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 | 92. If the first digit of a three-digit number is increased by $n$, and the second and third digits are decreased by $n$, then the resulting number will be $n$ times the original number. Find the number $n$ and the original number. | 178 | 55 | 3 |
math | 908*. Does the equation
$$
x^{2}+y^{2}+z^{2}=2 x y z
$$
have a solution in non-negative integers? | (0,0,0) | 39 | 7 |
math | I1.2 Given that $f(x)=-x^{2}+10 x+9$, and $2 \leq x \leq \frac{a}{9}$. If $b$ is the difference of the maximum and minimum values of $f$, find the value of $b$. | 9 | 64 | 1 |
math | ## Task 4 - 251224
Determine all positive integers $n$ that have the following property: In the closed interval $\left[2^{n}, 2^{n+1}\right]$, there is at least one natural number divisible by $n^{3}$. | 1orn\geq8 | 63 | 6 |
math | ## problem statement
Find the differential $d y$.
$y=e^{x}(\cos 2 x+2 \sin 2 x)$ | 5e^{x}\cdot\cos2x\cdot | 31 | 12 |
math | $14 \cdot 75$ Find all natural numbers $a$ and
$$
b:\left[\frac{a^{2}}{b}\right]+\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b .
$$
(37th International Mathematical Olympiad Preliminary Question, 1996) | b=^{2}+1,\inN\text{or}=b^{2}+1,b\inN | 91 | 25 |
math | 7. There is a rectangular iron sheet of size $80 \times 50$. Now, a square of the same size is to be cut from each corner, and then it is to be made into an open box. What should be the side length of the square to be cut so that the volume of this open box is maximized? | 10 | 71 | 2 |
math | 9. Let the domain of the function $f(x)$ be $\mathbf{R}$. If there exists a constant $\omega>0$ such that $|f(x)| \leqslant \omega|x|$ for all real numbers $x$, then $f(x)$ is called a "conditionally bounded function". Now, given the following functions:
(1) $f(x)=4 x$;
(2) $f(x)=x^{2}+2$;
(3) $f(x)=\frac{2 x}{x^{2}-2 x+5}$;
(4) $f(x)$ is an odd function defined on the real number set $\mathbf{R}$, and for all $x_{1}, x_{2}$, we have
$$
f\left(x_{1}\right)-f\left(x_{2}\right) \leqslant 4\left|x_{1}-x_{2}\right| .
$$
Among these, the functions that are "conditionally bounded functions" are $\qquad$ (write down all the sequence numbers that meet the condition). | (1)(3)(4) | 233 | 7 |
math | ## Zadatak B-4.1.
Riješite jednadžbu
$$
\log _{2020} \sqrt{x}+\log _{2020} \sqrt[4]{x}+\log _{2020} \sqrt[8]{x}+\ldots+\log _{2020} \sqrt[2^{n}]{x}+\ldots=\frac{2}{\log _{2} x}+\log _{x} 505
$$
| 2020 | 117 | 4 |
math | Example 2.5.5 There are 6 different elements of class A and 4 different elements of class B. In a certain experiment, an even number of elements from class A must be taken, and no fewer than 2 elements from class B must be taken. How many experimental schemes are there? | 352 | 63 | 3 |
math | 3. [4] In each cell of a strip of length 100, there is a chip. You can swap any two adjacent chips for 1 ruble, and you can also swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the chips in reverse order?
(Egor Bakaev) | 50 | 78 | 2 |
math | ## Aufgabe 1 - 181241
Man ermittle alle ganzen Zahlen $a$ mit der Eigenschaft, dass zu den Polynomen
$$
\begin{aligned}
& f(x)=x^{12}-x^{11}+3 x^{10}+11 x^{3}-x^{2}+23 x+30 \\
& g(x)=x^{3}+2 x+a
\end{aligned}
$$
ein Polynom $h(x)$ so existiert, dass für alle reellen $x$ die Gleichung $f(x)=g(x) \cdot h(x)$ gilt.
