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 | II. Find all integer values of $a$ for which the equation $(a+1) x^{2}-\left(a^{2}+1\right) x$ $+2 a^{3}-6=0$ has integer roots. | -1,0,1 | 51 | 6 |
math | Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\
$\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\.
\\
If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$. | 25 | 111 | 2 |
math | Gashkov S.B.
Find all such $a$ and $b$ that $|a|+|b| \geqslant \frac{2}{\sqrt{3}}$ and for all $x$ the inequality $|a \sin x + b \sin 2x| \leq 1$ holds. | =\\frac{4}{3\sqrt{3}},b=\\frac{2}{3\sqrt{3}} | 71 | 25 |
math | Example 2 Find the direct proportion function $f(x)$ that satisfies the inequality $f[f(x)] \geqslant x-3, x \in \mathbf{R}$. | f(x)=-xorf(x)=x | 40 | 9 |
math | 3. Given $a, b, c \in \mathbf{R}$, and $a+b+c=3$. Then the minimum value of $3^{a} a+3^{b} b+3^{c} c$ is $\qquad$ | 9 | 55 | 1 |
math | 6. Given real numbers $a, b$ satisfy $a^{2} \geqslant 8 b$. Then
$$
(1-a)^{2}+(1-2 b)^{2}+(a-2 b)^{2}
$$
the minimum value is $\qquad$ | \frac{9}{8} | 64 | 7 |
math | 3.262. $\left(\operatorname{tg} 255^{\circ}-\operatorname{tg} 555^{\circ}\right)\left(\operatorname{tg} 795^{\circ}+\operatorname{tg} 195^{\circ}\right)$.
3.262. $\left(\tan 255^{\circ}-\tan 555^{\circ}\right)\left(\tan 795^{\circ}+\tan 195^{\circ}\right)$. | 8\sqrt{3} | 124 | 6 |
math | 8. (2003 Nordic Mathematical Contest) Find all triples of integers $(x, y, z)$ such that $x^{3}+y^{3}+z^{3}-3 x y z=$ 2003. | (668,668,667),(668,667,668),(667,668,668) | 51 | 37 |
math | 1. Find all solutions of the equation
$$
\frac{x-2}{y}+\frac{5}{x y}=\frac{4-y}{x}-\frac{|y-2 x|}{x y}
$$ | 1,2 | 48 | 3 |
math | Example 2-25 $n$ indistinguishable balls are placed into $m$ indistinguishable boxes, with no box left empty. How many ways are there to do this? | C(n-1,-1) | 39 | 7 |
math | 7. Let $a$ be a non-zero real number. In the Cartesian coordinate system $x O y$, the focal distance of the quadratic curve $x^{2}+a y^{2}+a^{2}=0$ is 4, then the value of $a$ is | \frac{1-\sqrt{17}}{2} | 60 | 13 |
math | \section*{Problem 18}
\(p(x)\) is the cubic \(x^{3}-3 x^{2}+5 x\). If \(h\) is a real root of \(p(x)=1\) and \(k\) is a real root of \(p(x)=5\), find \(h+k\)
| 2 | 69 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{x^{4}-1}{2 x^{4}-x^{2}-1}$ | \frac{2}{3} | 40 | 7 |
math | [Systems of Linear Equations]
Solve the system of equations with $n$ unknowns $\frac{x_{1}}{a_{1}}=\frac{x_{2}}{a_{2}}=\ldots=\frac{x_{n}}{a_{n}}, x_{1}+x_{2}+\ldots+x_{n}=A$. | x_{1}=\frac{Aa_{1}}{a_{1}+\ldots+a_{n}},\ldots,x_{n}=\frac{Aa_{n}}{a_{1}+\ldots+a_{n}};whena_{1}+\ldots+a_{n}\neq0,\quad(x_{1},\ldots,x_{n})=(a_{1}, | 73 | 85 |
math | # 2. Task 2*
The number 2017 has 7 ones and 4 zeros in its binary representation. When will the next year come, in which the number of the year in binary representation will have no more ones than zeros? (Enter the year.) Points for the task: 8.
