task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying
$a + b - c - d = 3$
$ab - 3bc + cd - 3da = 4$
$3ab - bc + 3cd - da = 5$
Find $11(a - c)^2 + 17(b -d)^2$. | 63 | 81 | 2 |
math | (Column 11 Arrange the three numbers $1.5^{-0.2}, 1.5^{0.7},\left(\frac{2}{3}\right)^{\frac{1}{3}}$ in ascending order. | (\frac{2}{3})^{\frac{1}{3}}<1.5^{-0.2}<1.5^{0.7} | 49 | 32 |
math | 4. Given that the pure imaginary numbers $x_{1}, x_{2}, \cdots, x_{1999}$ have a modulus of 1. Then the remainder when $x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{1998} x_{1999}+x_{1999} x_{1}$ is divided by 4 is $\qquad$ | 1 | 96 | 1 |
math | 1. (2 points) In trapezoid $A B C D$ with bases $A D=20$ and $B C=10$, circles constructed on sides $A B, B C$, and $C D$ as diameters intersect at one point. The length of diagonal $A C$ is 18. Find the length of $B D$. | 24 | 79 | 2 |
math | Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 8 | 111 | 1 |
math | 13. Let $n$ be a given positive integer, and $a_{1}, a_{2}, \cdots, a_{n}$ be a sequence of real numbers such that for each $m \leqslant n$, we have $\left|\sum_{k=1}^{m} \frac{a_{k}}{k}\right| \leqslant 1$.
Find the maximum value of $\left|\sum_{k=1}^{n} a_{k}\right|$. | 2n-1 | 109 | 4 |
math | 8. There is no less than 10 liters of milk in the bucket. How can you pour exactly 6 liters of milk from it using an empty nine-liter bucket and a five-liter bucket? | 6 | 41 | 1 |
math | The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=1, a_{n+1}=\frac{1}{16}\left(1+4 a_{n}+\sqrt{1+24 a_{n}}\right)$, find its general term formula. | a_{n}=\frac{1}{3}+\frac{1}{3\cdot2^{2n-1}}+\frac{1}{2^{n}} | 69 | 35 |
math | The points $A(-6,-1), B(2,3), C(-1,4)$ are given in a Cartesian coordinate system. Determine the point $D$ such that the quadrilateral $A B C D$ is an isosceles trapezoid $(A B \| C D)$. | D_{1}(-5,2) | 64 | 9 |
math | Example 1. Solve the equation $\sqrt{x^{4}+2 x+5}$
$$
+\sqrt{x^{2}+2 x+37}=10
$$ | x=-1 \pm 3 \sqrt{21} / 5 | 39 | 16 |
math | The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter? | 16 | 55 | 2 |
math | 8-7. Petya thought of four different digits, not equal to 0. Then he formed all possible four-digit numbers from these digits without repeating any digits. The sum of all these numbers turned out to be 73326. What 4 digits did Petya think of? | 1,2,3,5 | 63 | 7 |
math | Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy the equation
$$ f^{a b c-a}(a b c)+f^{a b c-b}(a b c)+f^{a b c-c}(a b c)=a+b+c $$
for all $a, b, c \geq 2$. (Here $f^{k}$ means $f$ applied $k$ times.) | f(n)=n-1 | 109 | 6 |
math | How many nonzero coefficients can a polynomial $ P(x)$ have if its coefficients are integers and $ |P(z)| \le 2$ for any complex number $ z$ of unit length? | 2 | 39 | 1 |
math | Problem 2. Consider the natural numbers $a, b, c, d$ such that $a<b<c<d$ and $a \cdot b \cdot c \cdot d=2014$. Find the last digit of the number $n=a^{2014}+b^{2014}+c^{2014}+d^{2014}$.
