task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | Find all pairs of real numbers $(a,b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$. | (28, -48), (2, 56) | 48 | 15 |
math | 18. Let the variable $x$ satisfy the inequality: $x^{2}+b x \leqslant-x(b<-1)$, and the minimum value of $f(x)=x^{2}+b x$ is $-\frac{1}{2}$, find the value of the real number $b$. | -\frac{3}{2} | 69 | 7 |
math | 15. From $1,2, \cdots, 2004$, choose $k$ numbers such that among the chosen $k$ numbers, there are definitely 3 numbers that can form the side lengths of a triangle (here it is required that the three side lengths of the triangle are all different). What is the minimum value of $k$ that satisfies the condition? | 17 | 80 | 2 |
math | Three, (25 points) Given that $a, b, c$ are positive integers, and the parabola $y=$ $a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A, B$. If the distances from $A, B$ to the origin are both less than 1, find the minimum value of $a+b+c$.
Translate the above text into English, please retain the origin... | 11 | 110 | 2 |
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 | 1. Determine all integers $n$ for which $\frac{5 n-23}{n-7}$ is an integer. | n\in{-5,1,3,4,5,6,8,9,10,11,13,19} | 27 | 32 |
math | Through the midpoint of the edge $A B$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length $a$, a plane is drawn parallel to the lines $B D_{1}$ and $A_{1} C_{1}$.
1) In what ratio does this plane divide the diagonal $D B_{1}$?
2) Find the area of the resulting section. | \frac{3}{5};\frac{7a^{2}\sqrt{6}}{16} | 93 | 23 |
math | Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where $m\geq n\geq p \geq 1$ are integer numbers.
[i]Ioan Bogdan[/i] | 21 | 94 | 2 |
math | For the school birthday party, Ana, Pedro, Miriam, and Fábio brought a total of 90 sweets. Their teacher observed that
- if Ana had brought two more sweets;
- if Pedro had brought two fewer sweets;
- if Miriam had brought twice as many and
- if Fábio had brought half as many,
the four friends would have all brought t... | A=18,P=22,M=10,F=40 | 94 | 16 |
math | 15. For any $n \in \mathbf{N}_{+}$, calculate the sum $S=\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$. | n | 55 | 1 |
math | For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime? | (2, 3) | 31 | 7 |
math | ## Task B-3.5.
Grandpa Ivo, a retired mathematician, turned every game with his grandson into a math problem. So, in response to the question of how to make a paper dragon, he gave a very unusual answer. The dragon has the shape of a kite, whose diagonals are determined by the vectors
$\overrightarrow{A C}=(5 a+4) \v... | B(-\frac{3}{5},-\frac{4}{5}),C(5,0) | 198 | 22 |
math | Example 4. The random variable $X$ is given by the probability density function
$$
p(x)= \begin{cases}1 & \text { if } 1 \leq x \leq 2 \\ 0 & \text { if } x < 1 \text{ or } x > 2\end{cases}
$$
Find the probability density function of the function $Y=X^{2}$. | (y)=\begin{cases}1/2\sqrt{y}&\text{for}1\leqy\leq4\\0&\text{for}y>4\end{cases} | 90 | 45 |
math | $1 \cdot 66$ number $\sin \frac{\pi}{18} \sin \frac{3 \pi}{18} \sin \frac{5 \pi}{18} \sin \frac{7 \pi}{18} \sin \frac{9 \pi}{18}$ is it a rational number? | \frac{1}{16} | 73 | 8 |
math | Let $n>0$ be an integer. We have a two-pan balance and $n$ weights, whose weights are $2^{0}, 2^{1}, \ldots, 2^{n-1}$. We want to place these $n$ weights on the balance one by one in such a way that the right pan is never heavier than the left pan. At each step, we choose one of the weights that have not yet been place... | 1\cdot3\cdot5\cdot\ldots\cdot(2n-1) | 130 | 20 |
math | Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$. | \angle BAC = 120^\circ | 71 | 12 |
math | Let's determine the value of the parameter $k$ such that the roots of the following equation, when listed in ascending order, have the same difference between each pair of consecutive roots.
$$
4 x^{4}-(4 k+13) x^{2}+(9 k+9)=0
$$ | k_{1}=-\frac{3}{4},k_{2}=\frac{77}{4} | 64 | 24 |
math | 10. What is the smallest integer $x$ larger than 1 such that $x^{2}$ ends in the same three digits as $x$ does? | 376 | 34 | 3 |
math | For which $x$ does the following inequality hold?