# | 2048 | 67 | 4 |
math | $11 \cdot 14 n$ is a non-negative integer. Given $f(0)=0, f(1)=1, f(n)=$ $f\left(\left[\frac{n}{2}\right]\right)+n-2\left[\frac{n}{2}\right]$. Determine $f(n)$. Find the maximum value of $f(n)$ for $0 \leqslant n \leqslant 1991$. (Here $[x]$ denotes the greatest integer not exceeding $x$)
(Japan Mathematical Olympiad, ... | 10 | 127 | 2 |
math | Exercise 4. Determine all natural numbers $n$ such that 21 divides $2^{2^{n}}+2^{n}+1$. | n\equiv2(\bmod6) | 32 | 9 |
math | 1. (5 points) Calculate: $0.15 \div 2.1 \times 56=$ | 4 | 25 | 1 |
math | Find all the pairs of positive integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+3n$, $B=2n^2+3mn+m^2$, $C=3n^2+mn+2m^2$ are consecutive in some order. | (m, n) = (k, k+1) | 67 | 14 |
math | Calculate the remainder of the Euclidean division of $2022^{2023^{2024}}$ by 19. | 8 | 31 | 1 |
math | 1. Find all triples of real numbers $(x, y, z)$ that satisfy
$$
\left\{\begin{array}{l}
x^{2}-y z=|y-z|+1, \\
y^{2}-z x=|z-x|+1, \\
z^{2}-x y=|x-y|+1 .
\end{array}\right.
$$ | (\frac{4}{3},\frac{4}{3},-\frac{5}{3}),(\frac{4}{3},-\frac{5}{3},\frac{4}{3}),(-\frac{5}{3},\frac{4}{3},\frac{4}{3}),(-\frac{4}{3},-\frac{4}{3},\frac{5}{} | 82 | 85 |
math | 4. Given the sequence of numbers $0,1,2,6,16,44,120, \ldots$ Extend it and write down the next two numbers. | 328 | 40 | 3 |
math | Task B-2.5. What is the last digit of the number $2012^{3}+3^{2012}$? | 9 | 32 | 1 |
math | 3. (Easy/Average) Given that $\tan x+\cot x=8$, find the value of $\sqrt{\sec ^{2} x+\csc ^{2} x-\frac{1}{2} \sec x \csc x}$. | 2\sqrt{15} | 55 | 7 |
math | 3.137. $\frac{\cos 7 \alpha-\cos 8 \alpha-\cos 9 \alpha+\cos 10 \alpha}{\sin 7 \alpha-\sin 8 \alpha-\sin 9 \alpha+\sin 10 \alpha}$. | \operatorname{ctg}\frac{17\alpha}{2} | 61 | 16 |
math | Four. (20 points) Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms as $S_{n}$, and it satisfies $a_{2}=2, S_{n}=\frac{n\left(1+a_{n}\right)}{2}\left(n \in \mathbf{N}_{+}\right)$.
(1) Find the general term of the sequence $\left\{a_{n}\right\}$.
