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 | 3. Given the set $M=\{(a, b) \mid a \leqslant-1, b \leqslant m\}$. If for any $(a, b) \in M$, it always holds that $a \cdot 2^{b}-b-3 a \geqslant 0$, then the maximum value of the real number $m$ is $\qquad$ | 1 | 87 | 1 |
math | ## Task 1 - V10721
All countries of the socialist camp together produced:
| Product | Pre-war year | 1959 |
| :--- | :--- | :--- |
| Electric power | 84.7 billion kWh | 418.2 billion kWh |
| Steel | 25.4 million t | 92.7 million t |
| Cement | 14.5 million t | 73.3 million t |
By how many percent did production incr... | 393.7,265,405.5 | 111 | 15 |
math | 17. (1993 3rd Macau Mathematical Olympiad) $x_{1}, x_{2}, \cdots, x_{1993}$ satisfy
$$
\begin{array}{l}
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993, \\
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}(k=1,2, \cdots, 1993) .
\end{array}
$$
Then what is the maximum possible value... | 1992 | 205 | 4 |
math | ## Task B-4.5.
Determine all prime numbers $p$ and $q$ for which the equation
$$
5 p^{3}-8 q+5 p-10=0
$$
holds. | p=2,q=5 | 47 | 6 |
math | For any positive integer $n$, define
$$
S(n)=\left[\frac{n}{10^{[\lg n]}}\right]+10\left(n-10^{[\lg n]}\left[\frac{n}{10^{[\operatorname{Ig} n]}}\right]\right) \text {. }
$$
Then among the positive integers $1,2, \cdots, 5000$, the number of positive integers $n$ that satisfy $S(S(n))=n$ is $\qquad$ | 135 | 117 | 3 |
math | There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say... | 24 | 167 | 2 |
math | 1. Given the function $S=|x-2|+|x-4|$.
(1) Find the minimum value of $S$;
(2) If for any real numbers $x, y$ the inequality
$$
S \geqslant m\left(-y^{2}+2 y\right)
$$
holds, find the maximum value of the real number $m$. | 2 | 85 | 1 |
math | 7. Given the equation $8 t^{3}-4 t^{2}-4 t+1=0$ has a root $x$ in $\left(0, \frac{\pi}{13}\right)$, then $x=$ | \sin\frac{\pi}{14} | 50 | 10 |
math | 3. (8 points) In a football team, 11 players are on the field, including 1 goalkeeper who does not participate in the formation of defenders, midfielders, and forwards. It is known that the number of defenders is between 3-5, the number of midfielders is between 3-6, and the number of forwards is between 1-3. Therefore... | 8 | 102 | 1 |
math | 2. The sum of the diagonals of a rhombus is equal to $8 \mathrm{~cm}$, and its area is equal to $7 \mathrm{~cm}^{2}$. Determine the perimeter of the rhombus? | 12\mathrm{~} | 52 | 7 |
math | Anya is waiting for a bus. Which event has the highest probability?
$A=\{$ Anya waits for the bus for at least one minute $\}$,
$B=\{$ Anya waits for the bus for at least two minutes $\}$,
$C=\{$ Anya waits for the bus for at least five minutes $\}$. | A | 69 | 1 |
math | The expression
$$
\left(\begin{array}{lllll}
1 & 1 & 1 & \cdots & 1
\end{array}\right)
$$
is written on a board, with 2013 ones in between the outer parentheses. Between each pair of consecutive ones you may write either "+" or ")(" (you cannot leave the space blank). What is the maximum possible value of the resulti... | 3^{671} | 92 | 6 |
math | 53. Point $K$ lies on the base $AD$ of trapezoid $ABCD$, such that $|AK|=\lambda|AD|$. Find the ratio $|AM|:|AD|$, where $M$ is the point of intersection with $AD$ of the line passing through the points of intersection of the lines $AB$ and $CD$ and the lines $BK$ and $AC$.