Nicolae Tomescu, Corabia | 5 | 94 | 1 |
math | XXVI - I - Task 1
At the ball, there were 42 people. Lady $ A_1 $ danced with 7 gentlemen, Lady $ A_2 $ danced with 8 gentlemen, ..., Lady $ A_n $ danced with all the gentlemen. How many gentlemen were at the ball? | 24 | 64 | 2 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{3 n-1}-\sqrt[3]{125 n^{3}+n}}{\sqrt[5]{n}-n}
$$ | 5 | 60 | 1 |
math | 12. Given $x, y, z \in \mathbf{Z}$, and
$$
x+y+z=3, x^{3}+y^{3}+z^{3}=3 \text {. }
$$
Then $x^{2}+y^{2}+z^{2}=$ $\qquad$ | 3or57 | 71 | 4 |
math | 8.268. $\operatorname{tg} \frac{3 x}{2}-\operatorname{tg} \frac{x}{2}=2 \sin x$.
8.268. $\tan \frac{3 x}{2}-\tan \frac{x}{2}=2 \sin x$. | x_{1}=2\pik,x_{2}=\\arccos\frac{-1+\sqrt{17}}{4}+2\pin,k,n\inZ | 67 | 38 |
math | 5. Antea and Barbara start their new job on the same day. Antea works three days in a row, then takes one day off and continues in the same manner (3 working days, then 1 day off). Barbara works seven days in a row, then takes three days off and continues in the same manner (7 working days, then 3 days off).
How many ... | 100 | 107 | 3 |
math | 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 | 68 | 2 |
math | Example 12 (2008 National High School Joint Competition Hubei Province Preliminary Test Question) Let the sequence $\{f(n)\}$ satisfy: $f(1)=1$, $f(2)=2, \frac{f(n+2)}{f(n)}=\frac{f^{2}(n+1)+1}{f^{2}(n)+1} \quad(n \geqslant 1)$.
(1) Find the recurrence relation between $f(n+1)$ and $f(n)$, i.e., $f(n+1)=g[f(n)]$;
(2) P... | 63<f(2008)<78 | 157 | 11 |
math | 5. Determine the largest three-digit number $\overline{a b c}$ such that $\overline{a b c}+\overline{b c a}+\overline{c a b}=1221$. Different letters represent different digits. | 821 | 53 | 3 |
math | Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem $k$ takes each $k$ minutes to solve. If for any given problem there is a $\frac13$ chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first? | \frac{19}{27} | 71 | 9 |
math | $9.254 \log _{3} \log _{0,2} \log _{32} \frac{x-1}{x+5}>0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$9.254 \log _{3} \log _{0.2} \log _{32} \frac{x-1}{x+5}>0$. | x\in(-\infty;-11) | 104 | 11 |
math | 11. (16 points) A and B are playing a game on a $20 \times 15$ chessboard. At the beginning, a queen is placed on a square of the chessboard except the top-right corner; starting with A, the two players take turns to move the queen. Each move can be a straight or diagonal move of several squares, but only to the right,... | 287 | 119 | 3 |
math | 1. $A B C$ is an isosceles triangle with $A B=2$ and $\measuredangle A B C=90^{\circ} . D$ is the midpoint of $B C$ and $E$ is on $A C$ such that the area of $A E D B$ is twice the area of $E C D$. Find the length of $D E$. | \frac{\sqrt{17}}{3} | 86 | 11 |
math | 10. Let $M$ be the set of all $\triangle ABC$ satisfying the conditions $BC=1, \angle A=2 \angle B$. If $\triangle ABC \in M$, and the angle bisector $CD$ of $\angle C$ divides the opposite side $AB$ into two segments in the ratio $AD:DB=t$, then $\triangle ABC$ is associated with the real number $t$. Determine the set... | \left(\frac{1}{2}, 1\right) | 98 | 14 |
math | 1. Verify that the number $7^{20}-1$ is divisible by $10^{3}$ and determine the last three digits of the number $7^{2015}$. | 943 | 41 | 3 |