(2) If $b_{n}=a_{n}^{\frac{i}{a_{n}+1}}$, find... | b_{2}=b_{8} | 207 | 8 |
math | 14. In the sequence $\left\{a_{n}\right\}$, $a_{1}=a_{2}=0$, and $a_{n+2}=6 a_{n+1}-9 a_{n}+8\left(n \in \mathbf{N}^{*}\right)$, find $a_{n}$ and $S_{n}$. | a_{n}=(\frac{4}{3}n-\frac{10}{3})\cdot3^{n-1}+2,\quadS_{n}=(2n-6)\cdot3^{n-1}+2(n+1) | 81 | 56 |
math | 21. Find the number of positive integers $n$ such that
$$
n+2 n^{2}+3 n^{3}+\cdots+2005 n^{2005}
$$
is divisible by $n-1$. | 16 | 55 | 2 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$$
\begin{aligned}
& M_{1}(-3 ;-1 ; 1) \\
& M_{2}(-9 ; 1 ;-2) \\
& M_{3}(3 ;-5 ; 4) \\
& M_{0}(-7 ; 0 ;-1)
\end{aligned}
$$ | 0 | 102 | 1 |
math | Determine $x^2+y^2+z^2+w^2$ if
$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{... | 36 | 254 | 2 |
math | 4. The roots of the equation $x^{2}-x+1=0$ are $\alpha, \beta$. There is a quadratic function $f(x)$ that satisfies $f(\alpha)=\beta, f(\beta)=\alpha, f(1)=$ 1, then the expression of $f(x)$ is $\qquad$ . | f(x)=x^{2}-2x+2 | 73 | 11 |
math | Find all functions from $\mathbb{R}-\{0,1\}$ to $\mathbb{R}$ such that
$$
f(x)+f\left(\frac{1}{1-x}\right)=x
$$ | f(x)=\frac{1}{2}(x+1-\frac{1}{x}-\frac{1}{1-x}) | 48 | 28 |
math | 4. Yumi has a flat circular chocolate chip cookie with radius $3 \mathrm{~cm}$. On the top of the cookie, there are $k$ circular chocolate chips, each with radius $0.3 \mathrm{~cm}$. No two chocolate chips overlap and no chocolate chip hangs over the edge of the cookie. For what value of $k$ is exactly $\frac{1}{4}$ of... | 25 | 99 | 2 |
math | 16. Let the product of all distinct positive divisors of 2005 be $a$, and the product of all distinct positive divisors of $a$ be $b$. Then $b=$ $\qquad$ | 2005^9 | 47 | 6 |
math | 19th USAMO 1990 Problem 1 A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is possible? Solution | 10^5 | 56 | 4 |
math | Four. (20 points) The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=3, a_{n}=$ $3^{a_{n-1}}(n \geqslant 2)$. Find the last digit of $a_{n}(n \geqslant 2)$. | 7 | 77 | 1 |
math | 35. By writing down 6 different numbers, none of which is 1, in ascending order and multiplying them, Olya got the result 135135. Write down the numbers that Olya multiplied. | 3\cdot5\cdot7\cdot9\cdot11\cdot13=135135 | 47 | 25 |
math | 5. Solve the inequality $\log _{1+x^{2}}\left(1+8 x^{5}\right)+\log _{1-3 x^{2}+16 x^{4}}\left(1+x^{2}\right) \leqslant 1+\log _{1-3 x^{2}+16 x^{4}}\left(1+8 x^{5}\right)$. Answer: $x \in\left(-\frac{1}{\sqrt[3]{8}} ;-\frac{1}{2}\right] \cup\left(-\frac{\sqrt{3}}{4} ; 0\right) \cup\left(0 ; \frac{\sqrt{3}}{4}\right) \c... | x\in(-\frac{1}{\sqrt[3]{8}};-\frac{1}{2}]\cup(-\frac{\sqrt{3}}{4};0)\cup(0;\frac{\sqrt{3}}{4})\cup{\frac{1}{2}} | 179 | 61 |
math | An isosceles triangle has angles of $50^\circ,x^\circ,$ and $y^\circ$. Find the maximum possible value of $x-y$.
