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 the lowest possible values from the function
\[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\]
for any real numbers $ x$. | 1005 | 125 | 4 |
math | 5. The function
$$
y=-4 x+3 \sqrt{4 x^{2}+1}
$$
has a minimum value of $\qquad$ . | \sqrt{5} | 36 | 5 |
math | ## Task A-3.1.
For which real numbers $x$ does the following hold
$$
5^{2 x}+4^{x}<29 \cdot 10^{x-1} ?
$$ | x\in\langle-1,1\rangle | 46 | 11 |
math | 4. (20 points) For what values of the parameter $a$ does the equation
$$
(a+1)(|x-2.3|-1)^{2}-2(a-3)(|x-2.3|-1)+a-1=0
$$
have exactly two distinct solutions? | -1 | 66 | 2 |
math | Problem 11.3. A natural number $n$ is called interesting if $2 n$ is a perfect square, and $15 n$ is a perfect cube. Find the smallest interesting number. | 1800 | 43 | 4 |
math | 1. Determine all real numbers $m$ for which the equation
$$
x^{4}-(3 m+2) x^{2}+m^{2}=0
$$
has 4 real roots that are consecutive terms of an arithmetic progression. | 6 | 53 | 1 |
math | (5) Given the function $f(x)$ satisfies the condition $f(a x-1)=\lg \frac{x+2}{x-3}(a \neq 0)$.
(1) Find the expression for $f(x)$;
(2) Find the domain of $f(x)$;
(3) Does there exist a real number $a$ such that $f(x)$ is an odd function or an even function? If so, find the value of $a$; otherwise, explain the reason. | 0.2 | 108 | 3 |
math | (Czech-Polish-Slovak Match 2018)(M-D) Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+x y\right)=f(x) f(y)+y f(x)+x f(x+y)
$$ | f(x)=1-xorf(x)=-xorf(x)=0 | 70 | 14 |
math | On the segment $A B$, a point $C$ is taken, and segments $A B$ and $C B$ serve as diameters of circles. Chord $A M$ is tangent to the smaller circle at point $D$. Line $B D$ intersects the larger circle at point $N, \angle D A B=\boldsymbol{\alpha}, A B=2 R$. Find the area of the quadrilateral $A B M N$. | R^{2}\cos\alpha(1+\sin\alpha) | 94 | 14 |
math | 5. For the positive integer $n$, define $a_{n}$ as the unit digit of $n^{(n+1)^{n-1}}$. Then $\sum_{n=1}^{2018} a_{n}=$ $\qquad$ . | 5857 | 58 | 4 |
math | The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle. | 287 | 45 | 3 |
math | 8.2. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved three seats to the right, Galia had moved one seat to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. ... | 3 | 94 | 1 |
math | 4. For the geometric sequence $a_{1}, a_{2}, a_{3}, a_{4}$, it satisfies $a_{1} \in(0,1), a_{2} \in(1,2), a_{3} \in(2,3)$, then the range of $a_{4}$ is
$\qquad$ | (2\sqrt{2},9) | 76 | 9 |
math | Example 5. A parade team whose number of people is a multiple of 5 and no less than 1000, if lined up in rows of 4, is short of 3 people; if lined up in rows of 3, is short of 2 people; if lined up in rows of 2, is short of 1 person. Find the minimum number of people in this parade team. | 1045 | 86 | 4 |
math | 3. Let $f(x)$ be a monotonic function defined on the interval $(0,+\infty)$. If for any $x \in(0,+\infty)$, we have
$$
f\left(f(x)-2 \log _{2} x\right)=4,
$$
then the solution set of the inequality $f(x)<6$ is $\qquad$ | 0<x<4 | 83 | 4 |
math | 8. In a certain school, there are 5000 students. Each student is assigned an ID number from 0001 to 5000 . No two students can have the same ID number. If a student is selected uniformly at random, what is the probability that the ID number of the student does not contain any 2 s among its digits? | \frac{729}{1250} | 77 | 12 |
math | Example 13 Let $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ all be natural numbers, and $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=x_{1} x_{2} x_{3} x_{4} x_{5}$. Try to find the maximum value of $x_{5}$. | 5 | 91 | 1 |
math | ii. (16 points) Find all natural numbers $n$ such that $2^{8}+2^{11}+2^{n}$ is a perfect square of a natural number. | 12 | 41 | 2 |
math | [131 The equation $x^{2}-m x+2 m-2=0$ with respect to $x$ has a solution in the interval $\left[0, \frac{3}{2}\right]$, find the range of real number $m$.