Taking $\lambda=1 / n, n=1,2,3, \ldots$, obt... | \frac{|AM|}{|AD|}=\frac{\lambda}{\lambda+1} | 138 | 20 |
math | Task 4.
Every day after lunch, 7 half-eaten pieces of white bread are left on the desks of the second-grade students. If these pieces are put together, they make up half a loaf of bread. How many loaves of bread will the second-grade students save in 20 days if they do not leave these pieces? How much money will the s... | 350 | 132 | 3 |
math | Example 9 Given a real number $k$, determine all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any $x, y \in \mathbf{R}$, we have $f\left(x^{2}+\right.$ $\left.2 x y+y^{2}\right)=(x+y)[f(x)+f(y)]$ and $|f(x)-k x| \leqslant\left|x^{2}-x\right|$. | f(x)=kx | 108 | 5 |
math | 12. (6 points) The sum of all simplest proper fractions for 1996 is
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 498 | 47 | 3 |
math | Find all integer solutions of the equation:
$$
x^{2}+y^{2}=3\left(u^{2}+v^{2}\right)
$$ | (0,0,0,0) | 34 | 9 |
math | 14. (12 points) Three people, A, B, and C, start from $A$ to $B$ at the same time. A and B turn back to $A$ after reaching $B$, and their speeds are reduced to half of their original speeds after turning back. A turns back first, and meets B face-to-face at $C$ after turning back, at which point C has already traveled ... | 5360 | 152 | 4 |
math | 3. [15] Let $A B C$ be a triangle with circumcenter $O$ such that $A C=7$. Suppose that the circumcircle of $A O C$ is tangent to $B C$ at $C$ and intersects the line $A B$ at $A$ and $F$. Let $F O$ intersect $B C$ at $E$. Compute $B E$. | \frac{7}{2} | 86 | 7 |
math | Let $f$ be a linear function. Compute the slope of $f$ if
$$\int_3^5f(x)dx=0\text{ and }\int_5^7f(x)dx=12.$$ | m = 3 | 49 | 5 |
math | Solve the following system of equations:
$$
\begin{aligned}
& x+y=a \ldots \\
& \frac{1}{x}+\frac{1}{y}=\frac{1}{b} \ldots
\end{aligned}
$$
Determine for what values of $a$ and $b$ the roots are real. What relationship must hold between $a$ and $b$ for one of the roots to be three times the other? | \frac{3a}{16} | 98 | 9 |
math | (4) Given that the coordinates of points $M$ and $N$ satisfy the system of inequalities $\left\{\begin{array}{l}x \geqslant 0, \\ y \geqslant 0, \\ x+2 y \leqslant 6, \\ 3 x+y \leqslant 12,\end{array} \quad \vec{a}=\right.$ $(1,-1)$, then the range of $\overrightarrow{M N} \cdot \vec{a}$ is . $\qquad$ | [-7,7] | 121 | 5 |
math | 2. $\int_{1}^{2} \sqrt{4-x^{2}} \mathrm{~d} x=$ | \frac{2 \pi}{3}-\frac{\sqrt{3}}{2} | 26 | 19 |
math | 14th ASU 1980 Problem 15 ABC is equilateral. A line parallel to AC meets AB at M and BC at P. D is the center of the equilateral triangle BMP. E is the midpoint of AP. Find the angles of DEC. | \angleDEC=90,\angleEDC=60 | 57 | 13 |
math | 3. (48th Slovenian Mathematical Olympiad) Mathieu first wrote down all the numbers from 1 to 10000 in order, then erased those numbers that are neither divisible by 5 nor by 11. Among the remaining numbers, what is the number at the 2004th position? | 7348 | 69 | 4 |
math | # Problem 1. (1 point)
Albert thought of a natural number and said about it: “If you divide 1 by my number and add $1 / 2$ to it, the result will be the same as if you add 2, divided by the number one greater than my guessed number, to $1 / 3$.” Little Bobby, thinking for a moment, replied: “I can't guess, as there ar... | 2,3 | 120 | 3 |
math | 32. When shooting at a target, Misha hit the bullseye several times, scored 8 points as many times, and hit the five several times. In total, he scored 99 points. How many shots did Misha take if $6.25\%$ of his shots did not score any points? | 16 | 69 | 2 |
math |
Problem 1. The faces of an orthogonal parallelepiped whose dimensions are natural numbers are painted green. The parallelepiped is partitioned into unit cubes by planes parallel to its faces. Find the dimensions of the parallelepiped if the number of cubes having no green face is one third of the total number of cubes... | (4,7,30),(4,8,18),(4,9,14),(4,10,12),(5,5,27),(5,6,12),(5,7,9),(6,6,8) | 72 | 56 |