math | B5. Simon has 2017 blue blocks numbered from 1 to 2017. He also has 2017 yellow blocks numbered from 1 to 2017. Simon wants to arrange his 4034 blocks in a row. He wants to do so in such a way that for every $k=1,2, \ldots, 2017$ the following conditions are met:
- to the left of the blue block with number $k$ there a... | 500 | 245 | 3 |
math | 14. $n$ couples are arbitrarily arranged in a row, find the total number of arrangements where no couple is adjacent. | (2n)!-C_{n}^{1}\cdot2(2n-1)!+C_{n}^{2}\cdot2^{2}\cdot(2n-2)!-\cdots+(-1)^{k}C_{n}^{k}2^{k}(2n-k)!+\cdots+(-1)^{n}C_{n}^{n}\cdot2^{n}\cdotn | 26 | 89 |
math | For which values of $a$ and $b$, integers $\geq 2, ab-1$ is divisible by the product $(a-1)(b-1)$? | (2,2)(3,3) | 37 | 9 |
math | For how many integers $n$ is $\frac{2 n^{3}-12 n^{2}-2 n+12}{n^{2}+5 n-6}$ equal to an integer? | 32 | 43 | 2 |
math | ## Task 10/85
We are looking for all quadruples $\left(n_{1} ; n_{2} ; n_{3} ; n_{4}\right)$ of natural numbers $n_{i}(i=1 ; 2 ; 3 ; 4)$ with $n_{1}<n_{2}<n_{3}<$ $n_{4}$ (where $0 \in N$), for which
$$
z=n_{1}!\cdot n_{2}!\cdot n_{3}!\cdot n_{4}!-32
$$
is a perfect square. | (0;1;2;4),(0;2;3;4),(1;2;3;4) | 129 | 25 |
math | 4. On a magnetic board, there are the first 16 natural numbers not divisible by 4. Elena intends to fill a rectangle with 3 rows and 5 columns with distinct numbers from the board such that the 3 row sums are equal and the 5 column sums are equal. She starts with the largest number on the board and succeeds in her goal... | 6 | 114 | 1 |
math | Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$. | 4 | 72 | 1 |
math | 4.22
$$
\frac{\cos ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)-\sin ^{2}\left(\frac{5 \pi}{4}-2 \alpha\right)}{\left(\cos \frac{\alpha}{2}+\sin \frac{\alpha}{2}\right)\left(\cos \left(2 \pi-\frac{\alpha}{2}\right)+\cos \left(\frac{\pi}{2}+\frac{\alpha}{2}\right)\right) \sin \alpha}
$$ | 4\cos2\alpha | 121 | 6 |
math | Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p<q \leqslant n$, Morgane writes the sum $p+q$ on the board. She then notes $\mathcal{P}(n)$ as the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times(2+5) \times(3+5)=280$.
Find all values of $n \geqslant 3$ for ... | 7 | 192 | 1 |
math | If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ? | 16 | 49 | 2 |
math | 4. A sequence of 2016 numbers is written. Each one, except the first and the last, is equal to the sum of its neighbors. Find the sum of all 2016 numbers. | 0 | 45 | 1 |
math | 4. In $\triangle A B C$, $A B=A C=2$, and there are 100 different points $P_{1}, P_{2}, \cdots, P_{100}$ on side $B C$. Let $m_{i}=A P_{i}^{2}+$ $B P_{i} \cdot P_{i} C(i=1,2, \cdots, 100)$, find the value of $m_{1}+m_{2}+\cdots+m_{100}$.
(1990 National Junior High School League Problem) | 400 | 131 | 3 |
math | 354. Find the expressions for the sums:
\[
\begin{aligned}
& 2^{2}+4^{2}+\ldots+(2 n)^{2} \\
& 1+3^{2}+\ldots+(2 n+1)^{2}
\end{aligned}
\] | \frac{2n(n+1)(2n+1)}{3} | 67 | 17 |
math | The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $i/p$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017... | 4028 | 144 | 4 |
math | Let's determine the mass of the Earth's atmosphere! Assume that the temperature is constant! (See the article published in the September issue.)
