integer with both properties? | There are positive integers that have these properties:
$\bullet$ I. The sum of the squares of their digits is $50,$ and
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of the largest integer with both properties? | 36 | dapo_math |
872 | 240 | For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself.... | For any set $S$, let $P(S)$ be its power set, the set of all its subsets. Consider all sets $A$ of 2015 arbitrary finite sets. Let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A)$, $T \in P(P(A))$, $S \in T$, and $S \subseteq T$. Note that by convention, a set may never contain itself.... | 872 | dapo_math |
37 | 241 | 设实数 $k, l, m$ 满足：函数 $y=(x+1)\left(x^{2}+k x+l\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点, 则 $k+l+m$ 的取值范围是$\left(-\infty, \frac{a}{b}\right]$，求$a+b$的值。 | 设实数 $k, l, m$ 满足：函数 $y=(x+1)\left(x^{2}+k x+l\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点, 则 $k+l+m$ 的取值范围是$\left(-\infty, \frac{a}{b}\right]$，求$a+b$的值。 | 37 | dapo_math |
5024 | 242 | If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor + \dots + \lfloor \frac{2^{2009}}{5} \rfloor$, find the remainder when $2^{2010}$ is divided by $N$. | If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor + \dots + \lfloor \frac{2^{2009}}{5} \rfloor$, find the remainder when $2^{2010}$ is divided by $N$. | 5024 | dapo_math |
7 | 243 | Find the largest value of $t$ such that \[\frac{13t^2 - 34t + 12}{3t - 2 } + 5t = 6t - 1.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | Find the largest value of $t$ such that \[\frac{13t^2 - 34t + 12}{3t - 2 } + 5t = 6t - 1.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 | dapo_math |
4030 | 244 | Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. (So $1 \in S$; $2 \in S$ if and only if... | Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. (So $1 \in S$; $2 \in S$ if and only if... | 4030 | dapo_math |
140 | 245 | The base three number $12012_3$ is equal to which base ten number? | The base three number $12012_3$ is equal to which base ten number? | 140 | dapo_math |
125 | 246 | Let $x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$ Find $(x+1)^{48}.$ | Let $x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$ Find $(x+1)^{48}.$ | 125 | dapo_math |
3 | 247 | 求最小的实数 C，使得对任意正实数 a_1, a_2, a_3, a_4, a_5（允许相同），总可以选择不同的下标 i,j,k,l，满足 \left|\df{a_i}{a_j}-\df{a_k}{a_l}\right|\leq C.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 | 求最小的实数 C，使得对任意正实数 a_1, a_2, a_3, a_4, a_5（允许相同），总可以选择不同的下标 i,j,k,l，满足 \left|\df{a_i}{a_j}-\df{a_k}{a_l}\right|\leq C.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 | 3 | dapo_math |
7 | 248 | Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (... | Let $a_1, a_2, \ldots, a_n$ be real numbers, and let $b_1, b_2, \ldots, b_n$ be distinct positive integers. Suppose there is a polynomial $f(x)$ satisfying the identity $$(1-x)^n f(x)=1+\sum_{i=1}^n a_i x^{b_i}.$$ Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \ldots, b_n$ and $n$ (... | 7 | dapo_math |
24 | 249 | Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain? | Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain? | 24 | dapo_math |
81 | 250 | If $x$ is a real number such that $3^x = 27x$, compute $\log_3 \left(\frac{3^{3^x}}{x^{3^3}}\right)$. | If $x$ is a real number such that $3^x = 27x$, compute $\log_3 \left(\frac{3^{3^x}}{x^{3^3}}\right)$. | 81 | dapo_math |
9 | 251 | 已知 $\forall x \in R, f(x)=2 x^{4}+m x^{3}+(m+6) x^{2}+m x+2>0$, 求正整数 $m$ 的最大值。 | 已知 $\forall x \in R, f(x)=2 x^{4}+m x^{3}+(m+6) x^{2}+m x+2>0$, 求正整数 $m$ 的最大值。 | 9 | dapo_math |
671 | 252 | The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $... | The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $... | 671 | dapo_math |
4 | 253 | 求最小的实数 \lambda，使得对任意正实数 x_1,x_2,x_3,x_4，均存在 1, 2, 3, 4 的一个排列 \sigma，满足(x_{\sigma(1)}x_{\sigma(2)}-x_{\sigma(3)}x_{\sigma(4)})^2\leq\lambda\ds{1\leq i<j\leq 4}{}(x_i^2-x_j^2)^2.（Martin）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 | 求最小的实数 \lambda，使得对任意正实数 x_1,x_2,x_3,x_4，均存在 1, 2, 3, 4 的一个排列 \sigma，满足(x_{\sigma(1)}x_{\sigma(2)}-x_{\sigma(3)}x_{\sigma(4)})^2\leq\lambda\ds{1\leq i<j\leq 4}{}(x_i^2-x_j^2)^2.（Martin）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 | 4 | dapo_math |
715 | 254 | $T$ is the smallest positive multiple of 14 whose digits are all 1s and 0s. What is the quotient when $T$ is divided by 14? | $T$ is the smallest positive multiple of 14 whose digits are all 1s and 0s. What is the quotient when $T$ is divided by 14? | 715 | dapo_math |