$$
\left(\log _{x} 3\right) \cdot\left(\log _{3 x} 3\right) \geq \log _{9 x} 3
$$ | 0<x<3^{-2};\quad3^{-\sqrt{2}}\leqx<3^{-1};\quad1<x\leq3^{\sqrt{2}} | 56 | 38 |
math | We know that we currently have six denominations of banknotes in circulation: 500, 1000, 2000, 5000, 10000, and 20000 forints. How many different amounts can be paid with three banknotes? | 49 | 67 | 2 |
math | 5.29 Simplify the expression
$$
\left(\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}-2\right)^{2} \text {, }
$$
where there are $k$ nested parentheses, and combine like terms to obtain a polynomial. Find the coefficient of $x^{2}$ in this polynomial. | 4^{k-1}\cdot\frac{4^{k}-1}{4-1} | 100 | 20 |
math | 3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. For example, $[3]=3,[\sqrt{2}]=1,[-\pi]=-4$. Let $x$ be a positive real number. If $\left[\log _{2} x\right]$ is even, then $x$ is called a lucky number. The probability that a number randomly selected from the interval $(0,1)$ is a lucky numbe... | \frac{1}{3} | 102 | 7 |
math | 30. Find the number of subsets $\{a, b, c\}$ of $\{1,2,3,4, \ldots, 20\}$ such that $a<b-1<c-3$. | 680 | 48 | 3 |
math | The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum? | 14 | 45 | 2 |
math | Find all the integers $x$ in $[20, 50]$ such that $6x + 5 \equiv -19 \mod 10,$ that is, $10$ divides $(6x + 15) + 19.$ | \{21, 26, 31, 36, 41, 46\} | 57 | 26 |
math | Határozzuk meg azokat a háromjegyű, 13-mal osztható számokat, amelyek középső jegye a másik két szám számtani közepe.
| 741,234,975,468 | 55 | 15 |
math | Find all positive integers $n$ such that $1!+2!+\ldots+n!$ is a perfect square. | n=1n=3 | 26 | 6 |
math | ## Problem 1.
Determine all natural numbers $a$ and $b$, where $a < b$, that are two-digit numbers, knowing that the greatest common divisor (gcd) of $a$ and $b$ is a prime number, 20 times smaller than the least common multiple (lcm) of $a$ and $b$. | =12,b=15;=20,b=25;=28,b=35;=44,b=55;=52,b=65;=68,b=85;=76,b=95 | 73 | 55 |
math | Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$. If $XA = \tfrac ab$, where $a$ and $b$ are coprime positive integers, find $a + b$.
[i]Proposed by Alexander Wang[/i] | 61 | 98 | 2 |
math | Andělka, Barča, Honzík, Vlád’a, and Matyáš participated in a competition throwing a shuttlecock. Each of them threw once, and the total length of their throws was 41 meters. Matyáš threw the shortest, which was 90 cm less than Andělka, and she threw 60 cm less than Vlád’a. Honzík threw the farthest and hit the tape mar... | =8.1,b=8,=9,v=8.7,=7.2 | 155 | 20 |
math | 7.6 $\quad 2^{x^{2}-1}-3^{x^{2}}=3^{x^{2}-1}-2^{x^{2}+2}$. | x_{1,2}=\\sqrt{3} | 39 | 11 |
math | For which positive integer values of $k$ will the expression
$$
N=3^{6 n-1}-k \cdot 2^{3 n-2}+1
$$
be divisible by 7 for every positive integer value of $n$? | k\equiv3\pmod{7} | 55 | 10 |
math | 23、Expression: $1^{2015}+2^{2015}+3^{2015}+\cdots+2013^{2015}+2014^{2015}$, the unit digit of the calculation result is | 5 | 63 | 1 |
math | 5. For every non-negative integer $n$, define $\operatorname{rad}(n)$ as follows: if $n=0$ or 1, then $\operatorname{rad}(n)=1$; if the prime factors of $n$ are $p_{1}, p_{2}, \cdots, p_{k}\left(p_{1}<p_{2}<\cdots<p_{k}\right)$, then $\operatorname{rad}(n)=p_{1} p_{2} \cdots p_{k}$. Find all non-negative integer coeffi... | f(x)=a x^{m} | 178 | 8 |
math | 1. If the system of inequalities about $x$ $\left\{\begin{array}{l}11-2 x \geqslant 1, \\ x-t>0\end{array}\right.$, the sum of all integer solutions is 12, then the range of values for $t$ is | 2\leqslant<3or-3\leqslant<-2 | 68 | 18 |
math | Problem 4. As is known, balance scales come to equilibrium when the weight on both pans is the same. On one pan, there are 9 identical diamonds, and on the other, 4 identical emeralds. If one more such emerald is added to the diamonds, the scales will be balanced. How many diamonds will balance one emerald? The answer ... | 3 | 80 | 1 |
math | 6. $100 n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ sum up to 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 smallest possible value of $S$. For what values of $x_{1}, x_{2}, \cdots, x_{n}$ is... | 1-2^{-\frac{1}{n}} | 142 | 11 |
math | 6. Find all values of the parameter $b$ such that the system
$$
\left\{\begin{array}{l}
x \cos a - y \sin a - 3 \leqslant 0 \\
x^{2} + y^{2} - 8 x + 2 y - b^{2} - 6 b + 8 = 0
\end{array}\right.