] | [-\frac{1}{2},4-2\sqrt{2}] | 59 | 16 |
math | Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$. | 7 | 91 | 3 |
math | $$
\begin{array}{l}
\text { 5. Let } \frac{\sin ^{4} \theta}{a}+\frac{\cos ^{4} \theta}{b}=\frac{1}{a+b}(a, b>0) \text {. } \\
\text { Then } \frac{\sin ^{8} \theta}{a^{3}}+\frac{\cos ^{8} \theta}{b^{3}}=
\end{array}
$$ | \frac{1}{(a+b)^{3}} | 104 | 12 |
math | For example, if $5 x, y$ are prime numbers, solve the equation $x^{y}-y^{x}=x y^{2}-19$.
(2004 Balkan Mathematical Olympiad) | 2,7;2,3 | 46 | 7 |
math | Yashchenko I.v.
In all entrances of the house, there is the same number of floors, and on each floor, there is the same number of apartments. The number of floors in the house is greater than the number of apartments per floor, the number of apartments per floor is greater than the number of entrances, and the number ... | 7 | 96 | 1 |
math | 9.126. $y=\sqrt{\log _{\frac{1}{2}}^{2}(x-3)-1}$. | x\in(3;\frac{7}{2}]\cup[5;\infty) | 30 | 20 |
math | The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$. | 722 | 95 | 3 |
math | 1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 64827. The answer should be presented as an integer. | 1120 | 42 | 4 |
math | Consider the set $A=\{1,2,3, \ldots, 2011\}$. How many subsets of $A$ exist such that the sum of their elements is 2023060? | 4 | 50 | 1 |
math | ## Task 5 - V00505
How many matches (5 cm long, 2 mm wide, and 2 mm high) can fit into a cube with a side length of 1 m? | 5000000 | 45 | 7 |
math | 6. The endpoints of a line segment $AB$ of fixed length 3 move on the parabola $y^{2}=x$, and $M$ is the midpoint of segment $AB$. Find the shortest distance from $M$ to the $y$-axis.
(1987 National College Entrance Examination) | \frac{5}{4} | 67 | 7 |
math | ## Task 3 - 050933
The positive integer $x$ ends with the digits $a$ and $b$ (in that order).
Determine all ordered pairs $(a, b)$ for which $x^{2}$ ends with the same digits $a$ and $b$ (also in terms of the order)! | (0;0),(0;1),(2;5),(7;6) | 72 | 17 |
math | Children at the camp earned points for completing tasks. These points could be further exchanged: five points for a sticker, six points for a badge. Jára calculated that if he wanted only stickers, one point would remain unused. If he chose only badges, three points would remain unused. In the end, he managed to use al... | 51 | 98 | 2 |
math | 19 (12 points) Whether a company invests in a project is decided by three decision-makers, A, B, and C. Each of them has one "agree," one "neutral," and one "disagree" vote. When voting, each person must and can only cast one vote, and the probability of each person casting any one of the three types of votes is $\frac... | \frac{13}{27} | 173 | 9 |
math | 14th ASU 1980 Problem 17 S is a set of integers. Its smallest element is 1 and its largest element is 100. Every element of S except 1 is the sum of two distinct members of the set or double a member of the set. What is the smallest possible number of integers in S? | 9 | 73 | 1 |
math | 10.1. What is the sum of the digits of the number $A=100^{40}-100^{30}+100^{20}-100^{10}+1$? | 361 | 51 | 3 |
math | $3 \cdot 25$ For each positive integer $n$, let
$$\begin{array}{l}
S_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} \\
T_{n}=S_{1}+S_{2}+S_{3}+\cdots+S_{n} \\
U_{n}=\frac{1}{2} T_{1}+\frac{1}{3} T_{2}+\frac{1}{4} T_{3}+\cdots+\frac{1}{n+1} T_{n}
\end{array}$$
Find integers $0<a, b, c, d<1000000$, such that
$$\begin{... | T_{1988}=1989 S_{1989}-1989, \quad U_{1988}=1990 S_{1989}-3978 | 219 | 47 |
math | Example 8 Let $Q$ be the set of all rational numbers, find all functions $f$ from $\mathrm{Q}$ to $\mathrm{Q}$ that satisfy the following conditions:
(1) $f(1)=2$;
(2) For any $x, y \in \mathrm{Q}$, $f(x y)=f(x) f(y)-f(x+y)+1$.
