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 | 57. Cut a line segment of length 14 into two segments, such that the two resulting segments and a line segment of length 10 can form a right triangle. Then the area of this right triangle is $\qquad$ - | 24 | 50 | 2 |
math | 15.26. Into how many parts do the planes of the faces divide the space a) of a cube; b) of a tetrahedron? | 15 | 34 | 2 |
math | 7. Given the following two sets of data:
First set: $20,21,22,25,24,23$; Second set: $22,24,23,25, a, 26$
If the variances of the two sets of data are equal, then the value of the real number $a$ is $\qquad$ . | 21or27 | 84 | 5 |
math | 4. In the Cartesian coordinate system $x O y$, let $F_{1}, F_{2}$ be the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, respectively. $P$ is a point on the right branch of the hyperbola, $M$ is the midpoint of $P F_{2}$, and $O M \perp P F_{2}, 3 P F_{1}=4 P F_{2}$. Then the... | 5 | 140 | 1 |
math | Problem 2. Solve the system of equations
$$
\left\{\begin{array}{l}
x(y+z)=5 \\
y(x+z)=10 \\
z(x+y)=13
\end{array}\right.
$$ | \\frac{2}{3},\\frac{3}{2},\pm6 | 50 | 17 |
math | Under the Christmas tree, there are 2012 cones. Winnie-the-Pooh and donkey Eeyore are playing a game: they take turns picking up cones for themselves. On his turn, Winnie-the-Pooh takes one or four cones, and Eeyore takes one or three. Pooh goes first. The player who cannot make a move loses. Which of the players can g... | Winnie-the-Pooh | 95 | 6 |
math | In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the fi nals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes? | 2520 | 65 | 4 |
math | In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.
[i]Proposed by Lewis Chen[/i] | 260 | 78 | 3 |
math | [Combinations and Permutations] [Case Enumeration]
How many ways can you choose from a full deck (52 cards)
a) 4 cards of different suits and ranks?
b) 6 cards such that all four suits are represented among them?
# | 8682544 | 53 | 7 |
math | Example 18. Find the sum: $1 \times 2+2 \times 3+3 \times 4$ $+\cdots+n(n+1)$ | \frac{n(n+1)(n+2)}{3} | 37 | 14 |
math | 1. (1993 National High School Mathematics Competition) What are the last two digits of the integer $\left[\frac{10^{93}}{10^{31}+3}\right]$? (Write the tens digit first, then the units digit) | 8 | 58 | 1 |
math | 8. A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of ... | \frac{9\pi}{2} | 115 | 9 |
math | 28. Determine the length of the path over which the amount of grain in the seeder's box will decrease by $14 \%$, if the box has a working width of 4 m, it was filled with 250 kg of grain, and the sowing rate is 175 kg per 1 ha. | 500 | 69 | 3 |
math | 1. The set $A=\left\{x \mid x \in \mathbf{Z}\right.$ and $\left.\frac{600}{5-x} \in \mathbf{Z}\right\}$ has $\qquad$ elements, and the sum of all elements is $\qquad$ . | 240 | 67 | 3 |
math | Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\] | a = 2 + \sqrt{3} | 56 | 11 |
math | 10. (3 points) There are three people, $A$, $B$, and $C$, who are a worker, a teacher, and an engineer, respectively. $A$ is older than the worker, $C$ is a different age from the teacher, and the teacher is younger than $B$. Therefore, the engineer is $\qquad$ . | B | 75 | 1 |
math | 【Question 1】
Calculate: $0.2 \times 63+1.9 \times 126+196 \times 9=$ $\qquad$.
| 2016 | 41 | 4 |
math | ## Task 4 - 320624
A rectangular children's room is $4 \mathrm{~m}$ and $40 \mathrm{~cm}$ long and $3 \mathrm{~m}$ and $30 \mathrm{~cm}$ wide. It has exactly one door, which is $90 \mathrm{~cm}$ wide. Thomas wants to install a new baseboard around the walls of this room. He calculates the required total length of the ... | 27.50\mathrm{DM} | 147 | 10 |
math | 79 Let the conjugate of the complex number $z$ be $\bar{z}$. If $z \cdot \bar{z}$ satisfies $z \cdot \bar{z}+(1-2 i) z+(1+2 i) \bar{z}=3$, then the minimum value of the sum of the real and imaginary parts of the complex number $z$ is $\qquad$ . | -7 | 86 | 2 |
math | 11.24 What is greater: $\operatorname{tg} 1$ or $\operatorname{arctg} 1 ?$ | \operatorname{tg}1>\operatorname{arctg}1 | 31 | 16 |
math | Find the number of digit of $\sum_{n=0}^{99} 3^n$.