$$
has at least one solution for any value of the parameter $a$. | b\in(-\infty;-\sqrt{17}]\cup[\sqrt{17}-6;+\infty) | 104 | 28 |
math | 13) (10 points) If there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+2 x_{2}+\cdots+$ $n x_{n}=2009$, and $x_{1}+x_{2}+\cdots+x_{n}=0$, where $x_{i}= \pm 7, i=1,2, \cdots$,
$n$. Determine the minimum value of $n$. | 34 | 112 | 2 |
math | Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements. | r = 200 | 56 | 7 |
math | ## Task 2 - 150712
Two vessels, called $A$ and $B$, together have a capacity of exactly 8 liters. A certain amount of water $W$ is distributed between the two vessels such that $A$ is half full and $B$ is completely full. If water is poured from $B$ into $A$ until $A$ is completely full, then $B$ is still one-sixth fu... | )A=5 | 142 | 4 |
math | Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit. | 1 | 76 | 3 |
math | For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$.
Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$.
[i](I. Gorodnin)[/i] | n = 3^{18} \times 673 | 77 | 15 |
math | 10. In the tetrahedron $P-ABC$, the three edges $PA$, $PB$, and $PC$ are pairwise perpendicular, and $PA=1$, $PB=PC=2$. If $Q$ is any point on the surface of the circumscribed sphere of the tetrahedron $P-ABC$, then the maximum distance from $Q$ to the plane $ABC$ is | \frac{3}{2}+\frac{\sqrt{6}}{6} | 86 | 17 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 10} \frac{\lg x-1}{\sqrt{x-9}-1}$ | \frac{1}{5\ln10} | 38 | 11 |
math | 10. (USA 6) ${ }^{\mathrm{IMO4}}$ Find the largest number obtainable as the product of positive integers whose sum is 1976 . | 2\cdot3^{658} | 39 | 9 |
math | 10. If $z_{1}, z_{2}, z_{3}$ satisfy $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1, z_{1}+z_{2}+z_{3}=1, z_{1} z_{2} z_{3}=1$, then $\left(z_{1}, z_{2}, z_{3}\right)$ $=$ . $\qquad$ | (1,\mathrm{i},-\mathrm{i}) | 101 | 10 |
math | Example 5. Solve the inequality
$$
|x-6|<x^{2}-5 x+9
$$ | (-\infty;1)\cup(3;+\infty) | 25 | 15 |
math | Task B-4.2. Three numbers form an increasing arithmetic sequence. If the second number is increased by 1, and the third by 10, the sequence becomes geometric. The smallest of these three numbers is 2. What are these numbers? | 2,5,8 | 53 | 5 |
math | ## Task B-3.1.
Determine all integers $x$ for which
$$
\log _{4}(x+12) \cdot \log _{x} 2 \geqslant 1
$$ | x\in{2,3,4} | 50 | 10 |
math | ## Task 12/84
The solution sought is that of the equation
$$
\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{1}{z^{2}}
$$
in natural numbers $x ; y ; z$, for which the product $x y z$ is minimal. | 15^{-2}+20^{-2}=12^{-2} | 71 | 16 |
math | 1. If $f(x)=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$, then the maximum value of $f(x)$ is $\qquad$ | 11 | 40 | 2 |
math | 3 [ Algebraic problems on the triangle inequality ]
How many (non-degenerate) triangles with a perimeter of 100 and integer side lengths exist?