| f(x)=x+1 | 85 | 6 |
math | Consider polynomials $P$ of degree $2015$, all of whose coefficients are in the set $\{0,1,\dots,2010\}$. Call such a polynomial [i]good[/i] if for every integer $m$, one of the numbers $P(m)-20$, $P(m)-15$, $P(m)-1234$ is divisible by $2011$, and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20,... | 460 | 199 | 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 | 21. Five participants of the competition became its prize winners, scoring 20, 19, and 18 points and taking first, second, and third places, respectively. How many participants won each prize place if together they scored 94 points? | 1,2,2 | 55 | 5 |
math | ## Task 1.
To each vertex of a regular $n$-gon, a non-negative real number is assigned such that the sum of all assigned numbers is 1. Let $A$ be the sum of the products of the numbers assigned to every three consecutive vertices. Determine the maximum possible value of $A$ and find all cases in which this value is ac... | \frac{1}{27} | 92 | 8 |
math | ## Task B-4.2.
Determine all natural numbers $x, y$ for which $1!+2!+\cdots+x!=y^{2}$. | (x,y)\in{(1,1),(3,3)} | 36 | 13 |
math | ## 14. Lacewing
One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), or maybe in a café (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has one chance in ... | \frac{3}{4} | 180 | 7 |
math | The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.... | 6 | 365 | 1 |
math | 5. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2, a_{2}=6$, and $\frac{a_{n+2}+a_{n}}{a_{n+1}+1}=2$. Then $\lim _{n \rightarrow+\infty} \frac{\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots+\sqrt{a_{n}}}{n^{2}}$ $=$ . $\qquad$ | \frac{1}{2} | 108 | 7 |
math | Problem 5. Solve the system of equations
$$
\left\{\begin{array}{l}
5 x^{2}+14 x y+10 y^{2}=17 \\
4 x^{2}+10 x y+6 y^{2}=8
\end{array}\right.
$$ | (-1,2),(11,-7),(-11,7),(1,-2) | 68 | 20 |
math | 8.131. $\sqrt{3}-\tan x=\tan\left(\frac{\pi}{3}-x\right)$ | x_{1}=\pik;x_{2}=\frac{\pi}{3}(3+1)\quadk,\inZ | 29 | 27 |
math | Find all reals $ k$ such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all $ a,b,c,d\geq \minus{} 1$.
[i]Edited by orl.[/i] | \frac{3}{4} | 85 | 7 |
math | 10. Find the smallest positive period of the function $y=|\sin 2 x|+|\cos 2 x|$. | \frac{\pi}{4} | 28 | 7 |
math | [ Rectangles and squares. Properties and characteristics ] [ Equilateral (equiangular) triangle ]
On the side $A B$ of the square $A B C D$, an equilateral triangle $A B M$ is constructed. Find the angle $D M C$.
# | 30 | 59 | 2 |
math | 6. In the planar quadrilateral $ABCD$, $AC$ intersects $BD$ at point $O$. If
$$
\begin{array}{l}
\overrightarrow{AC} + 2 \overrightarrow{DO} = 4 \overrightarrow{AB}, \\
3 AB = 2 CD, \\
\overrightarrow{BA} \cdot \overrightarrow{BD} = 9,
\end{array}
$$
then the maximum value of the area of $\triangle ACD$ is $\qquad$. | 4\sqrt{3} | 111 | 6 |
math | 12.159. The sides of the parallelogram are in the ratio $p: q$, and the diagonals are in the ratio m:n. Find the angles of the parallelogram. | \arccos\frac{(p^{2}+q^{2})(n^{2}-^{2})}{2pq(^{2}+n^{2})};\pi-\arccos\frac{(p^{2}+q^{2})(n^{2}-^{2})}{2pq(^{2}+n^{2})} | 43 | 74 |
math | Is the multiplication:
$$
79133 \times 111107=8794230231
$$
exact? (Without a calculator) | 8792230231 | 41 | 10 |
math | 1. Let $N=99999$. Then $N^{3}=$ | 999970000299999 | 19 | 15 |
math | 9 Given the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ with its left and right foci being $F_{1}$ and $F_{2}$, respectively, and point $P$ lies on the line $l: x-\sqrt{3} y+8+2 \sqrt{3}=0$. When $\angle F_{1} P F_{2}$ takes its maximum value, the ratio $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ is $\qquad$ | \sqrt{3}-1 | 126 | 6 |
math | 11. (20 points) Let the sequence of rational numbers $\left\{a_{n}\right\}$ be defined as follows:
$a_{k}=\frac{x_{k}}{y_{k}}$, where $x_{1}=y_{1}=1$, and
if $y_{k}=1$, then $x_{k+1}=1, y_{k+1}=x_{k}+1$;
if $y_{k} \neq 1$, then $x_{k+1}=x_{k}+1, y_{k+1}=y_{k}-1$.
