task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | $7 \cdot 8$ If a positive divisor is randomly selected from $10^{99}$, the probability that it is exactly a multiple of $10^{88}$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, find $m+n$.
The prime factorization of $10^{99}$ is $2^{99} \cdot 5^{99}$. The total number of positive divisors of $10^{99}$ is $(99+1)(99+... | 634 | 330 | 3 |
math | Example 2 Find the range of $y=\sqrt{x^{2}-2 x+5}-$ $\sqrt{x^{2}-6 x+13}$. | (-2,2) | 34 | 5 |
math | 43rd Swedish 2003 Problem 4 Find all real polynomials p(x) such that 1 + p(x) ≡ (p(x-1) + p(x+1) )/2. | p(x)=x^2+bx+ | 46 | 9 |
math | 2. Let set $A=\left\{x \mid x^{2}-[x]=2\right\}$ and $B=\{x|| x \mid<2\}$, where the symbol $[x]$ denotes the greatest integer less than or equal to $x$, then $A \cap B=$
$\qquad$ . | {-1,\sqrt{3}} | 72 | 7 |
math | Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$. | (3, 1, 70, 2) | 62 | 14 |
math | 6. In $\triangle A B C$,
$$
\tan A 、(1+\sqrt{2}) \tan B 、 \tan C
$$
form an arithmetic sequence. Then the minimum value of $\angle B$ is | \frac{\pi}{4} | 50 | 7 |
math | Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the ... | 504 | 83 | 3 |
math | 5. Given a regular tetrahedron $P-ABC$ with edge length $1$, points $E, F, G, H$ are on edges $PA, PB, CA, CB$ respectively. If $\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}$, and $\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}$, then $\overrightarrow{EG} \cdot \overrightarrow{FH}=$ $\qquad$ ... | \frac{5}{18} | 109 | 8 |
math | 4. (10 points) In a cave, there is a pile of peaches, which is the common property of three monkeys. The first monkey comes to the cave, divides the peaches into two parts in a ratio of 5:4, and takes the larger part; the second monkey comes after, divides the remaining peaches into two parts in a ratio of 5:4, and tak... | 20 | 128 | 2 |
math | Find all continuous functions $f : R \rightarrow R$ such, that $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$ for any real numbers $x$ and $y$ | f(x) = c - 2x | 52 | 10 |
math | 4. In the computer center, there are 200 computers, some of which (in pairs) are connected by cables, a total of 345 cables are used. We will call a "cluster" a set of computers such that a signal from any computer in this set can reach all other computers via cables (possibly through intermediate computers). Initially... | 153 | 108 | 3 |
math | 10. Let $a_{1}, a_{2}, a_{3}, a_{4}$ be 4 rational numbers such that
$$
\left\{a_{i} a_{j} \mid 1 \leqslant i<j \leqslant 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\},
$$
Find the value of $a_{1}+a_{2}+a_{3}+a_{4}$. | \frac{9}{4} | 123 | 7 |
math | 7.277. $\left\{\begin{array}{l}x^{y}=2, \\ (2 x)^{y^{2}}=64(x>0)\end{array}\right.$ | (\frac{1}{\sqrt[3]{2}};-3),(\sqrt{2};2) | 45 | 22 |
math | 4. Let's highlight the complete squares: $\left\{\begin{array}{l}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (y+1)^{2}+(z-3)^{2}=10 .\end{array}\right.$