| 3 | 141 | 1 |
math | 2. Find the number of roots of the equation: $2^{\lg \left(x^{2}-2023\right)}-\lg 2^{x^{2}-2022}=0$. | 4 | 45 | 1 |
math | 8.2. (England, 75). Solve the equation
$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+\left[\sqrt[3]{x^{3}-1}\right]=400
$$
in natural numbers. | 5 | 60 | 1 |
math | $\underline{\text { Folklore }}$
Vanya went to the shooting range with his dad. The deal was this: Vanya would be given 10 bullets, and for each hit on the target, he would receive three more bullets. Vanya made 14 shots, and he hit the target exactly half of the time. How many bullets did Vanya have left? | 17 | 78 | 2 |
math |
In cyclic quadrilateral $ABCD$ with $AB = AD = 49$ and $AC = 73$, let $I$ and $J$ denote the incenters of triangles $ABD$ and $CBD$. If diagonal $\overline{BD}$ bisects $\overline{IJ}$, find the length of $IJ$. | \frac{28}{5}\sqrt{69} | 75 | 13 |
math | Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that
$$ n=S(a)=S(b)=S(a+b) , $$
where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $
[i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i] | n=9k | 92 | 4 |
math | 3. In the tetrahedron $P-ABC$, $PC \perp$ plane $ABC$, $AB=8$, $BC=6$, $PC=9$, $\angle ABC=120^{\circ}$. Then the cosine value of the dihedral angle $B-AP-C$ is $\qquad$ | \frac{11 \sqrt{111}}{148} | 71 | 17 |
math | Find all the natural numbers $m$ and $n$, such that the square of $m$ minus the product of $n$ with $k$, is 2, where the number $k$ is obtained from $n$ by writing 1 on the left of the decimal notation of $n$. | m=11, n=7 | 62 | 8 |
math | 18. (15 points) Clothing retailers A and B wish to deal in clothing produced by a certain brand's manufacturing company. The design department of the company, without any information about A and B's sales, randomly provides them with $n$ different designs. A and B independently select the designs they approve of. What is the probability that at least one design is approved by both A and B?
| 1-\left(\frac{3}{4}\right)^{n} | 82 | 15 |
math | Problem 9. (12 points)
Ivan, a full-time student, started working at his long-desired job on March 1 of the current year. In the first two months, during the probationary period, Ivan's salary was 20000 rubles. After the probationary period, the salary increased to 25000 rubles per month. In December, for exceeding the plan, Ivan was awarded a bonus of 10000 rubles. In addition, while studying in a full-time budget-funded graduate program, Ivan received a scholarship of 2800 rubles per month throughout the year. What amount of personal income tax should Ivan pay to the budget? (Provide the answer as a whole number, without spaces and units of measurement) | 32500 | 164 | 5 |
math | 3. If the expansion of $(a+2 b)^{n}$ has three consecutive terms whose binomial coefficients form an arithmetic sequence, then the largest three-digit positive integer $n$ is $\qquad$ . | 959 | 44 | 3 |
math | 13.047. A musical theater announced a competition for admission to the orchestra. Initially, it was planned that the number of places for violinists, cellists, and trumpeters would be distributed in the ratio $1.6: 1: 0.4$. However, it was then decided to increase the intake, and as a result, 25% more violinists and 20% fewer cellists were admitted than originally planned. How many musicians of each genre were admitted to the orchestra if a total of 32 people were admitted? | 20 | 117 | 2 |
math | \section*{Problem 2 - 121242}
All pairs \((x, y)\) of integers are to be specified for which the equation is satisfied:
\[
x(x+1)(x+7)(x+8)=y^{2}
\] | (x,y)\in{(-9,\12),(-8,0),(-7,0),(-4,\12),(-1,0),(0,0),(1,\12)} | 59 | 41 |
math | The 24th question: Given an integer $n \geq 2$, for a permutation $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $\{1,2, \ldots, n\}$ and $1 \leq i \leq n$, let $x_{i}$ denote the maximum length of an increasing subsequence starting with $a_{i}$, and let $y_{i}$ denote the maximum length of a decreasing subsequence starting with $a_{i}$. Find the minimum possible value of $\sum_{\mathrm{i}=1}^{\mathrm{n}}\left|x_{i}-y_{i}\right|$. | [\frac{n}{2}] | 147 | 6 |
math | 8.197. $\frac{\cot 2z}{\cot z}+\frac{\cot z}{\cot 2z}+2=0$. | \frac{\pi}{3}(3k\1),k\inZ | 35 | 16 |
math | 1A. Determine all integers $x$ for which $\log _{2}\left(x^{2}-4 x-1\right)$ is also an integer. | x=-1orx=5 | 34 | 7 |
math | 4. Given the sequence $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$, and $a_{0}=3$. Then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ . | \frac{1}{3}\left(2^{n+2}-n-3\right) | 87 | 21 |
math | Our Slovak grandmother was shopping in a store that only had apples, bananas, and pears. Apples were 50 cents each, pears were 60 cents each, and bananas were cheaper than pears. The grandmother bought five pieces of fruit, including exactly one banana, and paid 2 euros and 75 cents.