How many terms in the first 2011 terms of this sequence ... | 213 | 151 | 3 |
math | 3 *. Find all pairs of natural numbers $x, y$ such that $x^{3}+y$ and $y^{3}+x$ are divisible by $x^{2}+y^{2}$. | 1 | 46 | 1 |
math | 4. Solve the system of equations $\left\{\begin{array}{l}4 \cos ^{2} x-4 \cos x \cos ^{2} 6 x+\cos ^{2} 6 x=0, \\ \sin x=\cos y .\end{array}\right.$ | (\frac{\pi}{3}+2\pik,\\frac{\pi}{6}+2\pin),(-\frac{\pi}{3}+2\pik,\\frac{5\pi}{6}+2\pin),k,n\inZ | 66 | 58 |
math | Question 71: Positive real numbers $x, y$ satisfy the following conditions: there exist $a \in[0, x], b \in[0, y]$, such that $a^{2}+y^{2}=2$, $b^{2}+x^{2}=1, a x+b y=1$, then the maximum value of $x+y$ is $\qquad$ _. | \sqrt{5} | 86 | 5 |
math | 5. (COL 1) Consider the polynomial $p(x)=x^{n}+n x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n}$ having all real roots. If $r_{1}^{16}+r_{2}^{16}+\cdots+r_{n}^{16}=n$, where the $r_{j}$ are the roots of $p(x)$, find all such roots. | r_{1}=r_{2}=\ldots=r_{n}=-1 | 102 | 17 |
math | ## Task A-2.1.
Determine all pairs $(m, n)$ of natural numbers such that $n$ divides $2m-1$ and $m$ divides $2n-1$. | (1,1),(3,5),(5,3) | 43 | 13 |
math | 3. Find all pairs of prime numbers $p$ and $q$, for which $p^{2}+p q+q^{2}$ is a perfect square. | (3,5)(5,3) | 35 | 9 |
math | 6. Given the sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=\sqrt{3}, x_{n+1}=\frac{x_{n}+1}{1-x_{n}} \text {. }
$$
Then $x_{1001}-x_{401}=$ | 0 | 71 | 1 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$
f(f(x)+y)+x f(y)=f(x y+y)+f(x)
$$
for all real numbers $x$ and $y$. | f(x)=x \text{ and } f(x)=0 | 57 | 13 |
math | 11. (20 points) Given the sequence $\left\{a_{n}\right\}$:
$$
1,1,2,1,2,3, \cdots, 1,2, \cdots, n, \cdots
$$
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Find all pairs of positive real numbers $(\alpha, \beta)$ such that
$$
\lim _{n \rightarrow+\infty} \frac{S_{n}}{n^{\alpha}}... | (\frac{3}{2},\frac{\sqrt{2}}{3}) | 136 | 17 |
math | Five integers form an arithmetic sequence. Whether we take the sum of the cubes of the first four terms, or the sum of the cubes of the last four terms, in both cases we get 16 times the square of the sum of the considered terms. Determine the numbers. | 0,0,0,0,0;64,64,64,64,64;0,16,32,48,64;64,48,32,16,0 | 56 | 52 |
math | 2. If the equality holds
$$
\log _{a} b-\log _{a b} b=\log _{a b^{2}} b-\log _{a b^{3}} b
$$
determine $\log _{a} b, \log _{a b} b, \log _{a b^{2}} b$ and $\log _{a b^{3}} b$. | \log_{}b=-\frac{2}{3},\log_{}b=-2,\log_{^{2}}b=2,\log_{^{3}}b=\frac{2}{3} | 89 | 43 |
math | Example 2 (2008 - Adapted from China National Training Team Test) If $\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y}>M \sqrt[3]{x^{3}+y^{3}+z^{3}}$ holds for any positive real numbers $x, y, z$, find the maximum value of the real number $M$.