Add all the equations $\left\{\begin{array}{c}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (x-2)^{2}+(y+1)^{2}+(z-3)^{2}=14 .\end{array}\right.$
And... | (0;0;0),(0;-2;0),(0;0;6),(0;-2;6),(4;0;0),(4;-2;0),(4;0;6),(4;-2;6) | 253 | 49 |
math | 10.1. (12 points) Find the minimum value of the function $f(x)=x^{2}+3 x+\frac{6}{x}+\frac{4}{x^{2}}-1$ on the ray $x>0$. | 3+6\sqrt{2} | 55 | 8 |
math | Example 6.28. A radio system, having 1000 elements (with a failure rate $\lambda_{i}=10^{-6}$ failures/hour), has passed testing and has been accepted by the customer. It is required to determine the probability of failure-free operation of the system in the interval $t_{1}<\left(t=t_{1}+\Delta t\right)<t_{2}$, where $... | 0.37 | 100 | 4 |
math | 4. Solve in the set of real numbers the equation
$$
\frac{x-a_{1}}{a_{2}+\ldots+a_{n}}+\frac{x-a_{2}}{a_{1}+a_{3}+\ldots+a_{n}}+\ldots+\frac{x-a_{n}}{a_{1}+\ldots+a_{n-1}}=\frac{n x}{a_{1}+\ldots+a_{n}}
$$
where $n \geq 2$ and $a_{i}>0, i \in\{1,2, \cdots, n\}$. | a_{1}+a_{2}+\cdots+a_{n} | 133 | 16 |
math | Example 7.8 Now we use red, blue, and yellow to color the 12 edges of a cube $V$, such that the number of edges colored red, blue, and yellow are $3, 3, 6$, respectively. Find the number of distinct edge-coloring patterns of the cube. | 784 | 65 | 3 |
math | 12.56*. In triangle $ABC$, angle $C$ is twice as large as angle $A$ and $b=2a$. Find the angles of this triangle. | 30,90,60 | 38 | 8 |
math | 2. The largest natural number that cannot be expressed as the sum of four distinct composite numbers is $\qquad$ .
翻译完成，保留了原文的格式和换行。 | 26 | 36 | 2 |
math | [ Algebraic inequalities (miscellaneous).] [ Case analysis $]$
$x, y>0$. Let $S$ denote the smallest of the numbers $x, 1 / y, y+1 / x$. What is the maximum value that $S$ can take? | \sqrt{2} | 57 | 5 |
math | 14.23. How many natural numbers less than a thousand are there that are not divisible by 5 or 7? | 686 | 27 | 3 |
math | 4. 156 Solve the system of equations
$$\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y \\
\frac{4 y^{2}}{1+4 y^{2}}=z \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.$$
for all real solutions, and prove that your solution is correct. | (x, y, z)=(0,0,0) \text { and }(x, y, z)=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) | 101 | 48 |
math | 7. Let $f(x)$ be a monotonic function defined on the interval $(0,+\infty)$. For any $x>0$, we have
$$
f(x)>-\frac{4}{x}, f\left(f(x)+\frac{4}{x}\right)=3 \text {. }
$$
Then $f(8)=$ $\qquad$ | \frac{7}{2} | 79 | 7 |
math | 8. Find the smallest odd number $a$ greater than 5 that satisfies the following conditions: there exist positive integers $m_{1}, n_{1}, m_{2}, n_{2}$, such that
$$
a=m_{1}^{2}+n_{1}^{2}, a^{2}=m_{2}^{2}+n_{2}^{2} \text {, }
$$
and $m_{1}-n_{1}=m_{2}-n_{2}$. | 261 | 108 | 3 |
