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 | Solve the equation $\left(a x^{2}+b x+14\right)^{2}+\left(b x^{2}+a x+8\right)^{2}=0$ in the set of integers, where $a$ and $b$ are integers. | -2,=-6,b=-5 | 60 | 8 |
math | Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$. | 6528 | 97 | 4 |
math | # Problem 6.
In the alphabet of the inhabitants of the magical planet ABV2020, there are only three letters: A, B, and V, from which all words are formed. In any word, two identical letters cannot be adjacent, and each of the three letters must be present in any word. For example, the words ABV, VABAVAB, and BVBVAB ar... | 1572858 | 120 | 7 |
math | 3. Find a two-digit number, the digits of which are different and the square of which is equal to the cube of the sum of its digits. | 27 | 31 | 2 |
math | 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f(x)+y-f(x) y=f(x+f(y))-f(x f(y))
$$ | f(x)=x | 60 | 4 |
math | 2. Given that $f(x)$ is an odd function defined on $\mathbf{R}$, and the function $y=f(x+1)$ is an even function. When $-1 \leqslant x \leqslant 0$, $f(x)=x^{3}$. Then $f\left(\frac{9}{2}\right)=$ $\qquad$ . | \frac{1}{8} | 83 | 7 |
math | Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$.
[asy]
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pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x);
draw(B--C--F--D--C^^D--A--E--B--A, linew... | 578 | 250 | 3 |
math | 4. Given complex numbers $z, z_{1}, z_{2}\left(z_{1} \neq z_{2}\right)$ satisfy $z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i}$, and $\left|z-z_{1}\right|=\left|z-z_{2}\right|=4$, then $|z|=$ $\qquad$ . | 2\sqrt{3} | 94 | 6 |
math | Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$
(a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying
\[2\le r \le n-2, \quad n-r+1 < p_r\]
and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. ... | m \ge 4 | 223 | 6 |
math | 6-178 Let $R$ be the set of all real numbers, and $R^{+}$ be the subset of all positive real numbers. $\alpha, \beta$ are given real numbers. Find all functions $f: R^{+} \rightarrow R$ such that for all $x$ and $y$ in $R^{+}$, we have
$$
f(x) f(y)=y^{\alpha} f\left(\frac{x}{2}\right)+x^{\beta} f\left(\frac{y}{2}\right... | f(x)=0 | 121 | 4 |
math | Example 2 Given the polynomial $p(n)=n^{3}-n^{2}-5 n+2$. Find all integers $n$, such that $p^{2}(n)$ is the square of a prime number. (2002 Australian National Mathematics Competition) | -3,-1,0,1,3 | 56 | 10 |
math | 20. Find the sum of the fourth powers of the real roots of the equation
$$
x^{4}-1000 x^{2}+2017=0
$$ | 1991932 | 41 | 7 |
math | 6.118. $\left\{\begin{array}{l}\sqrt{\frac{x+y}{2}}+\sqrt{\frac{x-y}{3}}=14, \\ \sqrt{\frac{x+y}{8}}-\sqrt{\frac{x-y}{12}}=3 .\end{array}\right.$ | (124;76) | 67 | 8 |
math | 10.1. Natural numbers starting from 1 are written in a row. This results in a sequence of digits: 1234567891011121314... What digit is in the 2021st position? | 1 | 59 | 1 |
math | 5.1. In parallelogram $O A B C$, the vertices $O(0 ; 0 ;)$, $A(3 ; 6)$, and $B(8 ; 6)$ are given. Find the ratio of the lengths of the diagonals $O B$ and $A C$, and also write the equations of the sides of the parallelogram and the diagonal $A C$. | OB:AC=\sqrt{2.5};0(OC),6(AB),2x(OA),2x-10(BC),-3x+15(AC) | 86 | 40 |
math | $7 \cdot 73$ Let $n \in N$, and make $37.5^{n}+26.5^{n}$ a positive integer, find the value of $n$.