Translating the text as requested, while retaining the original line breaks and formatting. | 5\cdot10^{18}\mathrm{~} | 46 | 13 |
math | 9. Given that $\alpha, \beta$ are the two roots of the quadratic equation $2 x^{2}-t x-2=0$ with respect to $x$, and $\alpha<\beta$, if the function $f(x)=$ $\frac{4 x-t}{x^{2}+1}$.
(1) Find the value of $\frac{f(\alpha)-f(\beta)}{\alpha-\beta}$;
(2) For any positive numbers $\lambda_{1}, \lambda_{2}$, prove that $\lef... | 2|\alpha-\beta| | 190 | 6 |
math | (7) There are 14 small balls of the same size and shape, 7 of which are red and 7 are white. They are placed in two boxes, A and B, where box A contains 4 red balls and 3 white balls, and box B contains 3 red balls and 4 white balls. Now, a ball is randomly drawn from box A and placed into box B, then a ball is randoml... | \frac{25}{8} | 129 | 8 |
math | 3. If real numbers $a, b, c$ make the quadratic function
$$
f(x)=a x^{2}+b x+c
$$
satisfy $|f(x)| \leqslant 1$ for $0 \leqslant x \leqslant 1$, then the maximum value of $2|a|+$ $\sqrt{2}|b|+|c|$ is . $\qquad$ | 17+8\sqrt{2} | 95 | 9 |
math | 6. First, three, and then four people shook hands with each other. How many handshakes were there? Find the pattern in counting the number of handshakes and determine their number for 7 people. | 21 | 44 | 2 |
math | 7. (4 points) Given a right triangle $A B C$, where angle $A$ is 60 degrees, and the hypotenuse $A B$ is $2+2 \sqrt{3}$. Through vertex $B$, a line $p$ parallel to $A C$ is drawn. Points $D$ and $E$ are placed on line $p$ such that $A B=B D, B C=B E$. $F$ is the intersection point of lines $A D$ and $C E$. Find what th... | 2\sqrt{2}+\sqrt{6}+9+5\sqrt{3},\sqrt{3}+1+\sqrt{2},9+5\sqrt{3}+2\sqrt{6}+3\sqrt{2},1+\sqrt{3}+\sqrt{6} | 213 | 65 |
math | Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$. | 35 | 85 | 2 |
math | 1. a) Solve in $Z$ the equation: $5 \cdot(2 \cdot|3 x-4|+4)-30=10$ | 2 | 35 | 1 |
math | 7.2. The road from point A to point B first goes uphill and then downhill. A cat takes 2 hours and 12 minutes to travel from A to B, and the return trip takes 6 minutes longer. The cat's speed going uphill is 4 km/h, and downhill is 5 km/h. How many kilometers is the distance from A to B? (Provide a complete solution, ... | 10 | 89 | 2 |
math | The original problem: When a natural number is multiplied by the number one greater than itself, the product is in the form $A B C D$, where $A, B, C, D$ are different digits. Starting from the number three less than the original, the product is in the form $C A B D$. Starting from the number thirty less than the origi... | 91\cdot92=8372,\quad88\cdot89=7832,\quad61\cdot62=3782 | 219 | 37 |
math | 8. An $8 \times 8$ chessboard is colored in the usual way, with 32 black squares and 32 white squares. A "path" consists of 8 white squares, one in each row, and adjacent white squares share a common vertex. The number of such paths is $\qquad$. | 296 | 67 | 3 |
math | Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal. | n = 4 | 83 | 5 |