保持源文本的换行和格式如下：
Example 2 (2008 - Adapted from China National Training Team Test) ... | 2 | 185 | 1 |
math | 8. Let the function
$$
f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n},
$$
where $a_{0}, a_{1}, \cdots, a_{n}$ are non-negative integers. It is known that $f(1)=4$, $f(5)=152$. Then $f(6)=$ $\qquad$ | 254 | 96 | 3 |
math | Example 3. If $n$ is a natural number, and $9 n^{2}+5 n+26$ equals the product of two consecutive natural numbers, find $n$. (1985 Shanghai Junior High School Mathematics Competition Question) | n=6 \text{ or } n=2 | 53 | 11 |
math | Let $n_1,n_2,\ldots,n_s$ be distinct integers such that
$$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample:
$(\text a)$ $|n_i|=1$ for some $i$
$(\text b)$ If further all $n_i$ are positive, then
$$\{n... | \{n_1, n_2, \ldots, n_s\} = \{1, 2, \ldots, s\} | 140 | 34 |
math | 1. There were 2013 empty boxes. In one of them, 13 new boxes (not nested inside each other) were placed. Thus, there became 2026 boxes. In one of them, 13 new boxes (not nested inside each other) were placed again, and so on. After several such operations, there became 2013 non-empty boxes. How many boxes are there in ... | 28182 | 101 | 5 |
math | Five, (20 points) Find the smallest natural number $k$, such that for any $x \in [0,1]$ and $n \in \mathbf{N}_{+}$, the inequality
$$
x^{k}(1-x)^{n}<\frac{1}{(1+n)^{3}}
$$
always holds. | 4 | 74 | 1 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=1, a_{2}=\frac{1}{3}, \\
a_{n} a_{n-1}+a_{n} a_{n+1}=2 a_{n-1} a_{n+1}\left(n \geqslant 2, n \in \mathbf{Z}_{+}\right) .
\end{array}
$$
Then the general term formula of the sequence is $\qquad$ . | a_{n}=\frac{1}{2n-1} | 124 | 14 |
math | Task 3. (15 points) At the quality control department of an oil refinery, Engineer Valentina Ivanovna received a research object consisting of about 200 oil samples (a container designed for 200 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low... | 77 | 170 | 2 |
math | Suppose that $n$ is positive integer and that $a$ is the integer equal to $\frac{10^{2 n}-1}{3\left(10^{n}+1\right)}$. If the sum of the digits of $a$ is 567, what is the value of $n$ ? | 189 | 70 | 3 |
math | 24th Putnam 1963 Problem B3 R is the reals. Find all f : R → R which are twice differentiable and satisfy: f(x) 2 - f(y) 2 = f(x + y) f(x - y). | f(Y)=AYorf(Y)=A\sinhaY | 56 | 12 |
math |
Zadatak B-2.6. Odredite sve kvadratne jednadžbe oblika $x^{2}+p x+q=0$ ako za koeficijente $p, q \in \mathbb{R}$ vrijedi $|p-q|=2012$, a zbroj kvadrata njezinih rješenja iznosi $2012^{2}$.
| \begin{aligned}x^{2}+2012x&=0\\x^{2}-2012x&=0\\x^{2}+2014x+4026&=0\\x^{2}-2010x-4022&=0\end{aligned} | 98 | 73 |
math | 2. (10 points) Among all natural numbers that are multiples of 20, the sum of those not exceeding 3000 and are also multiples of 14 is $\qquad$ | 32340 | 43 | 5 |
math | 2.12. (GDR, 77). How many pairs of values $p, q \in \mathbf{N}$, not exceeding 100, exist for which the equation
$$
x^{5}+p x+q=0
$$
has solutions in rational numbers? | 133 | 66 | 3 |
math | 7・15 Given a regular $(2n+1)$-sided polygon, three different vertices are randomly selected from its vertices. If all such selections are equally likely, find the probability that the center of the regular polygon lies inside the triangle formed by the randomly chosen three points. | \frac{n+1}{2(2n-1)} | 58 | 13 |
math | Example 6. Find the solution of the equation $y^{\prime}-y=\cos x-\sin x$, satisfying the condition: $y$ is bounded as $x \rightarrow+\infty$. | \sinx | 42 | 3 |
math | Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits. | n = 32768 | 29 | 9 |
math | $2 \cdot 65$ When integers $x$ and $y$ are not both zero, find the minimum value of $f=|5 x^{2}+11 x y-5 y^{2}|$.