(Shanghai Mathematical Competition, 1998) | 1,3,5,7 | 57 | 7 |
math | Example 3.26. Verify that the expression
$$
y\left(e^{x y}+6\right) d x+x\left(e^{x y}+6\right) d y
$$
is a total differential of some function $f(x, y)$, and find this function. | f(x,y)=e^{xy}+6xy+C | 66 | 12 |
math | 19. (2006 Zhejiang Province High School Mathematics Party Competition Training Test) Given $a, b, c \in \mathbf{R}^{+}$, and satisfying $\frac{k a b c}{a+b+c} \geqslant(a+b)^{2}$ $+(a+b+4 c)^{2}$, find the minimum value of $k$. | 100 | 83 | 3 |
math | 7. (10 points) The teacher is doing calculation practice with Jiajia, Fangfang, and Mingming. The teacher first gives each of them a number, and then asks them to each pick 3 cards with numbers on them. Jiajia picks 3, 6, 7, Fangfang picks 4, 5, 6, and Mingming picks 4, 5, 8. The teacher then asks them to each multiply... | 7 | 157 | 1 |
math | 6.221. $\left\{\begin{array}{l}x+y+z=2, \\ 2 x+3 y+z=1, \\ x^{2}+(y+2)^{2}+(z-1)^{2}=9 .\end{array}\right.$ | (-1;0;3),(3;-2;1) | 63 | 13 |
math | The line $x-2y-1=0$ insects the parabola $y^2=4x$ at two different points $A, B$. Let $C$ be a point on the parabola such that $\angle ACB=\frac{\pi}{2}$. Find the coordinate of point $C$. | (1, -2) \text{ or } (9, -6) | 68 | 17 |
math | Provide four consecutive natural numbers, each of which has a square divisor greater than 1. | 242=11^{2}\cdot2,\quad243=3^{2}\cdot27,\quad244=2^{2}\cdot61,\quad245=7^{2}\cdot5 | 18 | 49 |
math | 6th APMC 1983 Problem 2 Find all primes p, q such that p(p+1) + q(q+1) = n(n+1) for some positive integer n. | (p,q,n)=(3,5,6),(5,3,6),(2,2,3) | 43 | 22 |
math | 2. Solve in the set of real numbers the equation:
$$
\sqrt{|x-1|}+\sqrt{|x-2015|}=\sqrt{2014}
$$
(Problem E:14587 from G.M. nr. 12/2013) | x\in{1,2015} | 67 | 11 |
math | 10.5. 10.5 A - a four-digit number composed of non-zero digits, B the number written with the same digits in reverse order. It is known that the sum A+B is divisible by 109. What can the sum of the digits of A be? | 14,23,28 | 61 | 8 |
math | One, (20 points) If the two quadratic equations
$$
\begin{array}{l}
a^{2} x^{2}+a x-1=0, \\
x^{2}-a x-a^{2}=0
\end{array}
$$
have a common solution, find all possible values of $a$. | \frac{-1+\sqrt{5}}{2}, \frac{-1-\sqrt{5}}{2}, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2} | 71 | 48 |
math | 10.2 A group of friends went for a morning run around a lake. During the run, one by one they realized they had miscalculated their strength, and switched from running to walking. One of the friends calculated that he had run one-eighth of the total distance that the entire group had run, and walked one-tenth of the to... | 9 | 86 | 1 |
math | $1 \cdot 10$ Among the first 1000 positive integers, how many can be expressed in the form $[2 x]+[4 x]$ $+[6 x]+[8 x]$? where $x$ is some real number. | 600 | 55 | 3 |
math | 10. Let $\triangle A B C$ be equilateral, and let $D, E, F$ be points on sides $B C, C A, A B$ respectively, with $F A=9, A E=E C=6, C D=4$. Determine the measure (in degrees) of $\angle D E F$. | 60 | 72 | 2 |
math | 4・145 Solve the system of equations
$$\left\{\begin{array}{l}
x_{1}+x_{2}+x_{3}=6, \\
x_{2}+x_{3}+x_{4}=9, \\
x_{3}+x_{4}+x_{5}=3, \\
x_{4}+x_{5}+x_{6}=-3, \\
x_{5}+x_{6}+x_{7}=-9, \\
x_{6}+x_{7}+x_{8}=-6, \\
x_{7}+x_{8}+x_{1}=-2, \\
x_{8}+x_{1}+x_{2}=2 .