How many cents could one banana cost? Determine all possibilities. (K. Jasenčáková)
Hint. Start by calculating with $s$ fruits whose prices you know. | 35,45,55 | 106 | 8 |
math | 52 (975). Find a two-digit number, the doubled sum of whose digits is equal to their product. | 63,44,36 | 25 | 8 |
math | 6. (10 points) The students in the youth summer camp come from 4 municipalities, with $\frac{1}{6}$ of the students from Shanghai, 24 students from Tianjin, the number of students from Beijing is $\frac{3}{2}$ times the sum of the number of students from Shanghai and Tianjin, and $\frac{1}{4}$ of the students are from Chongqing. How many students are there in the youth summer camp? | 180 | 98 | 3 |
math | 10. Make $\frac{a b^{2}}{a+b}(a \neq b)$ a prime number, the positive integer pair $(a, b)=$ $\qquad$ | (6,2) | 41 | 5 |
math | 9.3. On the board, there are $N$ prime numbers (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. For what largest $N$ is this possible | 4 | 48 | 1 |
math | Example 3 The general term of the sequence $1001,1004,1009$ is $a_{n}=n^{2}+1000$, where $n \in \mathbf{N}_{+}$. For each $n$, let $d_{n}$ denote the greatest common divisor of $a_{n}$ and $a_{n+1}$, find the maximum value of $d_{n}$, where $n$ takes all positive integers. | 4001 | 106 | 4 |
math | 3. If numbers $a_{1}, a_{2}, a_{3}$ are taken in increasing order from the set $1, 2, \cdots, 14$, such that the following conditions are satisfied:
$$
a_{2}-a_{1} \geqslant 3 \text { and } a_{3}-a_{2} \geqslant 3 \text {. }
$$
Then the number of different ways to choose such numbers is $\qquad$ kinds. | 120 | 107 | 3 |
math | 3. Solve the equation
$$
2(x-6)=\frac{x^{2}}{(1+\sqrt{x+1})^{2}}
$$ | 8 | 31 | 1 |
math | Example 3. Find the real roots of the equation
$$
\mathrm{f}(\mathrm{x})=\mathrm{x}^{41}+\mathrm{x}^{3}+1=0
$$ | x=-0.9524838 | 44 | 11 |
math | ii. (25 points) Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has two roots that are natural numbers. | 76 | 57 | 2 |
math | 3. Solve the equation
$2 x+1+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+1) \cdot \sqrt{x^{2}+2 x+2}=0$. | -\frac{1}{2} | 59 | 7 |
math | 12. Let $\mathbb{N}$ be the set of all positive integers. A function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies $f(m+$ $n)=f(f(m)+n)$ for all $m, n \in \mathbb{N}$, and $f(6)=2$. Also, no two of the values $f(6), f(9), f(12)$ and $f(15)$ coincide. How many three-digit positive integers $n$ satisfy $f(n)=f(2005)$ ? | 225 | 126 | 3 |
math | 51st Putnam 1990 Problem A4 Given a point P in the plane, let S P be the set of points whose distance from P is irrational. What is the smallest number of such sets whose union is the entire plane? | 3 | 51 | 1 |
math | 272. Find a three-digit number that is equal to the sum of the tens digit, the square of the hundreds digit, and the cube of the units digit.
$273 *$. Find the number $\overline{a b c d}$, which is a perfect square, if $\overline{a b}$ and $\bar{c} \bar{d}$ are consecutive numbers, with $\overline{a b}>\overline{c d}$. | 357 | 99 | 3 |
math | Let $\{a_i\}_{i=0}^\infty$ be a sequence of real numbers such that \[\sum_{n=1}^\infty\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\cdots\] for all $|x|<1$. Find $a_{1000}$.