# | 208 | 33 | 3 |
math | 3. The number of positive integer pairs $(x, y)$ that satisfy the equation
$$
\begin{array}{l}
x \sqrt{y}+y \sqrt{x}-\sqrt{2006 x}-\sqrt{2006 y}+\sqrt{2006 x y} \\
\quad=2006
\end{array}
$$ | 8 | 82 | 1 |
math | 5. Positive real numbers $a, b, c$ form a geometric sequence $(q \neq 1), \log _{a} b, \log _{b} c, \log _{c} a$ form an arithmetic sequence. Then the common difference $d=$ | -\frac{3}{2} | 60 | 7 |
math | One face of a tetrahedron has sides of length $3$, $4$, and $5$. The tetrahedron’s volume is $24$ and surface area is $n$. When $n$ is minimized, it can be expressed in the form $n = a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers and b is not divisible by the square of any prime. Evaluate $a + b + c$. | 157 | 101 | 3 |
math | 4.34. All edges (including the base sides) of a triangular pyramid are equal. Find the ratio of the radius of the sphere inscribed in the pyramid to its height. | 1:4 | 38 | 3 |
math | 15. If the set $A=\{1,2, \cdots, n\}$ is arbitrarily divided into 63 pairwise disjoint subsets (which are non-empty and whose union is $A$) $A_{1}, A_{2}, \cdots, A_{63}$, then there always exist two positive integers $x$ and $y$ belonging to the same subset $A_{i}(1 \leqslant i \leqslant 63)$, such that $x>y$, and $31... | 2016 | 138 | 4 |
math | 9. Compute the sum of all real numbers $x$ such that
$$
2 x^{6}-3 x^{5}+3 x^{4}+x^{3}-3 x^{2}+3 x-1=0
$$ | -\frac{1}{2} | 52 | 7 |
math | ## Task 2 - 050922
28 students of a class participated in a sports festival. Each one takes part in at least one of the three disciplines: shot put, long jump, and 100-meter dash.
The number of those who participate in both shot put and long jump, but not in the 100-meter dash, is equal to the number of those who onl... | 20 | 206 | 2 |
math | 6 、 Calculate the value of $\cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7} \cos \frac{6 \pi}{7}$ is $\qquad$ . | \frac{1}{8} | 46 | 7 |
math | Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers.
[i]Proposed by bissue[/i] | 45 | 46 | 2 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{4 n^{2}+4 n-1}{4 n^{2}+2 n+3}\right)^{1-2 n}
$$ | \frac{1}{e} | 60 | 7 |
math | 4. Arrange the squares of positive integers $1,2, \cdots$ in a sequence: $149162536496481100121144 \cdots$, the digit at the 1st position is 1, the digit at the 5th position is 6, the digit at the 10th position is 4, the digit at the 2008th position is $\qquad$. | 1 | 103 | 1 |
math | 15. A bouncy ball falls from point $A$ to the ground, bounces up to point $B$, then falls to a platform $20 \mathrm{~cm}$ high, bounces up to point $C$, and finally falls to the ground. Each time it bounces, the height it reaches is $80 \%$ of the height from which it fell. It is known that point $A$ is $68 \mathrm{~cm... | 132 | 119 | 3 |
math | 4. Variant 1.
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 | 79 | 4 |
math | [ Varignon's Parallelogram ]
The diagonals of a convex quadrilateral are equal to $c$ and $d$ and intersect at an angle of $45^{\circ}$. Find the segments connecting the midpoints of opposite sides of the quadrilateral.
# | \frac{1}{2}\sqrt{^2+^2\cd\sqrt{2}} | 57 | 21 |
math | 10. Given that $a$ and $b$ are real numbers, the system of inequalities about $x$
$$
\left\{\begin{array}{l}
20 x+a>0, \\
15 x-b \leqslant 0
\end{array}\right.
$$
has only the integer solutions $2, 3, 4$. Then the maximum value of $ab$ is $\qquad$ | -1200 | 92 | 5 |
math | ## Task A-1.1.
