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 | Example 3 Set $A=\{0,1,2, \cdots, 9\},\left\{B_{1}\right.$, $\left.B_{2}, \cdots, B_{k}\right\}$ is a family of non-empty subsets of $A$, when $i \neq j$, $B_{i} \cap B_{j}$ has at most two elements. Find the maximum value of $k$. | 175 | 94 | 3 |
math | 3. Find all values of $m$ for which any solution of the equation
$$
2018 \cdot \sqrt[5]{6.2 x-5.2}+2019 \cdot \log _{5}(4 x+1)+m=2020
$$
belongs to the interval $[1 ; 6]$. | \in[-6054;-2017] | 78 | 13 |
math | Given the function
$$
f(x)=\frac{(x+a)^{2}}{(a-b)(a-c)}+\frac{(x+b)^{2}}{(b-a)(b-c)}+\frac{(x+c)^{2}}{(c-a)(c-b)}
$$
where $a, b$, and $c$ are distinct real numbers. Determine the range of the function. | 1 | 81 | 1 |
math | 231*. $x^{2}+x y+y^{2}+x+y-5=0$, if one solution is known: $x=1 ; y=1$. | (1,1),(1,-3),(-3,1) | 39 | 14 |
math | An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$? | 130 | 74 | 3 |
math | 7. There are a total of 10040 numbers written on the blackboard, including 2006 ones, 2007 twos, 2008 threes, 2009 fours, and 2010 fives. Each operation involves erasing 4 different numbers and writing the fifth type of number (for example, erasing 1, 2, 3, and 4 each once and writing one 5; or erasing 2, 3, 4, and 5 e... | 8 | 155 | 1 |
math | Problem 2. Let $a, b, c, d$ be natural numbers such that $a+b+c+d=2018$. Find the minimum value of the expression
$$
E=(a-b)^{2}+2(a-c)^{2}+3(a-d)^{2}+4(b-c)^{2}+5(b-d)^{2}+6(c-d)^{2}
$$ | 14 | 89 | 2 |
math | 1. $\left(12\right.$ points) Solve the equation $\left(x^{3}-3\right)\left(2^{\operatorname{ctg} x}-1\right)+\left(5^{x^{3}}-125\right) \operatorname{ctg} x=0$. | \sqrt[3]{3};\frac{\pi}{2}+\pin,\quadn\inZ | 69 | 22 |
math | 1. Calculate: $2015 \times 2015-2014 \times 2013=(\quad)$. | 6043 | 33 | 4 |
math | An angle $\theta$ with $0^{\circ} \leq \theta \leq 180^{\circ}$ satisfies $\sqrt{2} \cos 2 \theta=\cos \theta+\sin \theta$. Determine all possible values of $\theta$. | 15,135 | 57 | 6 |
math | . In the decimal writing of $A$, the digits appear in (strictly) increasing order from left to right. What is the sum of the digits of $9 A$? | 9 | 37 | 1 |
math | It is given that the roots of the polynomial $P(z) = z^{2019} - 1$ can be written in the form $z_k = x_k + iy_k$ for $1\leq k\leq 2019$. Let $Q$ denote the monic polynomial with roots equal to $2x_k + iy_k$ for $1\leq k\leq 2019$. Compute $Q(-2)$. | \frac{-1 - 3^{2019}}{2^{2018}} | 103 | 21 |
math | For example, $7 x, y, z$ are all positive integers, the equation $x+y+z=15$ has how many sets of solutions? | 91 | 33 | 2 |
math | # Task 1. (10 points)
How many natural numbers $n$ exist such that the equation $n x-12=3 n$ has an integer solution
# | 6 | 38 | 1 |
math | 2. Determine the minimum value of the expression
$$
V=x^{2}+\frac{2}{1+2 x^{2}}
$$
where $x$ is any real number. For which $x$ does the expression $V$ attain this value? | \frac{3}{2} | 55 | 7 |
math | $3 \cdot 25$ For each positive integer $n$, let
$$
\begin{array}{l}
S_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}, \\
T_{n}=S_{1}+S_{2}+S_{3}+\cdots+S_{n}, \\
U_{n}=\frac{1}{2} T_{1}+\frac{1}{3} T_{2}+\frac{1}{4} T_{3}+\cdots+\frac{1}{n+1} T_{n} .