[i]Proposed by David Altizio[/i] | 16 | 99 | 2 |
math | 14.1. A pedestrian walked a certain distance in 2.5 hours, and during any one-hour interval, he walked 5 km. Can we claim that the pedestrian walked the entire distance at an average speed of 5 km per hour?
$$
\text { (5-8 grades) }
$$ | 5.2 | 65 | 3 |
math | Let $ f$ be an infinitely differentiable real-valued function defined on the real numbers. If $ f(1/n)\equal{}\frac{n^{2}}{n^{2}\plus{}1}, n\equal{}1,2,3,...,$ Compute the values of the derivatives of $ f^{k}(0), k\equal{}0,1,2,3,...$ | f(0) = 1 | 80 | 8 |
math | 34 The maximum value of the function $y=x\left(1+\sqrt{1-x^{2}}\right)$ is $\qquad$ , and the minimum value is $\qquad$ . | \frac{3\sqrt{3}}{4},-\frac{3\sqrt{3}}{4} | 42 | 24 |
math | 2. For what values of $q$ is one of the roots of the equation $x^{2}-12 x+q=0$ the square of the other? | -64or27 | 36 | 6 |
math | Anička has saved 290 coins in her piggy bank, consisting of one-crown and two-crown coins. When she uses a quarter of all the two-crown coins, she collects the same amount as when she uses a third of all the one-crown coins.
What is the total amount Anička has saved?
(L. Růžičková) | 406 | 79 | 3 |
math | 7.2. Students in the seventh grade send each other New Year's stickers on Telegram. It is known that exactly 26 people received at least one sticker, exactly 25 - at least two stickers, ..., exactly 1 - at least 26 stickers. How many stickers did the students in this class receive in total, if it is known that no one received more than 26 stickers? (find all possible answers and prove that there are no others) | 351 | 97 | 3 |
math | 12. The rules of a "level-passing game" stipulate: on the $n$th level, a die must be rolled $n$ times, and if the sum of the points that appear in these $n$ rolls is greater than $2^{n}$, it counts as passing the level. Therefore, the probability of consecutively passing the first 3 levels is $\qquad$ | \frac{100}{243} | 84 | 11 |
math | Example 8. Given $1 x=(3+2 \sqrt{2})^{-1}$, $y=(3-2 \sqrt{2})^{-1}$, find the value of $(x+1)^{-1}+(y+1)^{-1}$. | 1 | 57 | 1 |
math | In triangle $A B C$, angle $C$ is twice angle $A$ and $b=2 a$. Find the angles of this triangle.
# | 30,90,60 | 32 | 8 |
math | The sum of the three smallest distinct divisors of some number $A$ is 8. How many zeros can the number $A$ end with?
# | 1 | 32 | 1 |
math | In a certain chess tournament, after the 7th round (i.e., 7 games), a player has 5 points. In how many ways could this result have been achieved? (A win is 1 point, a draw is $1 / 2$ point, a loss is 0 points.) | 161 | 64 | 3 |
math | Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer. | n = \overline{ab000} | 53 | 11 |
math | During a collection, a seven-member group each donated 10 forints per person. The seventh person donated 3 forints more than the average donation of the 7 people. How much did the seventh person donate? | 13.5\mathrm{Ft} | 45 | 10 |
math | 3. (BUL 1) Find all polynomials $f(x)$ with real coefficients for which
$$ f(x) f\left(2 x^{2}\right)=f\left(2 x^{3}+x\right) $$ | f(x) = 0 \text{ and } f(x) = (x^2 + 1)^k, k \in \mathbb{N}_0 | 52 | 35 |
math | 109 individuals purchased 109 books for a total value of $2845 \mathrm{Ft}$ to give as gifts. It turned out that only three different prices appeared among the books: $34 \mathrm{Ft}, 27.50 \mathrm{Ft}$, and $17.50 \mathrm{Ft}$. Determine how many books belonged to the first, second, and third price category, knowing that the number of books with the same price did not differ much from each other. | 35,36,38 | 114 | 8 |
math | ## Aufgabe 1
$$
\begin{array}{rcc}
35+8+7 & 57+6+9 & 3 \cdot 6+8 \\
71-6-7 & 44-8-8 & 28: 4-7
\end{array}
$$
| 26 | 71 | 2 |
math | Test $\mathbf{E}$ Function $f(x, y)$ satisfies for all non-negative integers $x, y$:
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f(x, f(x+1, y))$.