Determine all ordered pairs of integers $(x, y)$ for which
$$
x^{2} y+4 x^{2}-3 y=51
$$ | (x,y)=(0,-17),(2,35),(-2,35),(4,-1),(-4,-1) | 43 | 28 |
math | $\left[\begin{array}{l}\text { Algebraic inequalities (miscellaneous) } \\ {[\quad \text { Case analysis }}}\end{array}\right]$
$x, y>0$. 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} | 79 | 5 |
math | 3 [Trigonometric ratios in a right triangle]
Given a rhombus $A B C D$. The radii of the circles circumscribed around triangles $A B C$ and $B C D$ are 1 and 2. Find the distance between the centers of these circles.
# | \frac{3\sqrt{5}}{5} | 62 | 12 |
math | Let $a\geq b\geq c\geq d$ be real numbers such that $(a-b)(b-c)(c-d)(d-a)=-3.$
[list=a]
[*]If $a+b+c+d=6,$ prove that $d<0,36.$
[*]If $a^2+b^2+c^2+d^2=14,$ prove that $(a+c)(b+d)\leq 8.$ When does equality hold?
[/list] | d < 0.36 | 105 | 8 |
math | Example 6. Find the number of zeros of the function
$$
F(z)=z^{8}-4 z^{5}+z^{2}-1
$$
inside the unit circle $|z|<1$. | 5 | 47 | 1 |
math | ## 48. Math Puzzle $5 / 69$
Which sum is greater: $\sqrt{7}+\sqrt{10}$ or $\sqrt{3}+\sqrt{19}$? | \sqrt{7}+\sqrt{10}<\sqrt{3}+\sqrt{19} | 43 | 22 |
math | 4. How many different right-angled triangles exist, one of the legs of which is equal to $\sqrt{1001}$, and the other leg and the hypotenuse are expressed as natural numbers
ANSWER: 4. | 4 | 50 | 1 |
math | We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b... | 9 | 203 | 1 |
math | 4*. Find all such positive integers $k$ for which the number
$$
\underbrace{1 \ldots 1 \overbrace{2 \ldots 2}^{k}}_{2000}-\underbrace{2 \ldots 2}_{1001}
$$
is a perfect square. | 2 | 70 | 1 |
math | Example 7 How many real numbers $x$ are there such that the value of the function $y=\frac{x^{2}-2 x+4}{x^{2}-3 x+3}$ is an integer? | 7 | 45 | 1 |
math | The following expressions should be simplified to their simplest form:
$$
\begin{gathered}
\frac{12 a^{3} x^{4}+2 a^{2} x^{5}}{18 a b^{2} x+3 b^{2} x^{2}}, \frac{4-2 x+x^{2}}{x+2}-x-2 \\
\frac{1}{(a-b)(a-c)}+\frac{1}{(b-a)(b-c)}+\frac{1}{(c-a)(c-b)}
\end{gathered}
$$ | \frac{2^{2}x^{3}}{3b^{} | 124 | 16 |
math | Solve the following equation:
$$
\left(\sqrt{a+\sqrt{a^{2}-1}}\right)^{x}+\left(\sqrt{a-\sqrt{a^{2}-1}}\right)^{x}=2 a
$$ | 2or-2 | 54 | 4 |
math | 4. Given a tetrahedron $S-ABC$ with the base being an isosceles right triangle with hypotenuse $AB$, $SA=SB=SC=2, AB=2$. Suppose points $S, A, B, C$ all lie on a sphere with center $O$. Find the distance from point $O$ to the plane $ABC$. | \frac{\sqrt{3}}{3} | 79 | 10 |
math | C3. The addition sum $X C V+X X V=C X X$ is true in Roman numerals.