math | 8th Putnam 1948 Problem A4 Let D be a disk radius r. Given (x, y) ∈ D, and R > 0, let a(x, y, R) be the length of the arc of the circle center (x, y), radius R, which is outside D. Evaluate lim R→0 R -2 ∫ D a(x, y, R) dx dy. Solution | 4\pir | 88 | 3 |
math | Find the remainder of the division of the polynomial $P(x)=x^{6 n}+x^{5 n}+x^{4 n}+x^{3 n}+x^{2 n}+x^{n}+1$ by $Q(x)=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}$ $+x+1$, given that $n$ is a multiple of 7. | 7 | 97 | 1 |
math | 3. On the board, three two-digit numbers are written, one of which starts with 5, the second with 6, and the third with 7. The teacher asked three students to each choose any two of these numbers and add them. The first student got 147, and the answers of the second and third students are different three-digit numbers ... | 78 | 100 | 2 |
math | Example 2 If the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points, then the range of the real number $a$ is $\qquad$
(1998, National High School Mathematics Competition) | -1 \leqslant a \leqslant \frac{17}{8} | 65 | 21 |
math | 3. (5 points) This year, Lingling is 8 years old, and her grandmother is 60 years old. In \qquad years, her grandmother's age will be 5 times Lingling's age. | 5 | 47 | 1 |
math | Example 1 Find the range of the function $y=x^{2}+x \sqrt{x^{2}-1}$. ${ }^{[1]}$ (2013, Hubei Provincial Preliminary of the National High School Mathematics League) | \left(\frac{1}{2},+\infty\right) | 53 | 15 |
math | 2. The solution set of the equation $\sqrt{2 x+2-2 \sqrt{2 x+1}}+\sqrt{2 x+10-6 \sqrt{2 x+1}}=2$ is | [0,4] | 47 | 5 |
math | 2. Function $f$ is defined on the set of integers, satisfying
$$
f(n)=\left\{\begin{array}{ll}
n-3 & n \geqslant 1000 \\
f[f(n+5)] & n<1000
\end{array},\right.
$$
Find $f(84)$. | 997 | 79 | 3 |
math | 1. Given the sets
$$
\begin{array}{c}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and
and $M=N$, then
$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right)$ is equal to . $\qquad$ | -2 | 142 | 2 |
math | 5. (20 points) The clock shows the time as fifteen minutes past five. Determine the angle between the minute and hour hands at this moment.
---
Note: The translation maintains the original text's formatting and structure. | 67.5 | 45 | 4 |
math | 25 Let $n \geqslant 2$, find the maximum and minimum value of the product $x_{1} x_{2} \cdots x_{n}$ under the conditions $x_{i} \geqslant \frac{1}{n}(i=1,2, \cdots, n)$ and $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$. | \frac{\sqrt{n^{2}-n+1}}{n^{n}} \text{ and } n^{-\frac{n}{2}} | 99 | 30 |
math | Three. (25 points) Find all integer pairs $(a, b)$ that satisfy $a^{3}=7 \times 3^{b}-64$.
---
Please note that the translation retains the original formatting and structure of the text. | (-1,2),(5,3) | 51 | 9 |
math | For what value of the parameter $m$ is the sum of the squares of the roots of the equation $x^{2}-(m+1) x+m-1=0$ the smallest?
# | 0 | 42 | 1 |
math | 12. (10 points) A certain natural number minus 39 is a perfect square, and minus 144 is also a perfect square. Find this natural number.