\end{array}
$$
Try to find integers $0<a, b, c, d<1000000$, such t... | =1989,b=1989,=1990,=3978 | 224 | 23 |
math | $1 \cdot 2$ If an ordered pair of non-negative integers $(m, n)$ does not require a carry when adding $m+n$ (in decimal), then it is called "simple". Find the number of all simple ordered pairs of non-negative integers whose sum is 1492. | 300 | 63 | 3 |
math | Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x... | \sqrt{2} | 113 | 5 |
math | 47. a) $(a+1)(a-1)$; b) $(2 a+3)(2 a-3)$; c) $\left(m^{3}-n^{5}\right)\left(n^{5}+m^{3}\right)$; d) $\left(3 m^{2}-5 n^{2}\right)\left(3 m^{2}+5 n^{2}\right)$. | )^{2}-1;b)4^{2}-9;)^{6}-n^{10};)9^{4}-25n^{4} | 89 | 33 |
math | Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$. | 63 | 52 | 2 |
math | Nonnegative reals $x_1$, $x_2$, $\dots$, $x_n$ satisfies $x_1+x_2+\dots+x_n=n$. Let $||x||$ be the distance from $x$ to the nearest integer of $x$ (e.g. $||3.8||=0.2$, $||4.3||=0.3$). Let $y_i = x_i ||x_i||$. Find the maximum value of $\sum_{i=1}^n y_i^2$. | \frac{n^2 - n + 0.5}{4} | 114 | 15 |
math | Example 4 (53rd Romanian Mathematical Olympiad (Final)) Find all real numbers $a, b, c, d, e \in[-2,2]$, such that $a+b+c+d+e=0, a^{3}+b^{3}+c^{3}+d^{3}+e^{3}=0, a^{5}+b^{5}+c^{5}+d^{5}+e^{5}=10$.
| ,b,,,e\in{2,\frac{\sqrt{5}-1}{2},-\frac{\sqrt{5}+1}{2}} | 102 | 30 |
math | ## Problem 6.
It is known that the polynomial $\mathrm{p}(\mathrm{x})=\mathrm{x}^{3}-\mathrm{x}+\mathrm{k}$ has three roots that are integers.
Determine the number k. | 0 | 48 | 1 |
math | $1 \cdot 40$ form "words" by taking $n$ numbers from the alphabet $\{0,1,2,3,4\}$, such that the difference between every two adjacent numbers is 1. How many words can be formed in total? | m_{n}={\begin{pmatrix}14\times3^{k-1},n=2k+1,\\8\times3^{k-1},n=2k0\end{pmatrix}.} | 57 | 50 |
math | Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$. | 509 | 36 | 3 |
math | 5 Solve the exponential equation
$$
2^{x} \cdot 3^{5^{-x}}+\frac{3^{5^{x}}}{2^{x}}=6
$$ | 0 | 39 | 1 |
math | [ $[$ straight lines and planes in space]
Points $M$ and $N$ lie on the edges $B C$ and $A A 1$ of the parallelepiped $A B C D A 1 B 1 C 1 D 1$. Construct the point of intersection of the line $M N$ with the plane of the base $A 1 B 1 C 1 D 1$.