Determine $f(4,1981)$. | 2^{2^{1984}}-3 | 97 | 11 |
math | 11.3. Solve the equation $20[x]-14\{x\}=2014$ ([x] - the integer part of the number $x$, i.e., the greatest integer not exceeding $x,\{x\}$ - the fractional part of the number $x$ : $\{x\}=x-[x]$). | 101\frac{3}{7} | 74 | 10 |
math | 4.17. A sphere passes through the point $A(4,2,2)$ and has its center at the point $C(1,-1,-1)$. Write its equation. | (x-1)^2+(y+1)^2+(z+1)^2=27 | 41 | 20 |
math | Solve the following system of equations:
$x^{2}+y z=0$,
$v^{2}+y z=0$,
$(x+v) y=2$,
$(x+v) z=-2$. | \1,\1,\1,v=\1 | 47 | 9 |
math | One. (20 points) A batch of goods is prepared to be transported to a certain place, and there are three trucks, A, B, and C, available for hire. It is known that the cargo capacity of trucks A, B, and C remains constant each time, and trucks A and B would need $2a$ and $a$ trips, respectively, to transport this batch of goods alone. If trucks A and C work together for the same number of trips to transport the goods, truck A will transport a total of $180 \mathrm{t}$. If trucks B and C work together for the same number of trips to transport the goods, truck B will transport a total of 270 t. Now, if trucks A, B, and C work together for the same number of trips to transport the goods, how much should the cargo owner pay each truck owner (calculated at 20 yuan per ton)? | 2160 \text{ yuan, } 4320 \text{ yuan, } 4320 \text{ yuan} | 195 | 31 |
math | Solve the following system of equations:
$$
x+y+z=11
$$
$$
x^{2}+2 y^{2}+3 z^{2}=66 \text {. }
$$ | 6,3,2 | 44 | 5 |
math | 4. Calculate $\lim _{x \rightarrow 1} \frac{\sqrt{\operatorname{arctg} x}-\sqrt{\arccos \left(\frac{\sqrt{2}}{2} x\right)}}{\sqrt[3]{1+\ln [x(2 x-1)]}-\sqrt[3]{1+\ln \left(x^{3}-4 x+4\right)}}$.
Iulian Danielescu, Brăila | \frac{9}{8\sqrt{\pi}} | 100 | 11 |
math | 209. "Fibonacci Tetrahedron". Find the volume of the tetrahedron whose vertices are located at the points with coordinates $\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right)$ and $\left(F_{n+9}, F_{n+10}, F_{n+11}\right)$, where $F_{i}$ is the $i$-th term of the Fibonacci sequence: $1,1,2,3,5,8 \ldots$. | 0 | 160 | 1 |
math | Example 7 Find all positive integers $n$ that satisfy the following condition: there exist two complete residue systems modulo $n$, $a_{i}$ and $b_{i} (1 \leqslant i \leqslant n)$, such that $a_{i} b_{i} (1 \leqslant i \leqslant n)$ is also a complete residue system modulo $n$.