\end{array}\right.$$ | x_{1}=1, x_{2}=2, x_{3}=3, x_{4}=4, x_{5}=-4, x_{6}=-3, x_{7}=-2, x_{8}=-1 | 168 | 51 |
math | 11. Determine all functions \( f: \mathbf{N} \rightarrow \mathbf{N} \), such that
\[
f(a+b)=f(a)+f(b)+f(c)+f(d)
\]
for all non-negative integers \( a, b, c, d \) satisfying \( 2 a b=c^{2}+d^{2} \). | f(n)=kn^{2}(k\in{N}) | 80 | 13 |
math | Let $0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}$ be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\] \[\sin\beta-\cos\alpha=\cot\gamma\]
Compute the sum of all possible values of $\gamma$ in degrees.
[i]Proposed by Michael Ren[/i] | 45^\circ | 82 | 4 |
math | Let $1\le k\le n$ be integers. At most how many $k$-element subsets can we select from $\{1,2,\dots,n\}$ such that for any two selected subsets, one of the subsets consists of the $k$ smallest elements of their union? | n-k+1 | 61 | 4 |
math | 1. Calculate:
$$
\left(\frac{404445^{2}}{202222 \times 202223 \times 202224}-\frac{202223}{202222 \times 202224}-\frac{202222}{202223 \times 202224}\right) \times 12639=
$$
$\qquad$ | \frac{1}{8} | 115 | 7 |
math | 2.47 If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a perfect square, and $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$? | 675 | 52 | 3 |
math | 6. Let the set $A=\{0,1,2, \cdots, 9\},\left\{B_{1}, B_{2}, \cdots, B_{k}\right.$ be a family of non-empty subsets of $A$, and when $i \neq j$, $B_{i} \cap B_{j}$ has at most two elements. Then the maximum value of $k$ is $\qquad$ | 175 | 95 | 3 |
math | 3. Let $a, b, c$ be the three sides of a right triangle, with $c$ being the hypotenuse. The maximum value of $k$ such that the inequality $a^{2}(b+c)+b^{2}(c+a)$ $+c^{2}(a+b) \geqslant k a b c$ holds for all right triangles is $\qquad$. | 2+3\sqrt{2} | 84 | 8 |
math | 1st Iberoamerican 1985 Problem A2 P is a point inside the equilateral triangle ABC such that PA = 5, PB = 7, PC = 8. Find AB. Solution | \sqrt{129} | 44 | 7 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1} \frac{\sqrt{2^{x}+7}-\sqrt{2^{x+1}+5}}{x^{3}-1}
$$ | -\frac{\ln2}{9} | 55 | 8 |
math | Recently, the following game has become popular: $A$ says to $B$: "Write down your shoe size (as a whole number only!), add 9 to it. Multiply the resulting number by 2, and then add 5. Multiply the resulting sum by 50, and add 1708. Subtract your birth year from this sum. Tell me the final result!" $B$ (who has been ca... | 43,50 | 222 | 5 |
math | An arithmetic progression's first, second, fifth, and last terms are consecutive terms of a geometric progression, the sum of which is 80. Let's find these two progressions. | 2,6,10,\ldots542,6,18,\ldots54 | 38 | 22 |
math | Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 103 | 81 | 3 |
math | 5. In $\triangle A B C$, $A B=A C=5, B C=6$, and its orthocenter $H$ satisfies $\overrightarrow{A H}=m \overrightarrow{A B}+n \overrightarrow{B C}$. Then $m+n=$ $\qquad$. | \frac{21}{32} | 65 | 9 |
math | 3.108. $\frac{4 \sin \left(\frac{5}{2} \pi+\alpha\right)}{\operatorname{tg}^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{2}\right)-\operatorname{ctg}^{2}\left(\frac{3}{2} \pi+\frac{\alpha}{2}\right)}$.