| 160,208,400,2848 | 38 | 16 |
math | Example 3 If real numbers $x, y$ satisfy
$$
\begin{array}{l}
\frac{x}{3^{3}+4^{3}}+\frac{y}{3^{3}+6^{3}}=1, \\
\frac{x}{5^{3}+4^{3}}+\frac{y}{5^{3}+6^{3}}=1,
\end{array}
$$
then $x+y=$ $\qquad$
(2005, National Junior High School Mathematics Competition) | 432 | 112 | 3 |
math | Question 3 Given $n, k \in \mathbf{Z}_{+}, n>k$. Given real numbers $a_{1}, a_{2}, \cdots, a_{n} \in(k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots$, $x_{n}$ satisfy that for any set $I \subseteq\{1,2, \cdots, n\}$, $|I|=k$, there is $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$. Find the maximum val... | a_{1} a_{2} \cdots a_{n} | 171 | 15 |
math | Example 11. Find the domain of the function
\[
f(x)=\log \left(4-x^{2}\right)
\] | (-2,2) | 31 | 5 |
math | 7.5. Find all three-digit numbers $\mathrm{N}$ such that the sum of the digits of the number $\mathrm{N}$ is 11 times smaller than the number $\mathrm{N}$ itself (do not forget to justify your answer). | 198 | 53 | 3 |
math | ## Problem Statement
Calculate the definite integral:
$$
\int_{16 / 15}^{4 / 3} \frac{4 \sqrt{x}}{x^{2} \sqrt{x-1}} d x
$$ | 2 | 50 | 1 |
math | 9. (40 points) For real numbers $a, b$ and $c$ it is known that $a b + b c + c a = 3$. What values can the expression $\frac{a\left(b^{2}+3\right)}{a+b}+\frac{b\left(c^{2}+3\right)}{b+c}+\frac{c\left(a^{2}+3\right)}{c+a}$ take? | 6 | 100 | 1 |
math | The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In thr... | 450 | 108 | 3 |
math | Let the number of coefficients in the expansion of $(1+x)^{n}$ that leave a remainder of $r$ when divided by 3 be denoted as $T_{r}(n)$, $r \in\{0,1,2\}$. Calculate $T_{0}(2006), T_{1}(2006), T_{2}(2006)$. | T_{0}(2006)=1764,T_{1}(2006)=122,T_{2}(2006)=121 | 85 | 37 |
math | $14 \cdot 34$ Calculate the value of the sum $\sum_{n=0}^{502}\left[\frac{305 n}{503}\right]$.
(1st China Northeast Three Provinces Mathematics Invitational Competition, 1986) | 76304 | 62 | 5 |
math | 8. (10 points) There is a five-digit number, when it is divided by $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13$ these 12 natural numbers, the remainders are all different. This five-digit number is $\qquad$ | 83159 | 81 | 5 |
math | 6. Given the set $M=\{1,2, \cdots, k\}$, for $A \subseteq M$, denote the sum of all elements in $A$ as $S(A)$. If $M$ can be divided into two disjoint subsets $A$ and $B$, and $A \cup B=M, S(A)=2 S(B)$. Find all values of $k$.
(1994, Sichuan Province High School Mathematics Competition) | k=3m \text{ or } k=3m-1 | 101 | 15 |
math | 6. In tetrahedron $ABCD$, $\triangle ABD$ is an equilateral triangle, $\angle BCD=90^{\circ}, BC=CD=1, AC=\sqrt{3}, E$ and $F$ are the midpoints of $BD$ and $AC$ respectively. Then the cosine of the angle formed by line $AE$ and $BF$ is $\qquad$ | \frac{\sqrt{2}}{3} | 86 | 10 |
math | [ Projections of the bases, sides, or vertices of a trapezoid ] The Law of Sines
A quadrilateral $K L M N$ is inscribed in a circle of radius $R$, $Q$ is the point of intersection of its diagonals, $K L = M N$. The height dropped from point $L$ to side $K N$ is 6, $K N + L M = 24$, and the area of triangle $L M Q$ is ... | LM=4,KN=20,KL=MN=10;R=5\sqrt{5} | 123 | 24 |
math | There are $169$ lamps, each equipped with an on/off switch. You have a remote control that allows you to change exactly $19$ switches at once. (Every time you use this remote control, you can choose which $19$ switches are to be changed.)