[i]Proposed by Nathan Ramesh | 30^\circ | 44 | 4 |
math | $p, q, r$ are distinct prime numbers which satisfy
$$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$
for natural number $A$. Find all values of $A$. | 1980 | 60 | 4 |
math | The height of Cylinder A is equal to its diameter. The height and diameter of Cylinder B are each twice those of Cylinder A. The height of Cylinder $\mathrm{C}$ is equal to its diameter. The volume of Cylinder C is the sum of the volumes of Cylinders A and B. What is the ratio of the diameter of Cylinder $\mathrm{C}$ t... | \sqrt[3]{9}:1 | 81 | 8 |
math | Problem 5. Milan and Aleksandar had a large box of chocolate candies, which they were to divide in the following way: First, Milan took 1 candy, and Aleksandar took two candies, then Milan took three, and Aleksandar took four candies, and so on, each taking one more candy alternately.
When the number of candies left i... | 211 | 126 | 3 |
math | ## Task 1 - 160521
$$
\begin{array}{ccccc}
\mathrm{A} & \cdot & \mathrm{A} & = & \mathrm{B} \\
+ & & \cdot & & - \\
\mathrm{C} & \cdot & \mathrm{D} & = & \mathrm{E} \\
\hline \mathrm{F} & -\mathrm{G} & = & \mathrm{H}
\end{array}
$$
In the cryptogram above, digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are to be entered for t... | \begin{pmatrix}3&\cdot&3&=9\\+&\cdot&&-\\4&\cdot&2&=8\\\hline7&-&6&=1\\\hline\end{pmatrix} | 196 | 51 |
math | 7.1. In a hotel, rooms are arranged in a row in order from 1 to 10000. Masha and Alina checked into the hotel in two different rooms. The sum of the room numbers they are staying in is 2022, and the sum of the room numbers of all the rooms between them is 3033. In which room is Masha staying, if her room has a lower nu... | 1009 | 99 | 4 |
math | Task 3. A team of workers was working on pouring the rink on the large and small fields, with the area of the large field being twice the area of the small field. In the part of the team that worked on the large field, there were 4 more workers than in the part that worked on the small field. When the pouring of the la... | 10 | 108 | 2 |
math | 22. The base of a right prism is a right triangle. All edges of this prism have natural lengths, and the areas of some two of its faces are 13 and 30. Find the sides of the base of this prism. | 5,12,13 | 51 | 7 |
math | 4. find all pairs $(a, b)$ of integers with different divisors such that
$$
a^{2}+a=b^{3}+b
$$ | (,b)=(1,1),(-2,1),(-1,0),(5,3) | 35 | 22 |
math | 5. In the country of Lemonia, coins of denominations $3^{n}, 3^{n-1} \cdot 4, 3^{n-2} \cdot 4^{2}, 3^{n-3} \cdot 4^{3}, \ldots, 3 \cdot 4^{n-1}, 4^{n}$ piastres are in circulation, where $n-$ is a natural number. A resident of the country went to the bank without any cash on hand. What is the largest amount that the ba... | 2\cdot4^{n+1}-3^{n+2} | 124 | 15 |
math | 9. In $\triangle A B C$, $\angle A<\angle B<\angle C$, $\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=\sqrt{3}$.
Then $\angle B=$ | \frac{\pi}{3} | 54 | 7 |
math | Two (not necessarily different) numbers are chosen independently and at random from $\{1, 2, 3, \dots, 10\}$. On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?) | 30.25 | 74 | 5 |
math | 9. (10 points) There are 11 square formations, each consisting of the same number of soldiers. If 1 general is added, a large square formation can be formed. The minimum number of soldiers in one of the original square formations is $\qquad$.
| 9 | 57 | 1 |
math | Example 6. Calculate $\int_{1}^{e} \ln x d x$. | 1 | 19 | 1 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\frac{3 t^{2}+1}{3 t^{3}} \\
y=\sin \left(\frac{t^{3}}{3}+t\right)
\end{array}\right.