[2] | 1,2 | 93 | 3 |
math | II. (50 points)
Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$. Determine the maximum value of $P = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$. | \frac{10}{3} | 85 | 8 |
math | 3. (3 points) $\star+\square=24, \boldsymbol{\square}+\bullet=30, \bullet+\star=36, \boldsymbol{\square}=\underline{ }, \bullet=\underline{ }, \star=$ | 9,21,15 | 53 | 7 |
math | 6. Squirrel mom collects pine nuts, on sunny days she can collect 20 every day, and on rainy days she can only collect 12 every day. She collected 112 pine nuts in a row for several days, averaging 14 per day. How many of these days were rainy? | 6 | 65 | 1 |
math | Problem 1. Three brothers divided a certain amount of money such that the first brother received $\frac{1}{5}$ of the total amount, the second brother received $\frac{5}{8}$ of the total amount, and the third brother received the remainder of the money. Then, the third brother gave $\frac{3}{4}$ of his share to the first brother, and the remainder of his share to the second brother. What fraction of the total amount did the first brother receive? | \frac{53}{160} | 101 | 10 |
math | 1. Let $a$, $b$, $c$ be prime numbers, and satisfy $a^{5} \mid \left(b^{2}-c\right)$, where $b+c$ is a perfect square. Find the minimum value of $a b c$. | 1958 | 56 | 4 |
math | We write on the board the numbers $1, \frac{1}{2}, \ldots, \frac{1}{n}$. At each step, we choose two numbers $a$ and $b$ written on the board, erase them, and write in their place $a+b+a b$. What can be the last number that remains on the board? | n | 75 | 1 |
math | 9.5. For what minimum $\boldsymbol{n}$ in any set of $\boldsymbol{n}$ distinct natural numbers, not exceeding 100, will there be two numbers whose sum is a prime number? | 51 | 44 | 2 |
math | 3. Solve the equation $3^{1-2|x|}+3 \cdot 9^{1+|x|}=82$. (4 points) | \\frac{1}{2} | 34 | 7 |
math | Fomin S.B.
Two people toss a coin: one tossed it 10 times, the other - 11 times.
What is the probability that the second one got heads more times than the first one? | \frac{1}{2} | 44 | 7 |
math | Find all positive integers $n$ such that $2^{n}+12^{n}+2011^{n}$ is a perfect square. | 1 | 33 | 1 |
math | One, (40 points) Given a positive integer $n$, there are $3n$ numbers satisfying:
$$
0 \leqslant a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{3 n},
$$
and $\left(\sum_{i=1}^{3 n} a_{i}\right)^{3} \geqslant k_{n}\left(\sum_{i=1}^{n} a_{i} a_{n+i} a_{2 n+i}\right)$ always holds.
Find the best possible value of $k_{n}$ (expressed in terms of $n$). | 27 n^{2} | 148 | 6 |
math | 7. In $\triangle A B C$, the side lengths opposite to $\angle A 、 \angle B 、 \angle C$ are $a 、 b 、 c$, respectively, and
$$
\begin{array}{l}
\sin C \cdot \cos \frac{A}{2}=(2-\cos C) \sin \frac{A}{2}, \\
\cos A=\frac{3}{5}, a=4 .
\end{array}
$$
Then the area of $\triangle A B C$ is . $\qquad$ | 6 | 118 | 1 |
math | Determine maximum real $ k$ such that there exist a set $ X$ and its subsets $ Y_{1}$, $ Y_{2}$, $ ...$, $ Y_{31}$ satisfying the following conditions:
(1) for every two elements of $ X$ there is an index $ i$ such that $ Y_{i}$ contains neither of these elements;
(2) if any non-negative numbers $ \alpha_{i}$ are assigned to the subsets $ Y_{i}$ and $ \alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $ x\in X$ such that the sum of $ \alpha_{i}$ corresponding to all the subsets $ Y_{i}$ that contain $ x$ is at least $ k$. | \frac{25}{31} | 161 | 9 |
math | 22. [12] Let $x<y$ be positive real numbers such that
$$
\sqrt{x}+\sqrt{y}=4 \text { and } \sqrt{x+2}+\sqrt{y+2}=5 .
$$
Compute $x$. | \frac{49}{36} | 57 | 9 |
math | Example 1. In how many different ways can three people be selected for three different positions from ten candidates? | 720 | 22 | 3 |
math | 46. Calculate in the most rational way:
$$
333\left(\frac{71}{111111}+\frac{573}{222222}-\frac{2}{7 \cdot 37 \cdot 3}\right)
$$ | \frac{3}{14} | 63 | 8 |
math | $\left.\begin{array}{ll}{\left[\begin{array}{l}\text { Irrational Equations } \\ \text { [Completing the Square. Sums of Squares] }\end{array}\right]}\end{array}\right]$
Solve the equation
$$
\left(x^{2}+x\right)^{2}+\sqrt{x^{2}-1}=0
$$ | -1 | 88 | 2 |
math | 12. (12 points) Given $S=\frac{1}{9}+\frac{1}{99}+\frac{1}{999}+\cdots+\frac{1}{1000 \text { nines }}$, then the 2016th digit after the decimal point of $S$ is | 4 | 72 | 1 |
math | 297. Find the variance of the random variable $X$, given by the distribution function
$$
F(x)=\left\{\begin{array}{ccc}
0 & \text { for } & x \leqslant-2 \\
x / 4+1 / 2 & \text { for } & -2 < x \leqslant 2 \\
1 & \text { for } & x > 2
\end{array}\right.