3.108. $\frac{4 \sin \left(\frac{5}{2} \pi+\alpha\right)}{\tan^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{2}\right)-\cot^{2}\le... | \sin^{2}\alpha | 164 | 6 |
math | 5. Find all positive integers $n$ such that the sum $1+2+3+\cdots+n$ is a three-digit number composed of the same digit. | 36 | 35 | 2 |
math | For how many ordered pairs of positive integers $(a,b)$ with $a,b<1000$ is it true that $a$ times $b$ is equal to $b^2$ divided by $a$? For example, $3$ times $9$ is equal to $9^2$ divided by $3$.
[i]Ray Li[/i] | 31 | 78 | 2 |
math | Juca has fewer than 800 marbles. He likes to separate the marbles into groups with the same number of marbles. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finall... | 419 | 144 | 3 |
math | In a set of $n$ numbers, one of the numbers is 0, and another is 1.
a) What is the smallest possible variance of such a set?
b) What should the set be for this to happen? | all,exceptthefirstlast,equalto\frac{1}{2} | 48 | 17 |
math | Example 4.2.4. Given positive real numbers $x_{1}, x_{2}, \ldots, x_{n} \in[a, b]$, find the maximum value of
$$\left(x_{1}-x_{2}\right)^{2}+\left(x_{1}-x_{3}\right)^{2}+\cdots+\left(x_{1}-x_{n}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\cdots+\left(x_{n-1}-x_{n}\right)^{2}$$ | \max (F)=\left\{\begin{array}{l}
m^{2}(a-b)^{2} \text { if } n=2 m, m \in \mathbb{N} \\
m(m+1)(a-b)^{2} \text { if } n=2 m+1, m \in \mathbb{N}
\end{array}\right.} | 127 | 85 |
math | 5. Given the set
$$
A=\{n|n \in \mathbf{N}, 11| S(n), 11 \mid S(n+1)\} \text {, }
$$
where $S(m)$ denotes the sum of the digits of the natural number $m$. Then the smallest number in set $A$ is $\qquad$ . | 2899999 | 80 | 7 |
math | Find all natural numbers $x$, satisfying the conditions: the product of the digits of the number $x$ is equal to $44x - 86868$, and the sum of the digits is a cube of a natural number. | 1989 | 51 | 4 |
math | 4. If $\triangle A B C$ is an obtuse triangle, then the range of $\operatorname{arccos}(\sin A)+\operatorname{arccos}(\sin B)+\operatorname{arccos}(\sin C)$ is $\qquad$ | \left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) | 60 | 20 |
math | In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to thes... | 36 | 76 | 2 |
math | Determine all pairs $(x, y)$ of real numbers that satisfy the following system of inequalities:
$$
\begin{aligned}
x^{4}+8 x^{3} y+16 x^{2} y^{2}+16 & \leq 8 x^{2}+32 x y \\
y^{4}+64 x^{2} y^{2}+10 y^{2}+25 & \leq 16 x y^{3}+80 x y
\end{aligned}
$$ | (\frac{2}{\sqrt{11}},\frac{5}{\sqrt{11}}),(-\frac{2}{\sqrt{11}},-\frac{5}{\sqrt{11}}),(\frac{2}{\sqrt{3}},\frac{1}{\sqrt{3}}),(-\frac{2}{\sqrt{3}},-\frac{1}{} | 117 | 85 |
math | (1) 2011 is such a four-digit number, the sum of its digits is 4; there are $\qquad$ such four-digit numbers whose digits sum to 4. | 20 | 41 | 2 |
math | ## Task B-1.4.
Three friends, Ante, Bojan, and Vinko, are guessing an unknown six-digit number composed of the digits $1,2,3,4,5,6$, with no repeated digits. Ante said the number is 123456, Bojan said 245163, and Vinko said 463215. None of them guessed the exact number, but Ante correctly guessed the positions of 3 di... | 243156 | 139 | 6 |
math | # 5. The factory paints cubes in 6 colors (each face in its own color, the set of colors is fixed). How many varieties of cubes can be produced? | 30 | 36 | 2 |
math | 7.296
$2 \log _{a}^{\frac{1}{2}} b \cdot\left(\left(\log _{a} \sqrt[4]{a b}+\log _{b} \sqrt[4]{a b}\right)^{\frac{1}{2}}-\left(\log _{a} \sqrt[4]{\frac{b}{a}}+\log _{b} \sqrt[4]{\frac{a}{b}}\right)^{\frac{1}{2}}\right), a, b>1$. | 2, | 123 | 2 |
math | ## Task A-4.2.