(a) Given that at the beginning some lamps are on, can you turn all the lamps of... | 9 | 113 | 1 |
math | 11.1. On the plate, there are various pancakes with three fillings: 2 with meat, 3 with cottage cheese, and 5 with strawberries. Svetlana ate them all sequentially, choosing each subsequent pancake at random. Find the probability that the first and last pancakes eaten were with the same filling. | \frac{14}{45} | 68 | 9 |
math | Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$. | (18, 1) | 45 | 7 |
math | Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1) $ f(mn) \equal{} f(m)\plus{}f(n)$,
(2) $ f(2008) \equal{} 0$, and
(3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$. | f(n) = 0 | 96 | 7 |
math | 1. Inside square $A B C D$, a point $E$ is chosen so that triangle $D E C$ is equilateral. Find the measure of $\angle A E B$. | 150 | 39 | 3 |
math | [Example 5.2.6] Given that $f(x)$ is an odd function defined on $[-1,1]$, and $f(1)=$ 1, if $a, b \in[-1,1], a+b \neq 0$, then $\frac{f(a)+f(b)}{a+b}>0$. If the inequality
$$
f(x) \leqslant m^{2}-2 a m+1
$$
holds for all $x \in[-1,1], a \in[-1,1]$, find the range of real numbers $m$. | \leqslant-2or\geqslant2or=0 | 130 | 17 |
math | 9. (16 points) Given a moving point $P$ on the parabola $y^{2}=4 x$, and the focus $F(1,0)$. Find the maximum value of the inradius $r$ of $\triangle O P F$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{2 \sqrt{3}}{9} | 82 | 12 |
math | Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. | 24 | 65 | 2 |
math | Example 1. Reduce the general equations of a line to canonical form
\[
\left\{\begin{array}{l}
2 x-3 y-3 z-9=0 \\
x-2 y+z+3=0
\end{array}\right.
\] | \frac{x}{9}=\frac{y}{5}=\frac{z+3}{1} | 59 | 22 |
math | Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal). | n = 6 | 48 | 5 |
math | 142. Find $\lim _{x \rightarrow \infty}(\sqrt{x+1}-\sqrt{x})$. | 0 | 27 | 1 |
math | 8. [2] Define the sequence $\left\{x_{i}\right\}_{i \geq 0}$ by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k>2$. Find $x_{2013}$. | 9 | 89 | 1 |
math | Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5.
[i]Proposed by Matthew Lerner-Brecher[/i] | 816 | 62 | 3 |
math | Example 6 Given the function $f(x)=x+\frac{t}{x}(t>0)$ and the point $P(1,0)$, draw two tangent lines $P M$ and $P N$ from $P$ to the curve $y=f(x)$, with the points of tangency being $M$ and $N$.
(1) Let $|M N|=g(t)$, find the expression for the function $g(t)$.
(2) Does there exist a $t$ such that $M$, $N$, and $A(0,... | 6 | 276 | 1 |
math | ## Task B-1.2.
When asked how many minutes she spends on social networks daily, Iva answered: "The nonuple of that number is between 1100 and 1200, and the tridecuple is between 1500 and 1600." How many minutes does Iva spend on social networks daily? | 123 | 77 | 3 |
math | 1. $[\mathbf{3}]$ What is the sum of all of the distinct prime factors of $25^{3}-27^{2}$ ? | 28 | 34 | 2 |
math | Example 6. Find $\int e^{\alpha x} \sin \beta x d x$. | \frac{\alpha\sin\betax-\beta\cos\betax}{\alpha^{2}+\beta^{2}}e^{\alphax}+C | 21 | 35 |
math | 6. A line has equation $y=k x$, where $k \neq 0$ and $k \neq-1$. The line is reflected in the line with equation $x+y=1$. Determine the slope and the $y$-intercept of the resulting line, in terms of $k$. | Slopeis\frac{1}{k};y-interceptis\frac{k-1}{k} | 66 | 22 |
math | Problem 6. Point $A$ on the plane is located at the same distance from all points of intersection of two parabolas given in the Cartesian coordinate system on the plane by the equations $y=-3 x^{2}+2$ and $x=-4 y^{2}+2$. Find this distance. | \frac{\sqrt{697}}{24} | 66 | 13 |
math | Find all the triplets of real numbers $(x , y , z)$ such that :
$y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$ | (0, 0, 0), (2, 2, 2), (-2, -2, -2) | 79 | 27 |
math | 6. Let $x$ and $y$ be non-negative integers such that $69 x+54 y \leq 2008$. Find the greatest possible value of $x y$.