math | 6. Let the side length of rhombus $A_{1} A_{2} A_{3} A_{4}$ be $1, \angle A_{1} A_{2} A_{3}=$ $\frac{\pi}{6}, P$ be a point in the plane of rhombus $A_{1} A_{2} A_{3} A_{4}$. Then the minimum value of $\sum_{1 \leqslant i<j \leqslant 4} \overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{j}}$ is $\qquad$ . | -1 | 134 | 2 |
math | A sandwich and a meal plate cost on average $R \$ 5.00$ and $R \$ 7.00$, respectively. In how many ways can one buy sandwiches and meal plates with $R \$ 90.00$, without leaving any change? | 3 | 57 | 1 |
math | 2. Find all solutions to the equation
$$
m^{2}-2 m n-3 n^{2}=5
$$
where $m, n$ are integers. | =4,n=1;=-4,n=-1;=2,n=-1;=-2,n=1 | 37 | 23 |
math | Example 6 Solve the system of equations: $\left\{\begin{array}{l}a+b+c=0, \\ a^{2}+b^{2}+c^{2}=1, \\ a^{3}+b^{3}+c^{3}=4 a b c .\end{array}\right.$
(2017, Irish Mathematical Olympiad) | (,b,)=(\\frac{\sqrt{2}}{2},\\frac{\sqrt{2}}{2},0),(\\frac{\sqrt{2}}{2},0,\\frac{\sqrt{2}}{2}),(0,\\frac{\sqrt{2}}{2},\\frac{\sqrt{2}}{} | 81 | 70 |
math | Given $k \in \mathbb{N}^+$. A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have
(i) $168 \in I_i$;
(ii) $\forall x, y \in I_i$, we have $x-y \in I_i$.
Determine the number of $k-chain$ in total. | (k+1)^2 \binom{k+3}{3} | 130 | 14 |
math | Example 5 Given
$$
A=\left\{z \mid z^{18}=1\right\} \text { and } B=\left\{\omega \mid \omega^{48}=1\right\}
$$
are sets of complex roots of unity,
$$
C=\{z w \mid z \in A, w \in B\}
$$
is also a set of complex roots of unity. How many distinct elements are there in the set $C$? ${ }^{[3]}$ | 144 | 111 | 3 |
math | 10.337. Inside an equilateral triangle with side $a$, there are three equal circles, each of which touches two sides of the triangle and two other circles. Find the area of the part of the triangle that is outside these circles. | \frac{^{2}(2\sqrt{3}-6\pi+3\pi\sqrt{3})}{8} | 52 | 27 |
math | 13.449 The desired three-digit number starts with the digit 1. If it is erased and then written as the last digit of the number, the new three-digit number obtained will be greater than the desired number by $9 a^{1 / \lg a}$. Find this number. | 121 | 62 | 3 |
math | Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$. Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$. If the lines $AD$ and $OE$ meet at $F$, find $|AF|/|FD|$. | \frac{1}{2} | 83 | 7 |
math | Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$. | n+1 | 59 | 3 |
math | Question 53: Find all prime pairs $(p, q)$ such that $\left(3 p^{q-1}+1\right) \mid\left(11^{p}+17^{p}\right)$. | (p,q)=(3,3) | 50 | 7 |
math | 9. Let $a_{1}=1, a_{2}=3$, and for all positive integers $n$, $a_{n+2}=(n+3) a_{n+1}-$ $(n+2) a_{n}$, find all values of $n$ such that $a_{n}$ is divisible by 11. | 4,8orn\geqslant10 | 74 | 11 |
math | 9.1. What is the minimum sum of digits in the decimal representation of the number $f(n)=17 n^{2}-11 n+1$, where $n$ runs through all natural numbers?