Determine all natural numbers $n$ such that some three consecutive coefficients in the expansion of $(1+x)^{n}$ are in the ratio $3: 4: 5$. | 62 | 46 | 2 |
math | Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius. | \frac{\sqrt{3}}{2} | 66 | 10 |
math | Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$.
[i]Proposed by Brandon Wang[/i... | 177 | 107 | 3 |
math | 24. A positive integer is called frierdly if it is divisible by the sum of its digits. For example, 111 is friendly but 123 is not. Find the number of all two-digit friendly numbers. | 23 | 49 | 2 |
math | 4. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=$ $A A_{1}=2, A D=2 \sqrt{3}, M$ is a point in the plane $B A_{1} C_{1}$. Then the minimum value of $\overrightarrow{M A} \cdot \overrightarrow{M C}$ is $\qquad$ . | -\frac{16}{7} | 95 | 8 |
math | Task 4. (20 points) Find the smallest natural solution of the inequality $\left(\frac{2023}{2022}\right)^{27+18+12+8+\ldots+27 \cdot\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{72}$. | 5 | 89 | 1 |
math | 5. (8 points) It is defined that $1 ※ 2=0.1+0.2=0.3, 2 ※ 3=0.2+0.3+0.4=0.9, 5 ※ 4=0.5+0.6+0.7+0.8=2.6$. If $a ※ 15=16.5$, then $a$ equals $\qquad$ . | 4 | 99 | 1 |
math | 1. In a certain country, the alphabet consists of three letters: "M", "G", and "U". A word is any finite sequence of these letters, in which two consonants cannot stand next to each other and two vowels cannot stand next to each other. How many 200-letter words are there in this country that contain each of the three l... | 2^{101}-4 | 79 | 7 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$ f\left(x^{2}-y^{2}\right)=x f(x)-y f(y) $$
for all pairs of real numbers $x$ and $y$. | f(x)=c x, c \in \mathbb{R} | 61 | 15 |
math | 5. For the positive integer $n$, define $a_{n}$ as the unit digit of $n^{(n+1)^{n-2}}$. Then $\sum_{n=1}^{2018} a_{n}=$ $\qquad$ . | 5857 | 58 | 4 |
math | 4. Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are non-zero vectors, and $\boldsymbol{a}+3 \boldsymbol{b}$ is perpendicular to $7 \boldsymbol{a}-5 \boldsymbol{b}$, $\boldsymbol{a}-4 \boldsymbol{b}$ is perpendicular to $7 \boldsymbol{a}-2 \boldsymbol{b}$. Then the angle between vectors $\boldsymbol{a}$ and $\boldsy... | 60^{\circ} | 111 | 6 |
math | 4. In the expression $S=\sqrt{x_{1}-x_{2}+x_{3}-x_{4}}$, $x_{1}, x_{2}, x_{3}, x_{4}$ are a permutation of $1,2,3,4$. The number of different permutations that make $S$ a real number is $\qquad$ | 16 | 75 | 2 |
math | 10.4. Let's consider all 7! seven-digit numbers obtained from the number 1234567 by all possible permutations of its digits. How many of them give a remainder of 5 when divided by 7? Answer: 6!. | 6! | 56 | 2 |
math | Find all functions $f: \mathbb{R}\to [0;+\infty)$ such that:
\[f(x^2+y^2)=f(x^2-y^2)+f(2xy)\]
for all real numbers $x$ and $y$.