You may use $\log_{10} 3=0.4771$.
2012 Tokyo Institute of Technology entrance exam, problem 2-A | 48 | 57 | 2 |
math | Example 3 Given the set $\{1,2,3,4,5,6,7,8,9,10\}$. Find the number of subsets of this set that have the following property: each subset contains at least two elements, and the absolute difference between any two elements in each subset is greater than 1. | 133 | 70 | 3 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-10.5,10.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 462 | 59 | 3 |
math | Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have. | 94 | 73 | 2 |
math | 5. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1$, and for each $n \in \mathbf{N}^{*}, a_{n}, a_{n+1}$ are the roots of the equation $x^{2}+3 n x+b_{n}=0$, then $\sum_{k=1}^{20} b_{k}=$ $\qquad$ . | 6385 | 93 | 4 |
math | Example 10 Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$, satisfying the condition
$$
f[x+f(y)]=f(x) \cdot f(y)(x, y \in \mathbf{Q}) .
$$ | f(x)=0orf(x)=1 | 58 | 8 |
math | 3. In parallelogram $A B C D$ with side $A B=1$, point $M$ is the midpoint of side $B C$, and angle $A M D$ is 90 degrees. Find the side $B C$. | 2 | 53 | 1 |
math | 2. Let $k$ be a constant. If for all $x, y \in(0,1)$, we have
$$
x^{k}+y^{k}-x^{k} y^{k} \leqslant \frac{1}{x^{k}}+\frac{1}{y^{k}}-\frac{1}{x^{k} y^{k}},
$$
then the range of the real number $k$ is $\qquad$ . | (-\infty, 0] | 101 | 8 |
math | Solve the following equation:
$$
\frac{2\left(x^{2}-3\right)}{5-x^{2}}=\sqrt{\frac{5 x+6}{x+2}}
$$ | x_{2}=2\quad\text{}\quadx_{5}=2-\sqrt{17} | 43 | 22 |
math | 7. Let $x=\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi$, then $1+x^{4}+x^{8}+x^{12}+x^{16}=$ | 0 | 54 | 1 |
math | 9. (20 points) Find all values of $x$ and $y$ for which the following equality holds:
$$
(x-10)^{2}+(y-11)^{2}+(x-y)^{2}=\frac{1}{3}
$$ | 10\frac{1}{3},10\frac{2}{3} | 59 | 18 |
math | $1 \cdot 35$ A four-digit number is composed of four consecutive digits in sequence. If the first two digits from the left are swapped, the resulting number is a perfect square. Find the four-digit number.
The text above is translated into English, keeping the original text's line breaks and format. | 3456 | 64 | 4 |
math | Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$. Find $m+n$ | 95 | 47 | 2 |
math | Let $x$, $y$, and $z$ be positive real numbers. Prove that
$\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$.
When does equality hold? | \sqrt{\frac{xy}{x^2 + y^2 + 2z^2}} + \sqrt{\frac{yz}{y^2 + z^2 + 2x^2}} + \sqrt{\frac{zx}{z^2 + x^2 + 2y^2}} \leq \frac{3}{2} | 99 | 76 |
math | Example 3 Let $f_{1}(x)=\frac{2}{x+1}$, and $f_{n+1}(x)=$ $f_{1}\left[f_{n}(x)\right], n \in \mathbf{N}^{*}$. Let $a_{n}=\frac{f_{n}(2)-1}{f_{n}(2)+2}$, then $a_{99}=$ | -\frac{1}{2^{1011}} | 93 | 12 |
math | (25 points) (1) First, select $n$ numbers from $1,2, \cdots, 2020$, then choose any two numbers $a$ and $b$ from these $n$ numbers, such that $a \nmid b$. Find the maximum value of $n$.