# Answer. 2. | 2 | 50 | 1 |
math | 6. Let $\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}$, then $a_{2}+a_{4}+a_{6}+\cdots+a_{2 n}=$ | \frac{3^{n}-1}{2} | 74 | 11 |
math | 1. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the numbe... | 13 | 107 | 2 |
math | ## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$ | \sqrt{2}-1 | 124 | 6 |
math | 17. (5 points) Factory A and Factory B produce the same type of clothing. Factory A produces 2700 sets of clothing per month, with the time ratio for producing tops and pants being 2:1; Factory B produces 3600 sets of clothing per month, with the time ratio for producing tops and pants being 3:2. If the two factories c... | 6700 | 102 | 4 |
math | Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$. | S = \{p\}, \{2, F_n\}, \text{ or the set of all primes} | 63 | 25 |
math | Problem 6. For $x=\frac{\pi}{2 n}$, find the value of the sum
$$
\sin ^{2}(x)+\sin ^{2}(2 x)+\sin ^{2}(3 x)+\ldots+\sin ^{2}(n x)
$$ | \frac{n+1}{2} | 64 | 8 |
math | 8.3. Given an acute-angled triangle $A B C$. Point $M$ is the intersection point of its altitudes. Find the angle $A$, if it is known that $A M=B C$.
---
The text has been translated while preserving the original formatting and line breaks. | 45 | 61 | 2 |
math | Example 2 If a natural number $N$ is appended to the right of any natural number, the resulting number can be divided by $N$ (for example, 2 appended to 35 results in 352, which is divisible by 2), then $N$ is called a "magic number". Among the natural numbers less than 130, how many magic numbers are there? | 9 | 84 | 1 |
math | In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$? | 242 | 57 | 3 |
math | Example 6 Find all pairs of primes $(p, q)$ such that
$$
p q \mid\left(5^{p}+5^{q}\right) .{ }^{[2]}
$$
(2009, China Mathematical Olympiad) | (2,3),(3,2),(2,5),(5,2),(5,5),(5,313),(313,5) | 56 | 33 |
math | Question 2. $n$ is a positive integer greater than 1, $a_{1}, a_{2}, \ldots, a_{n}$ are $n$ distinct positive integers. Let $M=\left\{\left(a_{i}, a_{j}\right),\left[a_{i}, a_{j}\right] \mid 1 \leq i<j \leq n\right\}$, find the minimum number of elements in $M$.
untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
... | n | 153 | 1 |
math | 【Example 3】 6 people stand in a row, A must not stand at the head of the line.
Since A must not stand at the head of the line, A is a restricted element, and since the head of the line must not be occupied by A, the head of the line is a restricted position. | 5\cdot5! | 66 | 5 |
math | Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A=7\cdot 11^5$, the answer would be $7+11+5=23$. | 677 | 75 | 3 |
math | Zé Roberto and Humberto are playing the Millenium Game!
There are 30 empty boxes in a queue, and each box have a capacity of one blue stome.
Each player takes a blue stone and places it in a box (and it is a [i]move[/i]).
The winner is who, in its move, obtain three full consecutive boxes.
If Zé Roberto is the f... | a_{30} | 93 | 6 |
math | 4. Given point $P$ inside $\triangle A B C$, and satisfies
$$
\overrightarrow{A P}=\frac{1}{3} \overrightarrow{A B}+\frac{1}{4} \overrightarrow{A C} \text {, }
$$
and let the areas of $\triangle P B C$, $\triangle P C A$, and $\triangle P A B$ be $S_{1}$, $S_{2}$, and $S_{3}$, respectively. Then $S_{1}: S_{2}: S_{3}=$... | 5: 4: 3 | 126 | 7 |
math | 7. Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1, x_{n+1}=\frac{\sqrt{3} x_{n}+1}{\sqrt{3}-x_{n}}$. Then $\sum_{i=1}^{2017}\left|x_{i}\right|=$ | 3361 | 76 | 4 |
math | 4. Given 2019 indistinguishable coins. All coins have the same weight, except for one, which is lighter. What is the minimum number of weighings required to guarantee finding the lighter coin using a balance scale without weights? | 7 | 50 | 1 |
math | Determine the values of the digits $a, b, c, d, e$ in the following multiplications:
a) $a b c d e 7 \cdot 5=7 a b c d e$
b) $1 a b c d e \cdot 3=a b c d e 1$ | =1,b=4,=2,=8,e=5 | 67 | 14 |
math | There are relatively prime positive integers $m$ and $n$ so that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$ . Find $m + n$. | 19 | 60 | 2 |
math |
09.2. On a faded piece of paper it is possible, with some effort, to discern the following:
$$
\left(x^{2}+x+a\right)\left(x^{15}-\ldots\right)=x^{17}+x^{13}+x^{5}-90 x^{4}+x-90
$$
Some parts have got lost, partly the constant term of the first factor of the left side, partly the main part of the other factor. It wo... | 2 | 156 | 1 |
math | 3. What is the area of the region bounded by the curves $y=x^{2003}$ and $y=x^{1 / 2003}$ and lying above the $x$-axis? | \frac{1001}{1002} | 45 | 13 |
math | Begunni A.