# | P | 90 | 1 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}\operatorname{arctg}\left(x^{3}-x^{\frac{3}{2}} \sin \frac{1}{3 x}\right), x \neq 0 ; \\ 0, x=0\end{array}\right.$ | 0 | 89 | 1 |
math | 5. Let $f(x)$ be a function defined on $\mathbf{R}$ with a period of 2, which is even, strictly decreasing on the interval $[0,1]$, and satisfies $f(\pi)=1, f(2 \pi)=2$. Then the solution set of the inequality system $\left\{\begin{array}{l}1 \leq x \leq 2, \\ 1 \leq f(x) \leq 2\end{array}\right.$ is $\qquad$. | [\pi-2,8-2\pi] | 113 | 11 |
math | 4.- Find the integer solutions of the equation:
where p is a prime number.
$$
p \cdot(x+y)=x \cdot y
$$ | (0,0);(2p,2p);(p(p+1),p+1);(p(1-p),p-1);(p+1,p(p+1));(p-1,p(1-p)) | 31 | 50 |
math | # Problem 3. (3 points)
$4^{27000}-82$ is divisible by $3^n$. What is the greatest natural value that $n$ can take? | 5 | 41 | 1 |
math | $A$ says to $B$: "Take any number. Write its digits in reverse order and subtract the smaller number from the larger one. Multiply this difference by any number. In the resulting product, cross out any non-zero digit of your choice, and tell me the truncated number." $B$'s response: 35407, to which $A$ says that the cr... | 8 | 95 | 1 |
math | 1. There are 4 kg of a copper-tin alloy, in which $40\%$ is copper, and 6 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys need to be taken so that after melting, 8 kg of an alloy containing $p\%$ copper is obtained? Find all $p$ for which the problem has a solution. | 32.5\leqslantp\leqslant35 | 90 | 17 |
math | [ Ordinary fractions ]
How many representations does the fraction $\frac{2 n+1}{n(n+1)}$ admit as a sum of two positive fractions with denominators $n$ and $n+1$? | 1 | 45 | 1 |
math | Problem 4. Blagoya participated in a lottery where each ticket is marked with a three-digit number. He bought all the tickets marked with numbers where the product of the tens and units digits is equal to 12, and the sum of these two digits differs from the hundreds digit by 1. How many tickets and with which numbers d... | 8 | 75 | 1 |
math | Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$. | 2\sqrt{3} | 58 | 6 |
math | 1.042. $\frac{1.2: 0.375-0.2}{6 \frac{4}{25}: 15 \frac{2}{5}+0.8}=\frac{0.016: 0.12+0.7}{X}$. | \frac{1}{3} | 70 | 7 |
math | Example 7 Let $n$ be a positive integer. How many positive integer solutions does the equation $\frac{x y}{x+y}=n$ have?
(21st Putnam Mathematical Competition) | (2\alpha_{1}+1)(2\alpha_{2}+1)\cdots(2\alpha_{k}+1) | 41 | 31 |
math | Example 7 (1989 National High School League Question) If numbers are taken from $1,2, \cdots, 14$ in ascending order as $a_{1}, a_{2}, a_{3}$, such that $a_{2}-a_{1} \geqslant 3, a_{3}-a_{2} \geqslant 3$, then the number of different ways to satisfy the above conditions is $\qquad$ kinds. | C_{10}^{3} | 103 | 8 |
math | Question 223, Find the largest positive integer $n$, such that there exists a set with $n$ elements, where the set contains exactly 1 element divisible by $n$, exactly 2 elements divisible by $\mathrm{n}-1$, $\cdots$, exactly $\mathrm{n}-1$ elements divisible by 2, and $\mathrm{n}$ elements divisible by 1. | 5 | 79 | 1 |
math | What is the sum of the roots of the equation $10^{x}+10^{1-x}=10$? | 1 | 27 | 1 |
math | 4. Buratino buried two ingots on the Field of Wonders: a gold one and a silver one. On the days when the weather was good, the gold ingot increased by $30 \%$, and the silver one by $20 \%$. On the days when the weather was bad, the gold ingot decreased by $30 \%$, and the silver one by $20 \%$. After a week, it turned... | 4 | 114 | 1 |
math | $3-$}
It's very boring to look at a black-and-white clock face, so Clive painted the number 12 red at exactly noon and decided to paint the current hour red every 57 hours.