$$ | -^{4}\cdot\cos(\frac{^{3}}{3}+) | 75 | 17 |
math | 1. The measures of the angles formed around a point $O$ are expressed (in degrees) by powers of the number 5. Find the minimum number of angles under the given conditions. | 8 | 39 | 1 |
math | The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the ... | 249001 | 73 | 6 |
math | For all $n$, find the number $a_{n}$ of ways to pay $n$ euros using 1 and 2 euro coins (without considering the order). | a_{n}=[\frac{n}{2}]+1 | 35 | 13 |
math | The 24th All-Union Mathematical Olympiad has a problem:
There are 1990 piles of stones, with the number of stones being $1, 2, \cdots$, 1990. The operation is as follows: each time, you can choose any number of piles and take the same number of stones from each of them. How many operations are needed at least to take ... | 11 | 90 | 2 |
math | 9.207. $\left\{\begin{array}{l}0,2^{\cos x} \leq 1, \\ \frac{x-1}{2-x}+\frac{1}{2}>0 .\end{array}\right.$ | x\in(0;\frac{\pi}{2}] | 56 | 12 |
math | Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$, let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$.\\
Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygo... | 4 | 115 | 1 |
math | 4. Given $x, y \geqslant 0$, and $x+y \leqslant 2 \pi$. Then the function
$$
f(x, y)=\sin x+\sin y-\sin (x+y)
$$
has a maximum value of | \frac{3 \sqrt{3}}{2} | 59 | 12 |
math | 112. Make sure that the last digits of the numbers in the Fibonacci sequence $^{\circ}$ repeat periodically. What is the length of the period?
Ensure that the last digits of the numbers in the Fibonacci sequence $^{\circ}$ repeat periodically. What is the length of the period? | 60 | 61 | 2 |
math | 2. Given an integer $n>1$, let $a_{1}, a_{2}, \cdots, a_{n}$ be distinct non-negative real numbers, and define the sets
$$
A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\} .
$$
Find the minimum value of $\frac{|A|}{|B|}$. H... | \frac{2(2n-1)}{n(n+1)} | 156 | 16 |
math | Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.
Suggestion: Calculate the probability of the product being odd. What is the probability that the product of the numbers on the ... | \frac{1}{2} | 74 | 7 |
math | Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality \[|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_... | \{(1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 4), (4, 1), (4, 3)\} | 227 | 51 |
math | Example 23 Find the polynomial $f(x)$ that satisfies $f\left(x^{n}+1\right)=f^{n}(x)+1$.
untranslated text remains the same as requested. | f(x)=x^n+1 | 43 | 7 |
math | A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided? | 20 | 67 | 2 |
math | 1. Kolya came up with an entertainment for himself: he rearranges the digits in the number 2015, then places a multiplication sign between any two digits and calculates the value of the resulting expression. For example: $150 \cdot 2=300$, or $10 \cdot 25=250$. What is the largest number he can get as a result of such ... | 1050 | 91 | 4 |
math | Example 8 Find:
(1) $\mathrm{C}_{n}^{1}-3 \mathrm{C}_{n}^{3}+5 \mathrm{C}_{n}^{5}-7 \mathrm{C}_{n}^{7}+\cdots$;
(2) $2 C_{n}^{2}-4 C_{n}^{4}+6 C_{n}^{6}-8 C_{n}^{8}+\cdots$. | \begin{array}{l}
\text{(1) } \mathrm{C}_{n}^{1}-3 \mathrm{C}_{n}^{3}+5 \mathrm{C}_{n}^{5}-7 \mathrm{C}_{n}^{7}+\cdots = n \times 2^{\frac{n-1}{2}} \cos \frac{(n-1) \pi}{4} \\
\text{(2) } | 98 | 96 |
math | ( Inspired by USAMO 2014, P1 )
Let $P \in \mathbb{R}[X]$ be a monic polynomial of degree 2. Suppose that $P(1) \geq P(0)+3$, and that $P$ has two real roots (not necessarily distinct) $x_{1}$ and $x_{2}$. Find the smallest possible value for $\left(x_{1}^{2}+1\right)\left(x_{2}^{2}+1\right)$. | 4 | 114 | 1 |
math | 17.1.7 * Find the integer solution to the equation $\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\cdots+\left[\frac{x}{10!}\right]=1001$. | 584 | 54 | 3 |
math | Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{20} | 68 | 8 |
math | ## Task B-2.1.