1000 | 255 | In the decimal expression of $1^1 + 2^2 + 3^3 + · · · + 999^{999} + 1000^{1000},$ what are its a) first three digits from the left? b) first four digits? | In the decimal expression of $1^1 + 2^2 + 3^3 + · · · + 999^{999} + 1000^{1000},$ what are its a) first three digits from the left? b) first four digits? | 1000 | dapo_math |
1012 | 256 | 平面上有 2022 个点，满足任意三点不共线. 每个点被染为红色或蓝色，使得每三个（不同的）红点构成的三角形都至少包含一个蓝点. 求红点个数的最大可能值. | 平面上有 2022 个点，满足任意三点不共线. 每个点被染为红色或蓝色，使得每三个（不同的）红点构成的三角形都至少包含一个蓝点. 求红点个数的最大可能值. | 1012 | dapo_math |
15 | 257 | A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (Th... | A $2\times3$ rectangle has vertices at $(0,0)$, $(2,0)$, $(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (Th... | 15 | dapo_math |
2 | 258 | After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k:m, where k and m are integers. Please find the value of k + m. | After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k:m, where k and m are integers. Please find the value of k + m. | 2 | dapo_math |
499500 | 259 | For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\] | For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\] | 499500 | dapo_math |
0 | 260 | Let $a,$ $b,$ $c,$ $d,$ and $e$ be the distinct roots of the equation $x^5 + 7x^4 - 2 = 0.$ Find
\begin{align*}
&\frac{a^3}{(a - b)(a - c)(a - d)(a - e)} + \frac{b^3}{(b - a)(b - c)(b - d)(b - e)} \\
&\quad + \frac{c^3}{(c - a)(c - b)(c - d)(c - e)} + \frac{d^3}{(d - a)(d - b)(d - c)(d - e)} \\
&\quad + \frac{e^3}{(e ... | Let $a,$ $b,$ $c,$ $d,$ and $e$ be the distinct roots of the equation $x^5 + 7x^4 - 2 = 0.$ Find
\begin{align*}
&\frac{a^3}{(a - b)(a - c)(a - d)(a - e)} + \frac{b^3}{(b - a)(b - c)(b - d)(b - e)} \\
&\quad + \frac{c^3}{(c - a)(c - b)(c - d)(c - e)} + \frac{d^3}{(d - a)(d - b)(d - c)(d - e)} \\
&\quad + \frac{e^3}{(e ... | 0 | dapo_math |
1848 | 261 | Two knights placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of an $8 \times 8$ c... | Two knights placed on distinct squares of an $8 \times 8$ chessboard, where each square is a unit square, are said to attack each other if the distance between the centers of the squares on which the knights lie is $\sqrt{5}$. In how many ways can two identical knights be placed on distinct squares of an $8 \times 8$ c... | 1848 | dapo_math |
10 | 262 | Find the minimum value of $2x^2 + 2y^2 + 5z^2 - 2xy - 4yz - 4x - 2z + 15$ for real numbers $x$, $y$, $z$. | Find the minimum value of $2x^2 + 2y^2 + 5z^2 - 2xy - 4yz - 4x - 2z + 15$ for real numbers $x$, $y$, $z$. | 10 | dapo_math |
18 | 263 | Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \[\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2\] | Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \[\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2\] | 18 | dapo_math |
10 | 264 | Oscar buys 13 pencils and 3 erasers for $\$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost of one pencil and one eraser in cents? | Oscar buys 13 pencils and 3 erasers for $\$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost of one pencil and one eraser in cents? | 10 | dapo_math |
2 | 265 | In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), please find the value of \(k\). | In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), please find the value of \(k\). | 2 | dapo_math |
7 | 266 | Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$. | Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$. | 7 | dapo_math |
9 | 267 | An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$? | An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$? | 9 | dapo_math |
-14 | 268 | Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$ | Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$ | -14 | dapo_math |
1 | 269 | Compute $i^{-100}+i^{-99}+i^{-98}+\cdots+i^{-1}+i^0+i^1+\cdots+i^{99}+i^{100}$. | Compute $i^{-100}+i^{-99}+i^{-98}+\cdots+i^{-1}+i^0+i^1+\cdots+i^{99}+i^{100}$. | 1 | dapo_math |
149 | 270 | For any integer $k \geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\{x_n\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p(x_n)$ for $n \geq 0.$ Fi... | For any integer $k \geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\{x_n\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p(x_n)$ for $n \geq 0.$ Fi... | 149 | dapo_math |