a) How many numbers on the clock face will end up painted?
b) How many red numbers will there be if Clive paints them every 2005 hours? | )4;b)12 | 81 | 6 |
math | 7.4. Several knights and liars are standing in a circle, all of different heights. Each of them said the phrase: "I am taller than exactly one of my neighbors." Could there have been exactly 2023 liars among those standing in the circle? (Knights always tell the truth, while liars always lie). | No | 71 | 1 |
math | ## Task B-2.3.
One year, January 1 and April 1 were both on a Thursday. How many months in that year have five Fridays? Justify your answer. | 5 | 39 | 1 |
math | Solve the equation $x^{4}-6 x^{3}+10 x^{2}-4 x=0$.
List 8 | 2+\sqrt{2}, 2-\sqrt{2}, 0, 2 | 30 | 18 |
math | $19422 *$ Find the largest integer $x$ that makes $4^{27}+4^{1000}+4^{x}$ a perfect square. | 1972 | 40 | 4 |
math | 3. Let $a^{2}+b^{2}=1, c^{2}+d^{2}=1$ and $a c+b d=0$. Determine the value of the expression $a b+c d$? | 0 | 48 | 1 |
math | Example 1 (to $1^{\circ}$). Find the partial and total increments of the function $z=x^{2} y$ at the initial values $x=1, y=2$, if $\Delta x=0.1$; $\Delta y=-0.2$. | 0.178 | 60 | 5 |
math | 2. Let $\mathbb{R}_{+}$ denote the set of all positive numbers. Find all functions $f: \mathbb{R}_{+} \rightarrow \mathbb{R}$ such that $x f(y)-y f(x)=$ $f(y / x)$ for all $x, y \in \mathbb{R}_{+}$. | f(x)=(x-\frac{1}{x}) | 76 | 11 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{3}+5}-\sqrt{3 n^{4}+2}}{1+3+5+\ldots+(2 n-1)}
$$ | -\sqrt{3} | 66 | 5 |
math | 12 For any positive numbers $a_{1}, a_{2}, \cdots, a_{n}, n \geqslant 2$, find the minimum value of $\sum_{i=1}^{n} \frac{a_{i}}{S-a_{i}}$, where $S=$ $\sum_{i=1}^{n} a_{i}$. | \frac{n}{n-1} | 80 | 8 |
math | 36. A pair of natural numbers $a>b$ is called good if the least common multiple (LCM) of these numbers is divisible by their difference. Among all natural divisors of the number $n$, exactly one good pair was found. What can $n$ be? | 2 | 58 | 1 |
math | Ana, Bia, Cátia, Diana, and Elaine work as street vendors selling sandwiches. Every day, they stop by Mr. Manoel's snack bar and take the same number of sandwiches to sell. One day, Mr. Manoel was sick and left a note explaining why he wasn't there, but asking each of them to take $\frac{1}{5}$ of the sandwiches. Ana a... | 75 | 225 | 2 |
math | 25th BMO 1989 Problem 3 f(x) is a polynomial of degree n such that f(0) = 0, f(1) = 1/2, f(2) = 2/3, f(3) = 3/4, ... , f(n) = n/(n+1). Find f(n+1). | f(n+1)=\begin{cases}1&ifnisodd\\\frac | 80 | 17 |
math | 5. In $\triangle A B C$, $A B=1, A C=2, B-C=\frac{2 \pi}{3}$, then the area of $\triangle A B C$ is $\qquad$ | \frac{3\sqrt{3}}{14} | 47 | 13 |
math | | $\left[\begin{array}{l}\text { Cauchy's Inequality } \\ \text { [ Completing the square. Sums of squares }\end{array}\right]$ |
| :--- |
$a+b=1$. What is the maximum value of the quantity $a b$? | \frac{1}{4} | 63 | 7 |
math | ## 40. Tennis Tournament
199 people have registered to participate in a tennis tournament. In the first round, pairs of opponents are selected by lottery. The same process is used to select pairs in the second, third, and all subsequent rounds. After each match, one of the two opponents is eliminated, and whenever the... | 198 | 117 | 3 |