| 1010 | 68 | 4 |
math | 3. The equation $x^{2}+a x+4=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{2}-\frac{20}{3 x_{2}^{3}}=x_{2}^{2}-\frac{20}{3 x_{1}^{3}}
$$
Find all possible values of $a$. | -10 | 92 | 3 |
math | Problem 2. (3 points)
Alice the Fox thought of a two-digit number and told Pinocchio that this number is divisible by $2, 3, 4, 5$, and $6$. However, Pinocchio found out that exactly two of these five statements are actually false. What numbers could Alice the Fox have thought of? In your answer, indicate the number o... | 8 | 83 | 1 |
math | 1. (2003 Anhui Province Competition Question) Among three-digit numbers, if the digit in the tens place is smaller than the digits in the hundreds and units places, the number is called a concave number, such as 504, 746, etc., which are all concave numbers. Therefore, among three-digit numbers with no repeated digits,... | 240 | 92 | 3 |
math | [ Euler's function ]
Euler's function $\varphi(n)$ is defined as the number of integers from 1 to $n$ that are coprime with $n$. Find
a) $\varphi(17) ;$ b) $\varphi(p) ;$ c) $\varphi\left(p^{2}\right) ;$ d) $\varphi\left(p^{\alpha}\right)$. | )16;b)p-1;)p(p-1);)p^{\alpha-1}(p-1) | 89 | 25 |
math | 10.3. In the quadrilateral $A B C D$, $A B=2, B C=4, C D=5$. Find its area, given that it is both inscribed and circumscribed. | 2\sqrt{30} | 47 | 7 |
math | Solve $7^{x}-3 \cdot 2^{y}=1$ | 1,12,4 | 17 | 6 |
math | ## PROBLEM 1
Let $X \in M_{3}(C), \quad X=\left(\begin{array}{lll}a & b & c \\ 0 & a & b \\ 0 & 0 & a\end{array}\right)$. Calculate $X^{n}, n \in N$. | X^{n}=(\begin{pmatrix}^{n}&n^{n-1}n^{n-1}+\frac{n(n-1)}{2}^{n-2}\0&^{n}&n^{n-1}\0&0&^{n}\end{pmatrix}) | 69 | 64 |
math | 3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=0, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$
If $(\tan A+\tan B) \tan C=m... | \frac{1}{2} | 127 | 7 |
math | 7. Let the line $l$ passing through the fixed point $M(a, 0)$ intersect the parabola $y^{2}=4 x$ at points $P$ and $Q$. If $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$ is a constant, then the value of $a$ is $\qquad$ . | 2 | 85 | 1 |
math | 6. Let $x, y, z \in (0,1)$, satisfy
$$
\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2 \text {. }
$$
Find the maximum value of $x y z$.
(Tang Lihua, provided) | \frac{27}{64} | 81 | 9 |
math | Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying
\[\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.\]
Find the derivative of $f$ in an arbitrary point $x.$ | e^x | 113 | 3 |
math | 121. Solve the Cauchy problem for the equation $y^{\prime \prime}=1+x+x^{2}+$ $+x^{3}$, if $y=1$ and $y^{\prime}=1$ when $x=0$. | \frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{12}+\frac{x^{5}}{20}+x+1 | 56 | 42 |
math | 6.8 Find the first term and the common ratio of the geometric progression, given that $b_{4}-b_{2}=-\frac{45}{32}$ and $b_{6}-b_{4}=-\frac{45}{512}$. | b_{1}=6,q=\frac{1}{4} | 59 | 13 |
math | 3. Three people, A, B, and C, take the elevator from the 1st floor to floors 3 to 7 of the mall. A maximum of 2 people can get off at each floor. The number of ways they can get off the elevator is
$\qquad$ kinds. | 120 | 63 | 3 |
math | 6. Let the side length of a regular $n$-sided polygon be $a$, and the longest and shortest diagonals be $b$ and $c$ respectively. If $a=b-c$, then $n=$ $\qquad$ | 9 | 51 | 1 |
math | ${ }^{*} 7$. Find the smallest term in the sequence $\left\{a_{n}\right\}$. Where, $a_{\mathrm{n}}=n+\frac{1989}{n^{2}}$. | 23.769531 | 51 | 9 |