math | $\begin{array}{l}\text { 15. Find the value of } x^{2}+y^{2}+z^{2}+w^{3} \text { . If } \\ \frac{x^{2}}{\varepsilon^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}} \\ +\frac{w^{2}}{2^{2}-7^{2}}=1, \\ \frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}} \\ +\frac{w^{2}}{4^{2}-7^{2... | 36 | 324 | 2 |
math | 6. When the child was born, their parents were not yet 40 years old, but they were already adults. When the child turned 2 years old, the age of exactly one of the parents was divisible by 2; when the child turned 3 years old, the age of exactly one of the parents was divisible by 3, and so on. How long could such a pa... | 8 | 84 | 1 |
math | 4. Determine how many different values of $a$ exist for which the equation
$$
\left(a^{2}-5\right) x^{2}-2 a x+1=0
$$
has a unique solution | 2 | 47 | 1 |
math | The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle. | (13, 14, 15) | 42 | 13 |
math | Three. (25 points) Find the integer solutions of the indeterminate equation
$$
9 x^{2}-6 x-4 y^{4}+12 y^{3}-12 y^{2}+5 y+3=0
$$ | 1,2 | 54 | 3 |
math | We are given three distinct non-zero digits. On the board, we will write all three-digit numbers that can be formed from these digits, using all three digits for each number. The sum of the written numbers is 1776.
Which three digits did we work with? Determine all solutions. | 1,2,51,3,4 | 61 | 10 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{1-\cos \sqrt{x}}$ | 0 | 40 | 1 |
math | 4. Let's fix a point $O$ on the plane. Any other point $A$ is uniquely determined by its radius vector $\overrightarrow{O A}$, which we will simply denote as $\vec{A}$. We will use this concept and notation repeatedly in the following. The fixed point $O$ is conventionally called the initial point. We will also introdu... | \vec{C}=\frac{k\vec{B}-\vec{A}}{k-1} | 528 | 23 |
math | For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients? | p = 2 | 70 | 5 |
math | 1. Solve the inequality $\sqrt{x^{2}-4} \cdot \sqrt{2 x-1} \leq x^{2}-4$. | x\in{2}\cup[3;+\infty) | 32 | 14 |
math | The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.
Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$. | \frac{p^2 - 4}{4} | 89 | 12 |
math | 33. (1988 American Mathematical Invitational) If a positive divisor of $10^{99}$ is chosen at random, the probability that it is exactly a multiple of $10^{88}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime, find $m+n$. | 634 | 72 | 3 |
math | 13.107. Two brigades, working together, were supposed to repair a given section of a highway in 18 days. In reality, however, only the first brigade worked at first, and the second brigade, which has a higher labor productivity than the first, finished the repair of the road section. As a result, the repair of the give... | 45 | 119 | 2 |
math | Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$. | d \leq 3 | 53 | 7 |
math | 1. The ellipse $x^{2}+k y^{2}=1$ and the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$ have the same directrix. Then $k=$ $\qquad$ . | \frac{16}{7} | 58 | 8 |
math | 321. Spheres and a cube. Once, during transportation, it was required to pack a sphere with a diameter of 30 cm into a cubic box with a side of 32 cm. To prevent the sphere from moving during transportation, 8 identical small spheres had to be placed in the corners of the box. What is the diameter of such a small spher... | 63-31\sqrt{3}\approx9.308 | 78 | 16 |
math | 1. Let $f(x)$ be a function defined on $R$, for any real number $x$ we have $f(x+3) \cdot f(x-4)=-1$. Also, when $0 \leq x<7$, $f(x)=\log _{2}(9-x)$, then the value of $f(-100)$ is . $\qquad$ | -\frac{1}{2} | 83 | 7 |
math | 15. A1 (POL) ${ }^{\mathrm{IMO} 2}$ Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such that the inequality
$$ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} $$
holds for every $x_{1}, \ldots, x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ su... | \frac{1}{8} | 153 | 7 |
math | Problem 4. On a board, eight numbers equal to 0 are written, one after another. We call an operation the modification of four of the eight numbers as follows: two numbers increase by 3, one number increases by 2, and the fourth number increases by 1.