In the equation $x^{2}+m-3 x=m x-2$, determine the positive real number $m$ so that the total sum of all solutions of the equation and their squares is 44. | 4 | 53 | 1 |
math | ## Task 3 - 050713
The driver of a car registered in the GDR fled the scene after a traffic accident. After questioning several witnesses, the following information was obtained about the police registration number of the car:
a) The two letters of the license plate were AB or AD.
b) The two front digits were the sa... | 32 | 114 | 2 |
math | Question 15 Five monkeys divide a pile of peanuts. The first monkey divides the peanuts into five piles, eats one left over, and takes away one pile. The second monkey again divides the remaining peanuts into five piles, with exactly one left over, eats it, and takes away one pile. This continues until the fifth monkey... | 3121 | 98 | 4 |
math | Find all triples $(x, n, p)$ of positive integers $x$ and $n$ and prime numbers $p$ for which
$$
x^{3}+3 x+14=2 \cdot p^{n} \text {. }
$$ | (1,2,3) \text{ and } (3,2,5) | 54 | 19 |
math | The corridors of the labyrinth are the sides and diagonals of an n-sided convex polygon. How many light bulbs do we need to place in the labyrinth so that every passage is illuminated? | n-1 | 37 | 3 |
math | 5. Let $a$ be a natural number with 2019 digits and divisible by 9. Let $b$ be the sum of the digits of $a$, let $c$ be the sum of the digits of $b$, and let $d$ be the sum of the digits of $c$. Determine the number $d$.
## Third grade - B category | 9 | 79 | 1 |
math | 257. If it is known that the equation
$$
12 x^{5}-8 x^{4}-45 x^{3}+45 x^{2}+8 x-12=0
$$
admits the roots $+1 ; 1.5 ; -2$, write, without any calculations, the two missing roots. | \frac{2}{3},-\frac{1}{2} | 76 | 14 |
math | 11. If $a \pm b \mathrm{i}(b \neq 0)$ are the imaginary roots of the equation $x^{3}+q x+r=0$, where $a, b, q$ and $r$ are all real numbers, then $q$ can be expressed in terms of $a, b$ as $\qquad$ | b^{2}-3a^{2} | 76 | 9 |
math | Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For e... | 209601 | 162 | 6 |
math | 7. (2004 College Entrance Examination, Liaoning Province) A bag contains 10 identical balls, 5 of which are marked with the number 0, and 5 are marked with the number 1. If 5 balls are drawn from the bag, what is the probability that the sum of the numbers on the 5 balls drawn is less than 2 or greater than 3? | \frac{13}{63} | 84 | 9 |
math | 2. Let $a, b$ be positive real numbers,
$$
A=\frac{a+b}{2}, B=\frac{2}{\frac{1}{a}+\frac{1}{b}} \text {. }
$$
If $A+B=a-b$, then $\frac{a}{b}=$ | 3+2 \sqrt{3} | 66 | 8 |
math | 15. (6 points) There are three numbers $a, b, c$, $a \times b=24, a \times c=36, b \times c=54$, then $a+b+c=$ | 19 | 49 | 2 |
math | 5. Given points $A(1,1), B(1 / 2,0), C(3 / 2,0)$, the line passing through points $A, B$ and the line passing through points $A, C$ together with the line $y=a(0<a<1)$ enclose a planar region $G$. It is known that the probability of any point in the planar rectangular region $\{(x, y) \mid 0<x<2,0<y<1\}$ entering regio... | \frac{1}{2} | 131 | 7 |
math | The equation with integer coefficients $x^{4}+a x^{3}+b x^{2}+c x+d=0$ has four positive roots, counting multiplicities.