3 | 271 | On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and median score of the students' scores on this quiz. | On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and median score of the students' scores on this quiz. | 3 | dapo_math |
544 | 272 | A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth. All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets. A rectangular city park is bounded on the north and south... | A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth. All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets. A rectangular city park is bounded on the north and south... | 544 | dapo_math |
2695 | 273 | Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd intege... | Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd intege... | 2695 | dapo_math |
625 | 274 | In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result. | In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result. | 625 | dapo_math |
14 | 275 | The sides and vertices of a pentagon are labeled with the numbers $1$ through $10$. Each side of the pentagon must have the same sum of numbers. What is the smallest possible value of this sum? | The sides and vertices of a pentagon are labeled with the numbers $1$ through $10$. Each side of the pentagon must have the same sum of numbers. What is the smallest possible value of this sum? | 14 | dapo_math |
504 | 276 | The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known tha... | The Venusian prophet Zabruberson sent his pupils a $10000$-letter word, with each letter being either $A$ or $E$. This is known as the Zabrubic word. The pupils consider that for $1 \leq k \leq 10000$, each word comprised of $k$ consecutive letters of the Zabrubic word is a prophetic word of length $k$. It is known tha... | 504 | dapo_math |
30 | 277 | Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in ... | Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in ... | 30 | dapo_math |
30 | 278 | Suppose $\cos Q = 0.4$ in the diagram below. What is $QR$?
[asy]
pair P,Q,R;
P = (0,0);
Q = (6,0);
R = (0,6*tan(acos(0.4)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,18));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$12$",Q/2,S);
[/asy] | Suppose $\cos Q = 0.4$ in the diagram below. What is $QR$?
[asy]
pair P,Q,R;
P = (0,0);
Q = (6,0);
R = (0,6*tan(acos(0.4)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,18));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$12$",Q/2,S);
[/asy] | 30 | dapo_math |
47 | 279 | Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter such that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of th... | Points $A$ and $B$ are the endpoints of a diameter of a circle with center $C$. Points $D$ and $E$ lie on the same diameter such that $C$ bisects segment $\overline{DE}$. Let $F$ be a randomly chosen point within the circle. The probability that $\triangle DEF$ has a perimeter less than the length of the diameter of th... | 47 | dapo_math |
33 | 280 | In rectangle $ABCD$, point $M$ is the midpoint of $AB$, and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. | In rectangle $ABCD$, point $M$ is the midpoint of $AB$, and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. | 33 | dapo_math |
4 | 281 | Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2}... | Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2}... | 4 | dapo_math |
850 | 282 | Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + ... | Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \le x \le 15$. The probability that $\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + ... | 850 | dapo_math |
61 | 283 | Natasha has more than $\$1$ but less than $\$10$ worth of dimes. When she puts her dimes in stacks of 3, she has 1 left over. When she puts them in stacks of 4, she has 1 left over. When she puts them in stacks of 5, she also has 1 left over. How many dimes does Natasha have? | Natasha has more than $\$1$ but less than $\$10$ worth of dimes. When she puts her dimes in stacks of 3, she has 1 left over. When she puts them in stacks of 4, she has 1 left over. When she puts them in stacks of 5, she also has 1 left over. How many dimes does Natasha have? | 61 | dapo_math |
875 | 284 | Find the last three digits in the product $1 \cdot 3 \cdot 5 \cdot 7 \cdot \ldots \cdot 2009 \cdot 2011$. | Find the last three digits in the product $1 \cdot 3 \cdot 5 \cdot 7 \cdot \ldots \cdot 2009 \cdot 2011$. | 875 | dapo_math |