math | 2. Solve the equation $2 \log _{3} \operatorname{ctg} x=\log _{2} \cos x$. | \frac{\pi}{3}+2\pik,k\in\mathbb{Z} | 31 | 21 |
math | 356. A simple cryptarithm. Each letter in the following cryptarithm stands for a definite decimal digit:
$$
3(B I D F O R)=4(F O R B I D)
$$
Restore the original record. | 3(571428)=4(428571) | 50 | 18 |
math | Simplify the following expression:
$$
\left[\frac{(a+b)^{2}+2 b^{2}}{a^{3}-b^{3}}-\frac{1}{a-b}+\frac{a+b}{a^{2}+a b+b^{2}}\right] \cdot\left(\frac{1}{b}-\frac{1}{a}\right) \cdot
$$ | \frac{1}{} | 86 | 6 |
math | Problem 2. 2-1. Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-100)^{2}
$$
If the result is a non-integer, round it to the nearest integer and write it as the answer. | 44200 | 79 | 5 |
math | 9. If the equation $9^{-x^{x}}=4 \cdot 3^{-x^{x}}+m$ has real solutions for $x$, then the range of real values for $m$ is | \in[-3,0) | 44 | 7 |
math | Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$ | 1 | 57 | 1 |
math | Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$. | k^k | 111 | 3 |
math | 2. In a correspondence mathematics olympiad, out of 500 participants, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and finally, exactly 50 did not like the method of determining the winners of the olympiad. We will call an olympiad participant "significantly dis... | 60 | 121 | 2 |
math | 6.2. Mathematicians Andrey, Boris, and Viktor were solving problems from an olympiad. First, Andrey solved several problems, then Boris solved a third of the remaining problems. After this, a third of the problems remained unsolved, which Viktor completed. What part of all the problems did Andrey solve? | \frac{1}{2} | 67 | 7 |
math | A positive integer $n > 1$ is juicy if its divisors $d_1 < d_2 < \dots < d_k$ satisfy $d_i - d_{i-1} \mid n$ for all $2 \leq i \leq k$. Find all squarefree juicy integers. | 2, 6, 42, 1806 | 65 | 14 |
math | ## 73. Man and Dog.
- Walking the dog, - a mathematician friend once told me, - gives me plenty of food for thought. Once, for example, my dog, after waiting for me to go out, looked to see which way I was going to head, and when I started down the path, he raced to the end of it. Then he returned to me, ran to the en... | 16 | 184 | 2 |
math | 7. Find the solution to $2^{x} \equiv x^{2}(\bmod 3)$. | x \equiv \pm 2(\bmod 6) | 24 | 13 |
math | ## Task 2 - 070522
For a two-digit number $z$, it is known that the units digit represents a number three times as large as the tens digit. If the digits are swapped, the resulting number is 36 greater than the original.
What is $z$ in the decimal system? | 26 | 68 | 2 |
math | 6. Solve the equation $2^{x}+3^{y}=z^{2}$ in natural numbers. | (4,2,5) | 23 | 7 |
math | Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\ope... | 164 | 134 | 3 |
math | Task 3. Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ wi... | \lfloor\frac{n^{2}}{4}\rfloor | 132 | 14 |
math | 7. Given
$$
z_{n}=(1+\mathrm{i})\left(1+\frac{\mathrm{i}}{\sqrt{2}}\right) \cdots\left(1+\frac{\mathrm{i}}{\sqrt{n}}\right)\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Then the value of $\left|z_{2014}-z_{2015}\right|$ is $\qquad$ . | 1 | 103 | 1 |
math | Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$. | 48 | 26 | 2 |
math | Nine, (1) On the parabola $\mathrm{y}=\mathrm{x}^{2}-1$, find two distinct points such that these two points are symmetric with respect to the line $\mathbf{x}+\mathrm{y}=0$;