a) What is the minimum number of operations we need to perform to ob... | 4 | 139 | 1 |
math | 5. Given $a_{n}=6^{n}+8^{n}$. Then $a_{84} \equiv$ $\qquad$ $(\bmod 49)$ | 2 | 40 | 1 |
math | There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they? | 32, 125 | 36 | 7 |
math | 5. (10 points) A convoy of trucks is delivering supplies to a disaster victim resettlement point. Each truck has a carrying capacity of 10 tons. If each tent is allocated 1.5 tons of supplies, there will be less than one truck's worth of supplies left over. If each tent is allocated 1.6 tons of supplies, there will be ... | 213 | 104 | 3 |
math | # 9. Problem 9
Vasya throws three dice (cubes with numbers from 1 to 6 on the faces) and adds the numbers that come up. Additionally, if all three numbers are different, he can throw all three dice again and add the numbers that come up to the already accumulated sum, and continue doing so until at least two of the th... | 23.625 | 92 | 6 |
math | 52. When the number POTOП was taken as an addend 99999 times, the resulting number had the last three digits 285. What number is denoted by the word POTOП? (Identical letters represent identical digits.) | 51715 | 56 | 5 |
math | Problem 3. (3 points)
Let $f(x)$ be a quadratic trinomial with integer coefficients. Given that $f(\sqrt{3}) - f(\sqrt{2}) = 4$. Find $f(\sqrt{10}) - f(\sqrt{7})$. | 12 | 60 | 2 |
math | ## Task 5 - 020525
On a straight line, consecutively mark the segments $A B=3 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and $C D=4 \mathrm{~cm}$! How large is the distance between the midpoints of the segments $A B$ and $C D$? Justify your answer by calculation! | 8.5 | 87 | 3 |
math | 14. (15 points) For a positive integer $n$, let $f(n)$ be the sum of the digits in the decimal representation of the number $3 n^{2}+n+1$.
(1) Find the minimum value of $f(n)$;
(2) When $n=2 \times 10^{k}-1\left(k \in \mathbf{N}_{+}\right)$, find $f(n)$;
(3) Does there exist a positive integer $n$ such that
$$
f(n)=201... | 2012 | 123 | 4 |
math | 3. (USA 3) Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $A B C$ is an equilateral triangle whose side is 86 meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after cov... | 12 | 135 | 2 |
math | For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$? | 9 | 87 | 1 |
math | 7. The diagonals of a trapezoid are perpendicular to each other, and one of them is 17. Find the area of the trapezoid if its height is 15.
Answer. $\frac{4335}{16}$. | \frac{4335}{16} | 57 | 11 |
math | 6.- Determine the function $f: N \rightarrow N$ (where $N=\{1,2,3, \ldots\}$ is the set of natural numbers) that satisfies, for any $s, n \in N$, the following two conditions:
a) $f(1)=1, f\left(2^{s}\right)=1$.
b) If $n<2^{\text {s }}$, then $f\left(2^{\mathrm{s}}+n\right)=f(n)+1$.