math | 1. Emily's broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily's clock also d... | 12 | 85 | 2 |
math | Consider 2018 lines in the plane, no two of which are parallel and no three of which are concurrent. Let $E$ be the set of their intersection points. We want to assign a color to each point in $E$ such that any two points on the same line, whose segment connecting them contains no other point of $E$, are of different c... | 3 | 91 | 1 |
math | We painted the lateral surface of a truncated cone up to half of its slant height blue, and the rest red. The blue surface area is twice as large as the red one. How many times larger is the radius of the base circle compared to the radius of the top circle? | 5r | 57 | 2 |
math | XLVII OM - I - Problem 1
Determine all integers $ n $ for which the equation $ 2 \sin nx = \tan x + \cot x $ has solutions in real numbers $ x $. | 8k+2 | 45 | 4 |
math | 2. If you take three different digits, form all six possible two-digit numbers using two different digits, and add these numbers, the result is 462. Find these digits. Provide all variants and prove that there are no others. | (6,7,8),(4,8,9),(5,7,9) | 49 | 19 |
math | Analyzing the 4-digit natural numbers:
a) How many of them have all different digits?
b) How many have the digit 1 exactly once and all different digits?
c) How many have the digit 1? | 3168 | 46 | 4 |
math | Let there be positive integers $a, c$. Positive integer $b$ is a divisor of $ac-1$. For a positive rational number $r$ which is less than $1$, define the set $A(r)$ as follows.
$$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$
Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ grea... | \frac{a}{a + b} | 112 | 10 |
math | A finite set of integers is called bad if its elements add up to 2010. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest integer $n$ such that the set $\{502,503,504, \ldots, 2009\}$ can be partitioned into $n$ Benelux-sets.
(A partition of a set $S$ into $n$ subsets is a ... | 2 | 125 | 1 |
math | One, (40 points) Find the smallest real number $\lambda$, such that there exists a sequence $\left\{a_{n}\right\}$ with all terms greater than 1, for which for any positive integer $n$ we have $\prod_{i=1}^{n+1} a_{i}<a_{n}^{\lambda}$. | 4 | 76 | 1 |
math | 8. $\frac{\tan 96^{\circ}-\tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}{1+\tan 96^{\circ} \tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}=$ $\qquad$ | \frac{\sqrt{3}}{3} | 85 | 10 |
math | 48 blacksmiths need to shoe 60 horses. Each blacksmith spends 5 minutes on one horseshoe. What is the least amount of time they should spend on the work? (Note, a horse cannot stand on two legs.)
# | 25 | 52 | 2 |
math | Problem 6.2. Petya and Vasya decided to get as many fives as possible on September 1 and 2.
- On September 1, they got a total of 10 fives, and Petya got more fives than Vasya;
- On September 2, Vasya got 3 fives, while Petya did not get any;
- By the end of these two days, Vasya had more fives than Petya.