(1 mark)
設 $x$ 、 $y$ 爲滿足 $69 x+54 y \leq 2008$ 的非負整數。求 $x y$ 的最大可能值。
(1 分) | 270 | 98 | 3 |
math | Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value of $mn$. | 2350 | 42 | 4 |
math | 2. Inside the area of $\measuredangle A O B$, a ray $O C$ is drawn such that $\measuredangle A O C$ is $40^{\circ}$ less than $\measuredangle C O B$ and is equal to one third of $\measuredangle A O B$. Determine $\measuredangle A O B$.
翻译完成，保留了原文的换行和格式。 | 120 | 87 | 3 |
math | Example 6 Let $a_{1}, a_{2}, \cdots, a_{n}$ be an increasing sequence of positive integers. For a positive integer $m$, define
$$
b_{m}=\min \left\{n \mid a_{n} \geqslant m\right\}(m=1,2, \cdots),
$$
i.e., $b_{m}$ is the smallest index $n$ such that $a_{n} \geqslant m$. Given that $a_{20}=2019$, find the maximum value... | 42399 | 180 | 5 |
math | In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions. | BC = \frac{b}{2} (\sqrt{5} - 1) | 80 | 19 |
math | 11.33. Solve the system of equations
$$
\left\{\begin{array}{l}
3\left(x+\frac{1}{x}\right)=4\left(y+\frac{1}{y}\right)=5\left(z+\frac{1}{z}\right) \\
x y+y z+z x=1
\end{array}\right.
$$ | (1/3,1/2,1)(-1/3,-1/2,-1) | 81 | 22 |
math | The operation $\nabla$ is defined by $a \nabla b=(a+1)(b-2)$ for real numbers $a$ and $b$. For example, $4 \nabla 5=(4+1)(5-2)=15$. If $5 \nabla x=30$, what is the value of $x$ ? | 7 | 80 | 1 |
math | Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares. | \{1, 2, 3, 6, 7, 15\} | 24 | 22 |
math | 1. The number $a_{n}$ is formed by writing down the first $n$ squares of consecutive natural numbers in sequence. For example, $a_{11}=149162536496481100$ 121. Determine how many numbers divisible by twelve are among the numbers $a_{1}, a_{2}, \ldots, a_{100000}$. | 16667 | 94 | 5 |
math | 9. Let $A B C D$ be a trapezoid with $A B \| C D$ and $A D=B D$. Let $M$ be the midpoint of $A B$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle B C D$ and the diagonal $A C$. Suppose that $B C=27, C D=25$, and $A P=10$. If $M P=\frac{a}{b}$ for relatively prime positive integers $a$ a... | 2705 | 127 | 4 |
math | In a school, there are $m$ teachers and $n$ students. We assume that each teacher has exactly $k$ students, and that any two students always have exactly $\ell$ teachers in common. Determine a relation between $m, n, k, \ell$. | n(n-1)\ell=k(k-1) | 57 | 11 |
math | 7. Let $x_{i} \in \mathbf{R}, x_{i} \geqslant 0(i=1,2,3,4,5), \sum_{i=1}^{5} x_{i}=1$, then $\max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4}\right.$ $\left.+x_{5}\right\}$ the minimum value equals $\qquad$ . | \frac{1}{3} | 114 | 7 |
math | 60. Meteorological observations. At a weather station, it was noticed that during a certain period of time, if it rained in the morning, it was clear in the evening, and if it rained in the evening, it was clear in the morning. In total, there were 9 rainy days, with clear evenings 6 times and clear mornings 7 times. H... | 11 | 86 | 2 |
math | 7. Given the parabola $y^{2}=4 x$, with its focus at $F$, a line passing through the focus $F$ and with an inclination angle of $\theta\left(0<\theta<\frac{\pi}{2}\right)$ intersects the parabola at points $A$ and $B$, $A O$ (where $O$ is the origin) intersects the directrix at point $B^{\prime}$, and $B O$ intersects ... | \frac{8}{\sin ^{3} \theta} | 138 | 14 |