Find the smallest possible value of the coefficient $b$ under these conditions. | 6 | 53 | 1 |
math | 1258. Compute the integrals:
1) $\int_{0}^{\frac{\pi}{2}} \sin ^{3} x d x$;
2) $\int_{0}^{\ln 2} \sqrt{e^{x}-1} d x$
3) $\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} d x$;
4) $\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} d x$. | \frac{2}{3},2-\frac{\pi}{2},\frac{1}{8}\pi^{4},\sqrt{3}-\frac{\pi}{3} | 117 | 38 |
math | Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems? | 65 | 52 | 4 |
math | We define a sequence with $a_{1}=850$ and
$$
a_{n+1}=\frac{a_{n}^{2}}{a_{n}-1}
$$
for $n \geq 1$. Determine all values of $n$ for which $\left\lfloor a_{n}\right\rfloor=2024$.
Here, the floor $\lfloor a\rfloor$ of a real number $a$ denotes the greatest integer less than or equal to $a$. | 1175 | 112 | 4 |
math | Problem 3. On a circle, 999 numbers are written, each of which is 1 or -1, and not all numbers are equal. All products of ten consecutive numbers written around the circle are calculated, and the obtained products are summed.
a) (3 points) What is the smallest possible sum?
b) (3 points) And what is the largest possi... | )-997b)995 | 96 | 9 |
math | 2. Eight knights are randomly placed on a chessboard (not necessarily on distinct squares). A knight on a given square attacks all the squares that can be reached by moving either (1) two squares up or down followed by one squares left or right, or (2) two squares left or right followed by one square up or down. Find t... | 0 | 85 | 1 |
math | Ann and Max play a game on a $100 \times 100$ board.
First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once.
Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column... | 500000 | 182 | 6 |
math | 3. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,6,7,8$ (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 2160 | 78 | 4 |
math | 11.9. Simplify the expression
$$
\operatorname{tg} 20^{\circ}+\operatorname{tg} 40^{\circ}+\sqrt{3} \operatorname{tg} 20^{\circ} \operatorname{tg} 40^{\circ}
$$ | \sqrt{3} | 70 | 5 |
math | Find pairs of positive integers $(x, y)$, satisfying
$$
x>y \text {, and }(x-y)^{x y}=x^{y} y^{x} \text {. }
$$
(2013, Taiwan Mathematical Olympiad Training Camp) | (x, y) = (4, 2) | 57 | 11 |
math | We inscribe a regular hexahedron and a regular octahedron in a sphere; how do the radii of the spheres that can be inscribed in the hexahedron and octahedron compare? | \frac{r}{3}\sqrt{3} | 46 | 11 |
math | 11. [15] Find the largest positive integer $n$ such that $1!+2!+3!+\cdots+n$ ! is a perfect square. Prove that your answer is correct. | 3 | 45 | 1 |
math | Ex. 17. A quadrilateral is inscribed in a circle of radius 25, the diagonals of which are perpendicular and equal to 48 and 40. Find the sides of the quadrilateral. | 5\sqrt{10},9\sqrt{10},13\sqrt{10},15\sqrt{10} | 47 | 30 |
math | Let $a$ be a real number. Find all real-valued functions $f$ such that
$$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$
when constants of integration are suitably chosen. | f(x) = k Ce^{kx} | 57 | 11 |
math | 22.7. (GDR, 77). Find all polynomials $P(x)$ that satisfy the identity
$$
x P(x-1) \equiv (x-2) P(x), \quad x \in \mathbf{R}
$$ | P(x)=(x^2-x) | 56 | 8 |
math | 12. Suppose the real numbers $x$ and $y$ satisfy the equations
$$
x^{3}-3 x^{2}+5 x=1 \text { and } y^{3}-3 y^{2}+5 y=5
$$
Find $x+y$. | 2 | 61 | 1 |
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