2 | 285 | Two strips of width 1 overlap at an angle of \(\alpha\) as shown. Find the area of the overlap (shown shaded) in terms of \(\alpha\). The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\). | Two strips of width 1 overlap at an angle of \(\alpha\) as shown. Find the area of the overlap (shown shaded) in terms of \(\alpha\). The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\). | 2 | dapo_math |
419 | 286 | Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd. | Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd. | 419 | dapo_math |
400 | 287 | Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]Two sides of the triangle have lengths 10 and 13. Find the maximum length of the third side.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]Two sides of the triangle have lengths 10 and 13. Find the maximum length of the third side.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 400 | dapo_math |
1260 | 288 | A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build using $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.) Please provide your answer as an integer. | A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build using $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.) Please provide your answer as an integer. | 1260 | dapo_math |
572 | 289 | 已知多项式
$$
f(z)=z^{3}+a z^{2}+b z+c(a, b. c \in \mathbf{Z})
$$
的所有根的模都是 20 或 15. 则这样的多
项式有 $\qquad$个。 | 已知多项式
$$
f(z)=z^{3}+a z^{2}+b z+c(a, b. c \in \mathbf{Z})
$$
的所有根的模都是 20 或 15. 则这样的多
项式有 $\qquad$个。 | 572 | dapo_math |
10 | 290 | 已知整数 $a, b, c, d$ 满足 $a+b+c+d=6$ ，则 $a b+a c+a d+b c+b d+c d$ 的正整数取值个数为 $\qquad$. | 已知整数 $a, b, c, d$ 满足 $a+b+c+d=6$ ，则 $a b+a c+a d+b c+b d+c d$ 的正整数取值个数为 $\qquad$. | 10 | dapo_math |
8 | 291 | In the diagram, $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ $FG,$ $GH,$ and $HK$ all have length $4,$ and all angles are right angles, with the exception of the angles at $D$ and $F.$
[asy]
draw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewidth(1));
draw((0,0)--(0.5... | In the diagram, $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ $FG,$ $GH,$ and $HK$ all have length $4,$ and all angles are right angles, with the exception of the angles at $D$ and $F.$
[asy]
draw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewidth(1));
draw((0,0)--(0.5... | 8 | dapo_math |
45 | 292 | Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angl... | Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angl... | 45 | dapo_math |
98 | 293 | In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? | In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? | 98 | dapo_math |
24 | 294 | 某班有 47 个学生，所用教室有 6 排，每排有 8个座位，用 $(i, j)$ 表示位于第 $i$ 排第 $j$ 列的座位。新学期准备调整座位，设某学生原来的座位为 $(i, j)$ ，如果调整后的座位为 $(m, n)$ ，则称该生作了移动 $[a$, $b]=[i-m, j-n]$ ，并称 $a+b$ 为该生的位置数。所有学生的位置数之和记为 $S$ 。求 $S$ 的最大可能值与最小可能值之差。 | 某班有 47 个学生，所用教室有 6 排，每排有 8个座位，用 $(i, j)$ 表示位于第 $i$ 排第 $j$ 列的座位。新学期准备调整座位，设某学生原来的座位为 $(i, j)$ ，如果调整后的座位为 $(m, n)$ ，则称该生作了移动 $[a$, $b]=[i-m, j-n]$ ，并称 $a+b$ 为该生的位置数。所有学生的位置数之和记为 $S$ 。求 $S$ 的最大可能值与最小可能值之差。 | 24 | dapo_math |
60 | 295 | Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$. | Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$. | 60 | dapo_math |
0 | 296 | 复数 $z_{1}, z_{2}, \\cdots, z_{100}$ 满足: $z_{1}=3+2 \\mathrm{i}, z_{n+1}=\\overline{z_{n}} \\cdot \\mathrm{i}^{n}(n=1,2, \\cdots, 99)$ （ i 为虚数单位), 请给出 $z_{99}+z_{100}$ 的实部和虚部之和。 | 复数 $z_{1}, z_{2}, \\cdots, z_{100}$ 满足: $z_{1}=3+2 \\mathrm{i}, z_{n+1}=\\overline{z_{n}} \\cdot \\mathrm{i}^{n}(n=1,2, \\cdots, 99)$ （ i 为虚数单位), 请给出 $z_{99}+z_{100}$ 的实部和虚部之和。 | 0 | dapo_math |
10 | 297 | Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors. | Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors. | 10 | dapo_math |
142 | 298 | For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1400}$ terminate? | For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1400}$ terminate? | 142 | dapo_math |
6 | 299 | Find the first digit after the decimal point of the number $\displaystyle \frac{1}{1009} + \frac{1}{1010} + \cdots + \frac{1}{2016}$. | Find the first digit after the decimal point of the number $\displaystyle \frac{1}{1009} + \frac{1}{1010} + \cdots + \frac{1}{2016}$. | 6 | dapo_math |
5 | 300 | 已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围。标准答案格式为$\left(-\infty, \frac{a}{b}\right)$，请给出区间端点a+b的和。 | 已知 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围。标准答案格式为$\left(-\infty, \frac{a}{b}\right)$，请给出区间端点a+b的和。 | 5 | dapo_math |
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