(Soviet Union University Mathematics Competition, 1977) | 5 | 63 | 1 |
math | [ Sums of numerical sequences and series of differences ] Pascal's Triangle and the Binomial Theorem Complex numbers help solve tasks_
Using the expansion $(1+i)^{n}$ by the Binomial Theorem, find:
a) $C_{100}^{0}-C_{100}^{2}+C_{100}^{4}-\ldots+C_{100}^{100}$
b) $C_{99}^{1}-C_{99}^{3}+C_{99}^{5}-\ldots-C_{99}^{99}$. | -2^{50} | 132 | 6 |
math | 1. Find the number of points in the $x O y$ plane that have natural coordinates $(x, y)$ and lie on the parabola $y=-\frac{x^{2}}{3}+70$. | 4 | 47 | 1 |
math | Bamal, Halvan, and Zuca are playing [i]The Game[/i]. To start, they‘re placed at random distinct vertices on regular
hexagon $ABCDEF$. Two or more players collide when they‘re on the same vertex. When this happens, all the colliding players lose and the game ends. Every second, Bamal and Halvan teleport to a random ver... | \frac{29}{90} | 404 | 9 |
math | 11. Let $A$ be an $n$-element subset of the set $\{1,2, \cdots, 2017\}$, and any two numbers in $A$ are neither coprime nor have a divisibility relationship. Find the maximum value of $n$. | 504 | 64 | 3 |
math | Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where... | 7 | 118 | 1 |
math | A2. What is the solution of the equation $24 \div(3 \div 2)=(24 \div 3) \div m$ ? | \frac{1}{2} | 34 | 7 |
math | $(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other di... | 1944 | 67 | 4 |
math | Example 1. Solve the equation
$$
3 e^{x} \tan y d x+\left(2-e^{x}\right) \sec ^{2} y d y=0
$$ | \operatorname{tg}y-C(2-e^{x})^{3}=0 | 43 | 18 |
math | 65. What whole number is equal to the sum of all the whole numbers preceding it? Is there only one such number? | 3 | 26 | 1 |
math | For which integers $n$ are the fractions reducible:
a) $\frac{n^{2}+2 n+4}{n^{2}+n+3}$
b) $\frac{n^{3}-n^{2}-3 n}{n^{2}-n+3}$? | )n=3k-1;b)n=3kn=3k+1 | 61 | 16 |
math | 22. (2004 National College Entrance Examination, Zhejiang Province) In a box, there are 10 balls of the same size, among which 3 balls are labeled 1, 4 balls are labeled 2, and 3 balls are labeled 5. The first time, one ball is randomly taken from the box, and after it is put back, another ball is randomly taken the se... | 5.2 | 152 | 3 |
math | 28 Find the smallest positive integer $n$, such that in decimal notation $n^{3}$ ends with the digits 888. | 192 | 29 | 3 |
math | 4. Find all real solutions to $x^{3}+(x+1)^{3}+(x+2)^{3}=(x+3)^{3}$. | 3 | 37 | 1 |
math | B. In the Cartesian coordinate system $x O y$, it is known that $O$ is the origin, point $A(10,100), B\left(x_{0}, y_{0}\right)$, where $x_{0} 、 y_{0}$ are integers, and points $O 、 A 、 B$ are not collinear. For all points $B$ that satisfy the above conditions, find the minimum area of $\triangle O A B$. | 5 | 102 | 1 |
math | 8. Two people, A and B, play a game: they toss a coin several times in a row. The game ends when the cumulative number of times the head (or tail) side is up reaches 5. When the game ends, if the cumulative number of times the head side is up reaches 5, A wins; otherwise, B wins. What is the probability that the game i... | \frac{93}{128} | 95 | 10 |
math | Example 2 Find the interval of monotonic increase for the function $f(x)=\log _{\frac{1}{2}}\left(x^{2}-2 x-3\right)$. | (-\infty,-1) | 41 | 7 |
math | 75. In the six-digit number $M=\overline{a b c 321}$, $a, b, c$ are three different numbers, all greater than 3. If $M$ is a multiple of 7, then the smallest value of $M$ is $\qquad$ | 468321 | 65 | 6 |
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