(2) For the parabola $y=2 x^{2}-1$ to always have two points symmetric with respect to $x+y=0$, find the range of $a$. | a>\frac{3}{4} | 93 | 8 |
math | 6. Antun has $80 \%$ more stickers than Branko. Branko has $\frac{3}{5}$ of the number of stickers that Darko has. If Branko gave 150 stickers to Darko, then Darko would have 3 times more stickers than Branko. How many stickers do all three of them have together? | 2010 | 74 | 4 |
math | Question 164, Point $\mathrm{P}$ moves on the circle $(\mathrm{x}-2)^{2}+(\mathrm{y}-1)^{2}=1$, vector $\overrightarrow{\mathrm{PO}}$ (where $\mathrm{O}$ is the origin of coordinates) rotates counterclockwise by $90^{\circ}$ around point $\mathrm{P}$ to get $\overrightarrow{\mathrm{PQ}}$, then the trajectory equation o... | (x-3)^{2}+(y+1)^{2}=2 | 110 | 16 |
math | 209. Indicate all such numbers $\alpha$ that the numbers $[\alpha],[2 \alpha]$, $[3 \alpha], \ldots,[N \alpha]$, where $N$ is a fixed natural number, are all distinct, and the numbers $\left[\frac{1}{\alpha}\right],\left[\frac{2}{\alpha}\right],\left[\frac{3}{\alpha}\right], \ldots,\left[\frac{N}{\alpha}\right]$ are al... | \frac{N-1}{N}\leqslant\alpha\leqslant\frac{N}{N-1} | 317 | 29 |
math | Task 2. A pharmacist has three weights, with which he weighed out 100 g of iodine for one customer, 101 g of honey for another, and 102 g of hydrogen peroxide for a third. He always placed the weights on one pan of the scales and the goods on the other. Could it be that each weight was less than 90 g?
[4 points] (A.V.... | 49.5,50.5,51.5 | 95 | 14 |
math | 16. (15 points) Two cars, A and B, start from points $A$ and $B$ respectively, heading towards each other. The two cars meet after 5 hours, at which point car A has passed the midpoint by 25 kilometers. After meeting, the two cars continue to travel, and 3 hours later, car A reaches point $B$. How many kilometers does ... | 15 | 89 | 2 |
math | 3. Let real numbers $x, y$ satisfy
$$
x^{2}+\sqrt{3} y=4, y^{2}+\sqrt{3} x=4, x \neq y \text {. }
$$
Then the value of $\frac{y}{x}+\frac{x}{y}$ is $\qquad$ | -5 | 73 | 2 |
math | (4) If $4n+1, 6n+1$ are both perfect squares, then the smallest positive integer value of $n$ is
$\qquad$ | 20 | 37 | 2 |
math | 10. Let $a>1$ be a positive real number, and $n \geqslant 2$ be a natural number, and the equation $[a x]=x$ has exactly $n$ distinct solutions, then the range of values for $a$ is . $\qquad$ | [1+\frac{1}{n},1+\frac{1}{n-1}) | 64 | 19 |
math | Example 6.24. Five televisions have been put on subscription service. It is known that for a group of five televisions, the expected number of failures per year is one. If the televisions have the same probability of working without failure, what is the probability that at least one repair will be needed within a year? | 0.67 | 68 | 4 |
math | What are the conditions for the roots of the equations $a_{1} x^{2}+b_{1} x+c_{1}=0$ and $a_{2} x^{2}+b_{2} x+c_{2}=0$ to be equal? | \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} | 57 | 36 |
math | Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the ... | 2 | 127 | 1 |
math | 4. How many elements does the multiplicative group of 2x2 matrices with elements in residue classes modulo 3 have? What about the subgroup of matrices that have a determinant of 1 modulo 3?
Supplement with exercises G.M.1/2014
NOTE: All subjects are mandatory.
Each subject is graded from 0 to 7 points.