How many f... | Petya\got\6\fives,\\Vasya\got\7 | 119 | 18 |
math |
A 2. Find the maximum positive integer $k$ such that for any positive integers $m, n$ such that $m^{3}+n^{3}>$ $(m+n)^{2}$, we have
$$
m^{3}+n^{3} \geq(m+n)^{2}+k
$$
| 10 | 73 | 2 |
math | 19. If $a \circ b=\frac{\sqrt{a^{2}+3 a b+b^{2}-2 a-2 b+4}}{a b+4}$, find $((\cdots((2010 \circ 2009) \circ 2008) \circ \cdots \circ 2) \circ 1)$. (2 marks)若 $a \circ b=\frac{\sqrt{a^{2}+3 a b+b^{2}-2 a-2 b+4}}{a b+4}$, 求 $((\cdots((2010 \circ 2009) \circ 2008) \circ \cdots \circ 2) \circ 1)$ 。（2 分） | \frac{\sqrt{15}}{9} | 175 | 11 |
math | 15. Let $x, y$ be real numbers, then $\max _{5 x^{2}+4 y^{2}=10 x}\left(x^{2}+y^{2}\right)=$ | 4 | 46 | 1 |
math | Example 20. Solve the equation
$$
3^{x-1}+5^{x-1}=34
$$ | x_1=3 | 28 | 5 |
math | ## Condition of the problem
Find the differential $d y$.
$$
y=\frac{\ln |x|}{1+x^{2}}-\frac{1}{2} \ln \frac{x^{2}}{1+x^{2}}
$$ | -\frac{2x\cdot\ln|x|}{(1+x^{2})^{2}}\cdot | 52 | 23 |
math | 8. For a regular triangular prism $A B C-A_{1} B_{1} C_{1}$ with all edges of length 3, a line segment $M N$ of length 2 has one endpoint $M$ moving on $A A_{1}$ and the other endpoint $N$ moving on the base $A B C$. Then, the trajectory (surface) of the midpoint $P$ of $M N$ and the three faces of the regular triangul... | \frac{\pi}{9} | 117 | 7 |
math | Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression. | -\frac{82}{9} | 34 | 8 |
math | Levin $M$.
Find all natural numbers divisible by 30 and having exactly 30 distinct divisors. | 720,1200,1620,4050,7500,11250 | 25 | 29 |
math | 5 (1246). By what percentage will the area of a rectangle increase if its length is increased by $20 \%$ and its width by $10 \%$? | 32 | 38 | 2 |
math | For which pairs of positive integers $a, b$ is the expression $a^{2}+b^{2}-a-b+1$ divisible by $a b$? | =b=1 | 36 | 3 |
math | In the pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten no less than three pikes (full or hungry). What is the maximum number of pikes that can become full? | 9 | 52 | 1 |
math | ## Task 26/83
How many (genuine) four-digit natural numbers are there that are divisible by 11 and whose cross sum is also divisible by 11? | 72 | 40 | 2 |
math | Problem 4. Students from a school were supposed to go on a trip. $\frac{2}{9}$ more students registered than the planned number. Before departure, $\frac{3}{11}$ of the registered students canceled due to illness, so 5 fewer students went on the trip than the planned number. How many students went on the trip? | 40 | 73 | 2 |
math | 16. A seven-digit number $\overline{m 0 A 0 B 9 C}$ is a multiple of 33. We denote the number of such seven-digit numbers as $a_{m}$. For example, $a_{5}$ represents the number of seven-digit numbers of the form $\overline{50 A 0 B 9 C}$ that are multiples of $\mathbf{3 3}$. Then $a_{2}-a_{3}=$ | 8 | 102 | 1 |
math | Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards $1, 3, 5,7,..., 2n-1$
b) Bruno has cards $2, 4,6, 8,...,2n$
During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gain... | \left\lfloor \frac{n}{2} \right\rfloor | 193 | 17 |
math | 4. A4 (USA) Let \( a, b \), and \( c \) be given positive real numbers. Determine all positive real numbers \( x, y \), and \( z \) such that
\[ x+y+z=a+b+c \]
and
\[ 4xyz - \left(a^2 x + b^2 y + c^2 z\right) = abc \] | (x, y, z)=\left(\frac{b+c}{2}, \frac{c+a}{2}, \frac{a+b}{2}\right) | 83 | 34 |
math | 4. (USA) Let $a, b, c$ be given positive constants.
Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a+b+c, \\
4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c
\end{array}\right.
$$
for all positive real numbers $x, y, z$. | (x, y, z)=\left(\frac{1}{2}(b+c), \frac{1}{2}(c+a), \frac{1}{2}(a+b)\right) | 94 | 40 |
math | Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\frac mn$, where $m^{}_{}$ and $n^{}_{}$ are relatively prime positive... | 164 | 138 | 3 |
math | 12. If $16^{\sin ^{2} x}+16^{\cos ^{2} x}=10$, then $\cos 4 x=$
$\qquad$ . | -\frac{1}{2} | 44 | 7 |
math | \section*{Problem 1 - 211041}
Determine all pairs \((a ; b)\) of positive integers \(a, b\) that have the property that exactly three of the following four statements (1), (2), (3), (4) are true and one is false! The statements are:
\[
\begin{array}{llll}
b \mid(a+1), & (1) & ; & a=2 b+5, \quad(2) \\
3 \mid(a+b), & (... | (,b)=(9,2)(,b)=(17,6) | 142 | 16 |
math | 4. Vitya and Vova collected 27 kg of waste paper together. If the number of kilograms of waste paper collected by Vitya were increased by 5 times, and that collected by Vova by 3 times, they would have 111 kg together. How many kilograms did each boy collect? | 15 | 67 | 2 |
math | 11. (20 points) Find the smallest integer \( n (n > 1) \), such that there exist \( n \) integers \( a_{1}, a_{2}, \cdots, a_{n} \) (allowing repetition) satisfying
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2013 .
$$
12. (20 points) Let positive integers \( a, b, c, d \) satisfy
$$
a^{2}=c(d+13), b^{2}=c(d-1... | 5 | 157 | 1 |
math | Example 2 Let $x, y, z, w$ be real numbers, not all zero. Find the maximum value of $P=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$. | \frac{\sqrt{2}+1}{2} | 59 | 12 |
math | 6. Dima went to school in the morning, but after walking exactly half the distance, he realized he had forgotten his mobile phone at home. Dima estimated (he had an A in mental arithmetic) that if he continued walking at the same speed, he would arrive at school 3 minutes before the first bell, but if he ran home for t... | 2 | 160 | 1 |
math | 5. Among the $n$ positive integers from 1 to $n$, those with the most positive divisors are called the "prosperous numbers" among these $n$ positive integers. For example, among the positive integers from 1 to 20, the numbers with the most positive divisors are $12, 18, 20$, so $12, 18, 20$ are all prosperous numbers a... | 10080 | 134 | 5 |
math | 45th Putnam 1984 Problem B6 Define a sequence of convex polygons P n as follows. P 0 is an equilateral triangle side 1. P n+1 is obtained from P n by cutting off the corners one-third of the way along each side (for example P 1 is a regular hexagon side 1/3). Find lim n→∞ area(P n ). Solution | \frac{\sqrt{3}}{7} | 85 | 10 |
math | 3. Given that $a, b$ are real numbers, satisfying
$$
t=\frac{a^{2}}{a^{2}+2 b^{2}}+\frac{b^{2}}{2 a^{2}+b^{2}} \text {. }
$$
Then the minimum value of $t$ is | \frac{2}{3} | 69 | 7 |
math | 2. Given real numbers $a, b$ satisfy $a^{3}+b^{3}+3 a b=1$. Then $a+b$ $=$ . $\qquad$ | 1 \text{ or } -2 | 40 | 8 |
math | The five-digit positive integer $15 A B 9$ is a perfect square for some digits $A$ and $B$. What is the value of $A+B$ ? | 3 | 37 | 1 |
math | Problem 11.4. Find all values of the real parameter $a$ such that the equation
$$
\log _{a x}\left(3^{x}+4^{x}\right)=\log _{(a x)^{2}}\left(7^{2}\left(4^{x}-3^{x}\right)\right)+\log _{(a x)^{3}} 8^{x-1}
$$
has a solution.
Emil Kolev | \in(0,+\infty)\backslash{1} | 103 | 14 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.