What can be the product of several different prime numbers if it is divisible by each of them, decreased by 1?
Find all possible values of this product.
# | 6,42,1806 | 39 | 9 |
math | Three nonnegative real numbers satisfy $a,b,c$ satisfy $a^2\le b^2+c^2, b^2\le c^2+a^2$ and $c^2\le a^2+b^2$. Prove the inequality
\[(a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3)\ge 4(a^6+b^6+c^6).\]
When does equality hold? | (a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3) \geq 4(a^6+b^6+c^6) | 101 | 41 |
math | 17. Let $x, y, z$ be integers, and
$$
x+y+z=3, x^{3}+y^{3}+z^{3}=3 \text {. }
$$
Then $x^{2}+y^{2}+z^{2}=$ $\qquad$ . | 3 \text{ or } 57 | 67 | 9 |
math | 6. Find the value of $r$ such that
$$
\begin{array}{l}
{\left[r+\frac{19}{100}\right]+\left[r+\frac{20}{100}\right]+\cdots} \\
+\left[r+\frac{91}{100}\right]=546 .
\end{array}
$$
Find $[100 r]$. (Where $[x]$ denotes the greatest integer not greater than $x$) | 743 | 106 | 3 |
math | Task B-3.2. Let the set $A=\{n: n \in \mathbb{N}, n<101\}$. How many four-element subsets of the set $A$ are there such that the difference between the largest and smallest element is 12? | 4840 | 61 | 4 |
math | ## Task B-1.4.
How many right triangles are there with the lengths of the legs being integers $a, b$, and the length of the hypotenuse $b+1$, where $b<100$? | 6 | 49 | 1 |
math | How many 8-digit numbers are there whose decimal notation is of the form $a b 2019 c d$ with $a>0$, and which are divisible by 360? | 20 | 42 | 2 |
math | 5. The exam consists of $N \geqslant 3000$ questions. Each of the 31 students has learned exactly 3000 of them, and every question is known by at least 29 students. Before the exam, the teacher openly laid out all the question cards in a circle. He asked the students to point to one of the questions and explained that ... | 3100 | 157 | 4 |
math | For any natural number $n$, we denote $\mathbf{S}(n)$ as the sum of the digits of $n$. Calculate $\mathbf{S}^{5}\left(2018^{2018^{2018}}\right)$. | 7 | 58 | 1 |
math | Problem 7.1. Each of the seven dwarfs thought of a natural number. They all know what the others have thought of. Snow White asked each of the dwarfs what number they thought of.
- The 1st dwarf remained silent.
- The 2nd dwarf said: “My number is the same as the number of the first dwarf.”
- The 3rd dwarf said: “My n... | 7or14 | 211 | 4 |
math | 18. (6 points) In a certain exam, the average score of 11 students, rounded to the first decimal place, is 85.3. It is known that each student's score is an integer. Therefore, the total score of these 11 students is $\qquad$ points. | 938 | 65 | 3 |
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