Calculate the maximum value of $f... | 2^{2001}-1 | 155 | 8 |
math | Question 184: Fill $n^{2}(n \geq 3)$ integers into an $n \times n$ grid, with one number in each cell, such that the product of the numbers in each row and each column is 30. How many ways are there to do this? | 2^{(n-1)^2}\cdot(\mathrm{n}!)^3 | 64 | 17 |
math | 9.189. $\log _{5} \sqrt{3 x+4} \cdot \log _{x} 5>1$. | x\in(1;4) | 33 | 8 |
math | 1. Determine all integers $n$ for which the equation $x^{2}+n x+n+5=0$ has only integer solutions. | -5,-3,7,9 | 31 | 8 |
math | Condition of the problem
Find the differential $d y$.
$$
y=\ln \left|x^{2}-1\right|-\frac{1}{x^{2}-1}
$$ | \frac{2x^{3}}{(x^{2}-1)^{2}}\cdot | 40 | 20 |
math | Task 1. Determine all pairs $(a, b)$ of positive integers for which
$$
a^{2}+b \mid a^{2} b+a \quad \text { and } \quad b^{2}-a \mid a b^{2}+b
$$ | (,+1) | 59 | 4 |
math | A teacher proposes 80 problems to a student, informing that they will award five points for each problem solved correctly and deduct three points for each problem not solved or solved incorrectly. In the end, the student ends up with eight points. How many problems did he solve correctly? | 31 | 56 | 2 |
math | ## Task Condition
Compose the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$$
y=\frac{x^{3}+2}{x^{3}-2}, x_{0}=2
$$ | \frac{3}{4}\cdotx+\frac{1}{6} | 52 | 16 |
math | Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\text... | 26 | 135 | 2 |
math | 14. Toss a coin, each time it lands on heads you get 1 point, and on tails you get 2 points. Repeatedly toss this coin, then the probability of getting exactly $n$ points is $\qquad$ | p_{n}=\frac{2}{3}+\frac{1}{3}(-\frac{1}{2})^{n}(n\in{N}_{+}) | 51 | 37 |
math | Example 2 The number of sets $A$ that satisfy $\{a, b\} \subseteq A \varsubsetneqq \{a, b, c, d\}$ is $\qquad$.
| 3 | 44 | 1 |
math | Example 12. Solve the inequality
$$
3 \cdot 7^{2 x}+37 \cdot 140^{x}<26 \cdot 20^{2 x}
$$ | x\geqslant\log_{7/20}\frac{2}{3} | 44 | 20 |
math | 4. Let the lengths of the two legs of a right triangle be $a$ and $b$, and the length of the hypotenuse be $c$. If $a$, $b$, and $c$ are all integers, and $c=\frac{1}{3} a b-(a+b)$, find the number of right triangles that satisfy the condition.