math | Are there any digit-matching equalities similar to $(30+25)^{2}=3025$ among two-digit, or four-digit natural numbers? | (98+01)^2=9801,(20+25)^2=2025,(30+25)^2=3025 | 35 | 39 |
math | Task 1. (5 points) Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=5$. | 527 | 45 | 3 |
math | 2.121. $\sqrt[4]{32 \sqrt[3]{4}}+\sqrt[4]{64 \sqrt[3]{\frac{1}{2}}}-3 \sqrt[3]{2 \sqrt[4]{2}}$. | \sqrt[12]{32} | 55 | 9 |
math | 15th Iberoamerican 2000 Problem A3 Find all solutions to (m + 1) a = m b + 1 in integers greater than 1. | (,,b)=(2,2,3) | 38 | 10 |
math | Problem 6. Yulia thought of a number. Dasha added 1 to Yulia's number, and Anya added 13 to Yulia's number. It turned out that the number obtained by Anya is 4 times the number obtained by Dasha. What number did Yulia think of? | 3 | 64 | 1 |
math | 7. The set of all natural numbers $n$ that make $3^{2 n+1}-2^{2 n+1}-6^{n}$ a composite number is | n>2 | 36 | 3 |
math | We wish to distribute 12 indistinguishable stones among 4 distinguishable boxes $B_{1}$, $B_{2}, B_{3}, B_{4}$. (It is permitted some boxes are empty.)
(a) Over all ways to distribute the stones, what fraction of them have the property that the number of stones in every box is even?
(b) Over all ways to distribute th... | \frac{12}{65} | 102 | 9 |
math | 3. For $n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum equals 1, let $S$ be the largest of the following numbers: $\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}$, $\cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}}$. Find the minimum possible value of $S$, and for what values of $x_{1}, x_{2}, \cdots, x... | 1-2^{-\frac{1}{n}} | 151 | 11 |
math | Solve the following equation over the set of integer pairs:
$$
(x+2)^{4}-x^{4}=y^{3} \text {. }
$$ | -1,0 | 34 | 4 |
math | 11.2. It is known that $\frac{1}{\cos (2022 x)}+\operatorname{tg}(2022 x)=\frac{1}{2022}$.
Find $\frac{1}{\cos (2022 x)}-\operatorname{tg}(2022 x)$. | 2022 | 74 | 4 |
math | Booin d.A.
Given two sequences: $2,4,8,16,14,10,2$ and 3, 6, 12. In each of them, each number is obtained from the previous one according to the same rule.
a) Find this rule.
b) Find all natural numbers that transform into themselves (according to this rule).
c) Prove that the number $2^{1991}$ will become a single... | 18 | 104 | 2 |
math | 7.288. $\left\{\begin{array}{l}5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=200 \\ 5^{2 \sqrt[3]{x}}+2^{2 \sqrt{y}}=689\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
5^{\sqrt[3]{x}} \cdot 2^{\sqrt{y}}=200 \\
5^{2 \sqrt[3]{x}}+2^{2 \sqrt{y}}=689
\end{array}\right.
\] | (27\log_{5}^{3}2;4\log_{2}^{2}5),(8;9) | 147 | 28 |
math | 13. Xiao Li and Xiao Zhang are running at a constant speed on a circular track. They start at the same time and place, with Xiao Li running clockwise and completing a lap every 72 seconds; Xiao Zhang running counterclockwise and completing a lap every 80 seconds. A quarter-circle arc interval is marked on the track, ce... | 46 | 104 | 2 |
math | 12.9 $f(x)=\sin 4 x \cos 4 x ; f^{\prime}\left(\frac{\pi}{3}\right)=$ ? | -2 | 36 | 2 |
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