[i]Laurentiu Panaitopol[/i] | f(x) = ax^2 \text{ for some } a \ge 0 | 70 | 18 |
math | A1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality
$$
f([x] y)=f(x)[f(y)] .
$$
holds for all $x, y \in \mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. | f(x)=C,whereC=0or1\leqC<2 | 78 | 17 |
math | The weight of a number is the sum of its digits. What is the smallest number that weighs 2000? | 299\ldots999 | 25 | 9 |
math | 14. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, $F_{1}$ and $F_{2}$ are the left and right foci, respectively. A line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If the maximum area of $\triangle F_{1} A B$ is 6, find the equation... | \frac{x^{2}}{8}+\frac{y^{2}}{6}=1 | 116 | 20 |
math | 12.8. Determine the continuous functions $f:\left[\frac{1}{e^{2}} ; e^{2}\right] \rightarrow \mathbb{R}$, for which
$$
\int_{-2}^{2} \frac{1}{\sqrt{1+e^{x}}} f\left(e^{-x}\right) d x-\int_{-2}^{2} f^{2}\left(e^{x}\right) d x=\frac{1}{2}
$$ | f()=\frac{1}{2}\cdot\sqrt{\frac{}{1+}},\forall\in[\frac{1}{e^{2}};e^{2}] | 107 | 37 |
math | 7. Real numbers $x, y, z$ satisfy
$$
x+y+z=1 \text {, and } x^{2}+y^{2}+z^{2}=3 \text {. }
$$
Then the range of $x y z$ is $\qquad$ | [-1,\frac{5}{27}] | 61 | 10 |
math | M5. For which integers $n$ is $\frac{16\left(n^{2}-n-1\right)^{2}}{2 n-1}$ also an integer? | -12,-2,0,1,3,13 | 40 | 14 |
math | Example 7 Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $a_{x}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $a_{x}$ is $\qquad$
(1991, National High School Mathematics Competition) | 1001 | 88 | 4 |
math | Solve the system of equations
$$
\begin{aligned}
x-x y+y & =1 \\
x^{2}+y^{2} & =17
\end{aligned}
$$ | x_{1}=1,y_{1}=4;x_{2}=1,y_{2}=-4;x_{3}=4,y_{3}=1;x_{4}=-4,y_{4}=1 | 42 | 42 |
math | $5 \cdot 7$ Fibonacci numbers are defined as
$$
a_{0}=0, a_{1}=a_{2}=1, a_{n+1}=a_{n}+a_{n-1}(n \geqslant 1) .
$$
Find the greatest common divisor of the 1960th and 1988th terms.
(29th International Mathematical Olympiad Shortlist, 1988) | 317811 | 99 | 6 |
math | 2. In the set of integers, solve the equation
$$
x^{2}+x y+y^{2}=x^{2} y^{2}
$$ | (0,0),(1,-1),(-1,1) | 34 | 14 |
math | 14. If the function $f(x)=-\frac{1}{2} x^{2}+\frac{13}{2}$ has a minimum value of $2a$ and a maximum value of $2b$ on the interval $\left.a, b\right]$, find $[a, b]$. | [1,3]or[-2-\sqrt{17},\frac{13}{4}] | 68 | 22 |
math | Find the $a, b, c$ strictly positive integers such that: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ | (2,3,6),(2,4,4),(3,3,3) | 38 | 19 |
math | 170. During spring, Zhenya lost $20 \%$ of his weight, then gained $30 \%$ during summer, lost $20 \%$ again during autumn, and gained $10 \%$ during winter. Did Zhenya gain or lose weight over the year? | 91.52 | 62 | 5 |
math | 32. Given that $A B C D$ is a square. Points $E$ and $F$ lie on the side $B C$ and $C D$ respectively. such that $B E=C F=\frac{1}{3} A B$. $G$ is the intersection of $B F$ and $D E$. If
$$
\frac{\text { Area of } A B G D}{\text { Area of } A B C D}=\frac{m}{n}
$$
is in its lowest term, find the value of $m+n$. | 23 | 122 | 2 |
math | 3.23 A team of workers completed a certain task. If the team is reduced by 20 people, then the same task will be completed 5 days later than with the original composition, and if the team is increased by 15 people, then the task will be completed 2 days earlier. How many workers were originally in the team and how many... | 60 | 84 | 2 |