(2010, National Junior High School Mathematics League, Tianjin Preliminary ... | 3 | 95 | 1 |
math | 1. If the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=\frac{2 a_{n}}{3 a_{n}+2}, n \in \mathbf{N}_{+}$, then $a_{2017}=$ $\qquad$ . | \frac{1}{3026} | 79 | 10 |
math | For a certain task, $B$ needs 6 more days than $A$, and $C$ needs 3 more days than $B$. If $A$ works for 3 days and $B$ works for 4 days together, they accomplish as much as $C$ does in 9 days. How many days would each take to complete the task alone? | 18,24,27 | 76 | 8 |
math | 3rd Irish 1990 Problem 1 Find the number of rectangles with sides parallel to the axes whose vertices are all of the form (a, b) with a and b integers such that 0 ≤ a, b ≤ n. | \frac{n^2(n+1)^2}{4} | 50 | 13 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(2 n-3)^{3}-(n+5)^{3}}{(3 n-1)^{3}+(2 n+3)^{3}}
$$ | \frac{1}{5} | 63 | 7 |
math | ## Subject IV. (30 points)
Around a point $O$, $n$ angles are constructed, the first having a measure of $x^{0}$, the second $(2 x)^{0}$, the third $(3 x)^{0}$, and so on, the $n$-th having a measure of $120^{0} .\left(n, x \in \mathbf{N}^{*}\right)$
a) Determine the number of angles constructed;
b) Calculate the me... | 96 | 195 | 2 |
math | Find all triplets $ (x,y,z) $ of positive integers such that
\[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \] | (6, 2, 10) | 50 | 11 |
math | Let $ \overline{MN}$ be a diameter of a circle with diameter $ 1$. Let $ A$ and $ B$ be points on one of the semicircular arcs determined by $ \overline{MN}$ such that $ A$ is the midpoint of the semicircle and $ MB\equal{}\frac35$. Point $ C$ lies on the other semicircular arc. Let $ d$ be the length of the line segme... | 14 | 193 | 2 |
math | 7. In the tetrahedron $S-ABC$,
$$
SA=SB=SC=\sqrt{21}, BC=6 \text{.}
$$
If the projection of point $A$ onto the plane containing the side $SBC$ is exactly the orthocenter of $\triangle SBC$, then the volume of the inscribed sphere of the tetrahedron $S-ABC$ is | \frac{4 \pi}{3} | 87 | 9 |
math | 6.079. $\left\{\begin{array}{l}x^{3}+y^{3}=9, \\ x y=2 .\end{array}\right.$ | (1;2)(2;1) | 40 | 9 |
math | Let's determine the value of $b$ such that in the equation
$$
x^{2}-b x+12=0
$$
the difference of the squares of the roots is 7. | \7 | 43 | 2 |
math | ## SUBJECT 2.
Determine the elements of the set
$$
S=\left\{(a, b, c, d) \left\lvert\, 2 a+3 b+5 c+7 d \leq 174-\frac{8}{a}-\frac{27}{b}-\frac{125}{c}-\frac{343}{d}\right., \text { with } a, b, c, d \in(0, \infty)\right\}
$$ | (,b,,)=(2,3,5,7) | 114 | 13 |
math | Find all integers $ (x,y,z)$, satisfying equality:
$ x^2(y \minus{} z) \plus{} y^2(z \minus{} x) \plus{} z^2(x \minus{} y) \equal{} 2$ | (x, y, z) = (k + 1, k, k - 1) | 55 | 22 |
math | A $d$ meter long shooting range, at which point will the sound of the weapon's firing and the sound of the bullet hitting the target arrive simultaneously, if the bullet's speed is $c \mathrm{~m} / \mathrm{sec}$, and the speed of sound is $s \mathrm{~m} / \mathrm{sec}$? Consider the scenario.
Consider the scenario. | \frac{(+)}{2}=\frac{1}{2}(1+\frac{}{}) | 82 | 20 |
math | Four, (50 points) Determine all positive integer triples $(x, y, z)$ such that $x^{3}-y^{3}=z^{2}$, where $y$ is a prime number, and $z$ is not divisible by 3 and $y$.
---
Please note that the translation retains the original format and line breaks as requested. | (8,7,13) | 75 | 8 |
math | 2. Variant 1. At the intersection of perpendicular roads, a highway from Moscow to Kazan and a road from Vladimir to Ryazan intersect. Dima and Tolya set out with constant speeds from Moscow to Kazan and from Vladimir to Ryazan, respectively. When Dima passed the intersection, Tolya had 3500 meters left to reach it. Wh... | 9100 | 129 | 4 |
math | 8・ 78 The increasing sequence of integers that are divisible by 3 and are 1 less than a perfect square is $3,15,24,48, \cdots$ What is the remainder when the 1994th term of this sequence is divided by 1000? | 63 | 67 | 2 |
math | Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$.
Find length of $XY$.
| \frac{1}{2} | 76 | 7 |
math | [Example 5.3.4] Solve the system of equations:
$$
\left\{\begin{array}{l}
\left(1-x^{2}\right) y_{1}=2 x, \\
\left(1-y_{1}^{2}\right) y_{2}=2 y_{1}, \\
\left(1-y_{2}^{2}\right) y_{3}=2 y_{2}, \\
y_{3}=x .