Working time... | 24 | 92 | 2 |
math | ## Task A-2.4. (4 points)
For which $x \in \mathbb{R}$ is the number $\sqrt[3]{4+4 x}$ greater than the number $1+\sqrt[3]{x}$? | x\in(-1,1)\cup(1,+\infty) | 51 | 16 |
math | Example 2. A discrete random variable $X$ is given by the distribution law
| $X$ | -2 | -1 | 0 | 1 | 2 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.1 | 0.2 | 0.15 | 0.25 | 0.3 |
Find the distribution law and the mathematical expectation of the random variable $Y=X^{2}$. | 2.05 | 114 | 4 |
math | We inscribe a sphere in an equilateral cone, to which we lay an tangent plane parallel to the base of the cone. We then inscribe another sphere in the resulting cone, and so on. Determine the sum of the volumes of the spheres if this procedure is continued to infinity. The slant height of the cone is $l=2$. | \frac{2\pi\sqrt{3}}{13} | 71 | 15 |
math | Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$. We are also given that $\angle ABC = \angle CDA = 90^o$. Determine the length of the diagonal $BD$. | \frac{2021 \sqrt{2}}{2} | 59 | 15 |
math |
Problem 10b.1. Solve the equation:
$$
3^{\log _{3}(\cos x+\sin x)+\frac{1}{2}}-2^{\log _{2}(\cos x-\sin x)}=\sqrt{2}
$$
| 15\k360 | 60 | 7 |
math | 2. Polynomials $P(x)$ and $Q(x)$ of equal degree are called similar if one can be obtained from the other by permuting the coefficients (for example, the polynomials $2 x^{3}+x+7$ and $x^{3}+2 x^{2}+7 x$ are similar). For what largest $k$ is it true that for any similar polynomials $P(x), Q(x)$, the number $P(2009)-Q(2... | 2008 | 118 | 4 |
math | $\left[\begin{array}{l}\text { Angles between lines and planes } \\ {[\underline{\text { Linear dependence of vectors }}]}\end{array}\right]$
The lateral face of a regular quadrilateral pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of... | 30 | 85 | 2 |
math | 9. (16 points) Let the real number $t \in [0, \pi]$. If the equation $\cos (x+t)=1-\cos x$ has a solution for $x$, find the range of values for $t$.
| \in[0,\frac{2\pi}{3}] | 53 | 13 |
math | B1. Find all real numbers $x$ that satisfy the inequality
$$
2 \sin ^{2} 2 x \geq 3 \cos 2 x
$$ | \frac{\pi}{6}+k\pi\leqx\leq\frac{5\pi}{6}+k\pi,k\in\mathbb{Z} | 39 | 39 |
math | Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct e... | 51 | 173 | 2 |
math | Suppose that $x$ and $y$ are real numbers that satisfy the system of equations
$2^x-2^y=1$
$4^x-4^y=\frac{5}{3}$
Determine $x-y$ | 2 | 53 | 1 |
math | 9. A chemistry student conducted an experiment: from a tank filled with syrup solution, he poured out several liters of liquid, refilled the tank with water, then poured out twice as much liquid and refilled the tank with water again. As a result, the amount of syrup in the tank decreased by $\frac{8}{3}$ times. Determ... | 250 | 98 | 3 |
math | Find the gcd of $n^{17}-n$ for $n$ an integer. | 510 | 19 | 3 |
math | 4. (4 points) Solve the equation:
$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
$$
where the symbol $[a]$ denotes the integer part of the number $a$.
# | \frac{7}{15};\frac{4}{5} | 58 | 15 |
math | Problem 14. In triangle $ABC$, the lengths of its heights are known: $h_{a} ; h_{b} ; h_{c}$. Find the area of this triangle. | \frac{1}{\sqrt{(\frac{1}{h_{}}+\frac{1}{h_{b}}+\frac{1}{h_{}})(\frac{1}{h_{}}+\frac{1}{h_{b}}-\frac{1}{h_{}})(\frac{1}{h_{}}+\frac{1}{h_{}}-\frac{1}{h_{} | 41 | 84 |
math | The $25$ member states of the European Union set up a committee with the following rules:
1) the committee should meet daily;
2) at each meeting, at least one member should be represented;
3) at any two different meetings, a different set of member states should be represented;
4) at $n^{th}$ meeting, for every $k<n$, ... | 2^{24} | 110 | 7 |
math | Task 3. For real numbers $x$ and $y$, we define $M(x, y)$ as the maximum of the three numbers $x y, (x-1)(y-1)$, and $x+y-2 x y$. Determine the smallest possible value of $M(x, y)$ over all real numbers $x$ and $y$ with $0 \leq x, y \leq 1$. | \frac{4}{9} | 90 | 7 |
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