In this question, however, the sum is actually the lettersum shown alongside, in which: each letter stands for one of the digits 0 to 9 , and stands for the same digit each time it occurs; different letters stand for different digits; $X C V$ and no nu... | 497+447=944 | 121 | 11 |
math | Example 5. For the equation $(1989 x)^{2}-1988 \cdot 1990 x$ $-1=0$, the larger root is $r$. For the equation $x^{2}+1989 x-1990=0$, the smaller root is $s$. Find the value of $r-s$. (Adapted from the 1984 Beijing Mathematics Competition) | 1991 | 95 | 4 |
math | ## Task Condition
Find the angle between the planes
$x+2 y-2 z-7=0$
$x+y-35=0$ | \frac{\pi}{4} | 31 | 7 |
math | 4. Solve the equation
$$
2^{x^{5}}+4^{x^{4}}+256^{4}=3 \cdot 16^{x^{3}}
$$ | 2 | 40 | 1 |
math | B2 Four workers divided an 1880 EUR fee. The first worker received a quarter, the second $5 \%$ of the total amount. The remainder was divided between the third and fourth workers in a ratio of $1: 3$. How much did each worker receive? | The\first\worker\receives\470\EUR,\the\\94\EUR,\the\third\329\EUR,\\the\fourth\worker\987\EUR | 59 | 45 |
math | Example 4. Solve the equation
$$
y^{\mathrm{IV}}-y=e^{\alpha x}+e^{-\alpha x}+\cos \beta x
$$ | C_{1}e^{x}+C_{2}e^{-x}+C_{3}\sinx+C_{4}\cosx+\frac{e^{\alphax}+e^{-\alphax}}{\alpha^{4}-1}+\frac{\cos\betax}{\beta^{4}-1} | 39 | 68 |
math | Let $n$ and $k$ be positive integers. Find all monic polynomials $f\in \mathbb{Z}[X]$, of degree $n$, such that $f(a)$ divides $f(2a^k)$ for $a\in \mathbb{Z}$ with $f(a)\neq 0$. | f = X^n | 72 | 5 |
math | ## 2. Math Puzzle $7 / 65$
A $a=100 \mathrm{~m}$ long holiday train is rolling at a speed of $v_{Z}=72 \frac{\mathrm{km}}{\mathrm{h}}$.
How great is the speed $v_{W}$ of the wind hitting the train perpendicularly, if the smoke trail of the locomotive is deflected by the distance $b=40 \mathrm{~m}$ at the end of the t... | 28.8\frac{\mathrm{}}{\mathrm{}} | 108 | 14 |
math | We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers?
[i] Proposed by Lewis Chen [/i] | 165 | 49 | 3 |
math | 8. For any real numbers $A, B, C$, the maximum value of the trigonometric expression $\sin ^{2} A \cos ^{2} B+\sin ^{2} B \cos ^{2} C+\sin ^{2} C \cos ^{2} A$ is
$\qquad$ . | 1 | 72 | 1 |
math | Calculate $\frac{\pi}{\sqrt{2}}$ to three decimal places. | 2.221 | 17 | 5 |
math | Task 4/84 All rational solutions $(x ; y)$ of the equation
$$
4 x^{2} y^{2}-4 x^{2} y-x^{2}+4 x=2
$$ | (1;0.5) | 46 | 7 |
math | The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball. | 3\sqrt{21} | 95 | 7 |
math | Test $3 P$ is a point inside $\triangle A B C$, line $A C$ and $B P$ intersect at $Q$, line $A B$ and $C P$ intersect at $R$, it is known that $A R=R B=C P, C Q=P Q$. Find $\angle B R C$. | 120 | 69 | 3 |
math | 5. Dima has 25 identical bricks of size $5 \times 14 \times 17$. Dima wants to build a tower from all his bricks, each time adding one more brick on top (each new brick adds 5, 14, or 17 to the current height of the tower). We will call a number $n$ constructible if Dima can build a tower of height exactly $n$. How man... | 98 | 99 | 2 |
math | 11.1. The numbers $2^{2019}$ and $5^{2019}$ are written consecutively. How many digits are written in total? | 2020 | 38 | 4 |
math | I2.1 If $\alpha, \beta$ are roots of $x^{2}-10 x+20=0$, find $a$, where $a=\frac{1}{\alpha}+\frac{1}{\beta}$.
I2.2 If $\sin \theta=a\left(0^{\circ}<\theta<90^{\circ}\right)$, and $10 \cos 2 \theta=b$, find $b$.
I2.3 The point $A(b, c)$ lies on the line $2 y=x+15$. Find $c$.
I2.4 If $x^{2}-c x+40 \equiv(x+k)^{2}+d$, fin... | \frac{1}{2},5,10,15 | 158 | 14 |
math | 4. The volume of a rectangular prism is 8 cubic centimeters, and the total surface area is 32 square centimeters. If the length, width, and height form a geometric sequence, then the sum of all the edges of this rectangular prism is $\qquad$ . | 32 | 58 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\ln \frac{\sin x}{\cos x+\sqrt{\cos 2 x}}
$$ | \frac{1}{\sinx\cdot\sqrt{\cos2x}} | 31 | 17 |
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