\end{array}\right.
$$ | \begin{aligned}y_{1}&=\tan\frac{2n\pi}{7},\\y_{2}&=\tan\frac{4n\pi}{7},\\y_{3}=\tan\frac{n\pi}{7}\cdot(n&=-3,-2,-1,0,1,2,3)\end{aligned} | 105 | 75 |
math | Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying
\[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95)
\]
for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$ | 1995 | 111 | 4 |
math | 9- 118 (1) Factorize $x^{12}+x^{9}+x^{6}+x^{3}+1$.
(2) For any real number $\theta$, prove: $5+8 \cos \theta+4 \cos 2 \theta+\cos 3 \theta \geqslant 0$. | (x^{4}+x^{3}+x^{2}+1)(x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1) | 78 | 43 |
math | 8. Given $a, b \in \mathbf{Z}$, and $a+b$ is a root of the equation
$$
x^{2}+a x+b=0
$$
Then the maximum possible value of $b$ is $\qquad$ | 9 | 57 | 1 |
math | 5. Given $f(x)=\frac{1}{1+x^{2}}$. Then
$$
\begin{array}{l}
f(1)+f(2)+\cdots+f(2011)+ \\
f\left(\frac{1}{2}\right)+f\left(\frac{1}{3}\right)+\cdots+f\left(\frac{1}{2011}\right) \\
=
\end{array}
$$ | 2010.5 | 98 | 6 |
math |
3. Determine all triples $(x, y, z)$ consisting of three distinct real numbers, that satisfy the following system of equations:
$$
\begin{aligned}
x^{2}+y^{2} & =-x+3 y+z, \\
y^{2}+z^{2} & =x+3 y-z, \\
x^{2}+z^{2} & =2 x+2 y-z .
\end{aligned}
$$
| (x,y,z)=(-\frac{3}{2},\frac{5}{2},-\frac{1}{2})(x,y,z)=(0,1,-2) | 97 | 36 |
math | Determine all functions $f:\mathbb{R}\to\mathbb{R}$ for which there exists a function $g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)=\lfloor g(x+y)\rfloor$ for all real numbers $x$ and $y$.
[i]Emil Vasile[/i] | f(x) = \frac{n}{2} | 80 | 11 |
math | 81. Let positive real numbers $x, y, z$ satisfy the condition $2 x y z=3 x^{2}+4 y^{2}+5 z^{2}$, find the minimum value of the expression $P=3 x+2 y+z$.
| 36 | 59 | 2 |
math | 1. A necklace consists of 80 beads of red, blue, and green colors. It is known that on any segment of the necklace between two blue beads, there is at least one red bead, and on any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads that can be in ... | 27 | 100 | 2 |
math | 3. We define coloring of the plane as follows:
- choose a natural number $m$,
- let $K_{1}, K_{2}, \ldots, K_{m}$ be different circles with non-zero radii such that $K_{i} \subset K_{j}$ or $K_{j} \subset K_{i}$ for $i \neq j$,
- points in the plane that are outside any of the chosen circles are colored differently fr... | 2019 | 143 | 4 |
math | 14. Garden plot. The length of a rectangular plot of land was increased by $35 \%$, and the width was decreased by $14 \%$. By what percentage did the area of the plot change? | 16.1 | 44 | 4 |
math | 35. Think of a number written in one column:
| 10 | 23 | 16 | 29 | 32 |
| ---: | ---: | ---: | ---: | ---: |
| 27 | 15 | 28 | 31 | 9 |
| 14 | 32 | 30 | 8 | 26 |
| 36 | 24 | 12 | 25 | 13 |
| 23 | 16 | 24 | 17 | 30 |
How to guess the thought number by the sum of the numbers (excluding this number) in the row or column ... | 30 | 168 | 2 |
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