math | Let's determine the value of the following expression with as few calculations as possible:
$$
\frac{49 \cdot 91^{3}+338 \cdot 343^{2}}{66^{3}-176 \cdot 121}: \frac{39^{3} \cdot 7^{5}}{1331000}
$$ | \frac{5}{13} | 85 | 8 |
math | A3. What is the value of $\left(\frac{4}{5}\right)^{3}$ as a decimal? | 0.512 | 26 | 5 |
math | 19. A triangle $A B C$ is inscribed in a semicircle of radius 5 . If $A B=10$, find the maximum value of $s^{2}$ where $s=A C+B C$. | 200 | 49 | 3 |
math | 8. The centroid of $\triangle A B C$ is $G$, the midpoints of each side are $D, E, F, \overrightarrow{G D}+\overrightarrow{G E}+\overrightarrow{G F}=$ | \overrightarrow{0} | 51 | 6 |
math | 4・214 Three people, A, B, and C, are playing a game: On three cards, integers $p$, $q$, and $r$ ($0<p<q<r$) are written. The three cards are mixed and distributed to A, B, and C, with each person getting one card. Then, according to the number on each person's card, the corresponding number of marbles is given to each ... | p=1,q=4,r=8 | 201 | 9 |
math | Three. (25 points) Let the quadratic function $f(x)=x^{2}+a x+b$, $F=\max _{|x| \leq 1} |f(x)|$. When $a, b$ traverse all real numbers, find the minimum value of $F$.
| \frac{1}{2} | 63 | 7 |
math | 9. The sum of the first $n$ terms of an arithmetic sequence is 2000, the common difference is 2, the first term is an integer, and $n>1$, the sum of all possible values of $n$ is $\qquad$ . | 4835 | 58 | 4 |
math | 11. (10 points) The cinema is screening four animated films: "Toy Story", "Ice Age", "Shrek", and "The Monkey King", with ticket prices of 50 yuan, 55 yuan, 60 yuan, and 65 yuan, respectively. Each audience member can watch at least one and at most two films. Due to time constraints, "Ice Age" and "Shrek" cannot both b... | 1792 | 128 | 4 |
math | ## Task 1 - 060621
A distance of $20 \mathrm{~m}$ is divided into three sections. The first section is twice as long as the second, and the length of the third section is three times the length of the first section. Calculate the lengths of the individual sections! | 2\frac{2}{9}\mathrm{~},4\frac{4}{9}\mathrm{~},13\frac{1}{3}\mathrm{~} | 67 | 37 |
math | \section*{Problem 4 - 041224}
Solve the system of equations
\[
\begin{array}{r}
\frac{\sin x+\sin y}{\sin x-\sin y}=\frac{5}{3} \\
x+y=90^{\circ}
\end{array}
\]
A solution with integer degree approximations should be provided. | x\approx76+k\cdot180\quad;\quady\approx14-k\cdot180 | 83 | 26 |
math | 7. Given a sequence of positive numbers $a_{1}, a_{2}, \ldots, a_{10}$, satisfying the relation $a_{n}\left(a_{n-1}+a_{n+1}\right)=2 a_{n-1} a_{n+1}\left(a_{n}+1\right)$ for $n=2,3, \ldots, 9$. Find $a_{5}$ if it is known that $a_{1}=1$ and $a_{10}=0.01$ | 0.04 | 119 | 4 |
math | 18. (USA 5) Inside triangle \( A B C \) there are three circles \( k_{1}, k_{2}, k_{3} \) each of which is tangent to two sides of the triangle and to its incircle \( k \). The radii of \( k_{1}, k_{2}, k_{3} \) are 1, 4, and 9. Determine the radius of \( k \). | 11 | 93 | 2 |
math | $8 \cdot 95$ Find the number of sequences that satisfy the following conditions: the number of terms is $n$, each term is 0 or 1 or 2, and 0 cannot be the preceding term or the following term of 2. | x_{n}=\frac{1}{2}[(1+\sqrt{2})^{n+1}+(1-\sqrt{2})^{n+1}] | 55 | 35 |
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