$$ | \frac{4}{3} | 101 | 7 |
math | 6.3. A row of numbers and asterisks is written on the board: $5, *, *, *, *, *, *, 8$. Replace the asterisks with numbers so that the sum of any three consecutive numbers equals 20. | 5,8,7,5,8,7,5,8 | 49 | 15 |
math | For a positive integer $n$, let $1 \times 2 \times \cdots \times n=n!$ If $\frac{2017!}{2^{n}}$ is an integer, then the maximum value of $n$ is $\qquad$ . | 2010 | 58 | 4 |
math | 7. In a trapezoid, the diagonals are equal to 3 and 5, and the segment connecting the midpoints of the bases is equal to 2. Find the area of the trapezoid. | 6 | 47 | 1 |
math | 18. The operation $\diamond$ is defined on two positive whole numbers as the number of distinct prime factors of the product of the two numbers. For example $8 \diamond 15=3$.
What is the cube of the value of $(720 \diamond 1001)$ ? | 216 | 64 | 3 |
math | Blinkov A. A:
The teams held a football tournament in a round-robin format (each team played one match against every other team, with 3 points for a win, 1 point for a draw, and 0 points for a loss). It turned out that the sole winner scored less than $50 \%$ of the maximum possible points for one participant. What is the minimum number of teams that could have participated in the tournament? | 6 | 92 | 1 |
math | 1. 50 students from fifth to ninth grade published a total of 60 photos on Instagram, each not less than one. All students of the same grade (same parallel) published an equal number of photos, while students of different grades (different parallels) published a different number. How many students published only one photo? | 46 | 67 | 2 |
math | If $ a\equal{}2b\plus{}c$, $ b\equal{}2c\plus{}d$, $ 2c\equal{}d\plus{}a\minus{}1$, $ d\equal{}a\minus{}c$, what is $ b$? | b = \frac{2}{9} | 59 | 9 |
math | Problem 5. Timofey placed 10 grid rectangles on a grid field, with areas of $1, 2, 3, \ldots, 10$ respectively. Some of the rectangles overlapped each other (possibly completely, or only partially). After this, he noticed that there is exactly one cell covered exactly once; there are exactly two cells covered exactly twice; there are exactly three cells covered exactly three times, and exactly four cells covered exactly four times. What is the maximum number of cells that could be covered at least five times? The area of a grid rectangle is the number of cells it contains. Each rectangle lies on the field exactly along the grid cells. (20 points) | 5 | 146 | 1 |
math | 378. Two bodies simultaneously leave one point: one with a speed of $v_{1}=5 t \mathrm{~m} / \mathrm{c}$, the other with a speed of $v_{2}=3 t^{2} \mathrm{~m} / \mathrm{c}$. At what distance from each other will they be after $20 \mathrm{c}$, if they move in a straight line in the same direction? | 7000(\mathrm{}) | 96 | 7 |
math | 6. For what values of the parameter a does the equation $\left(x^{2}-a\right)^{2}+2\left(x^{2}-a\right)+(x-a)+2=0$ have exactly one solution? Specify the solution for the found values of the parameter a. (20 points) | =0.75,\quadx_{1}=-0.5 | 66 | 15 |
math | ## 263. Math Puzzle $4 / 87$
In a parking lot, there are cars, mopeds, and motorcycles with sidecars. There are a total of 16 vehicles with 50 wheels (excluding spare wheels). There are as many cars as motorcycles with sidecars.
How many vehicles of each type are parked? | 6 | 72 | 1 |
math | 9. (Adapted from the 1st "Hope Cup" Senior High School Competition) Let the function $f(n)=k$, where $n$ is a natural number, and $k$ is the digit at the $n$-th position after the decimal point of the irrational number $\pi=3.1415926535 \cdots$, with the rule that $f(0)=3$. Let $F_{n}=$ $\underbrace{f\{f\{f\{f\{f}(n)\} \cdots\}\}$, then $F[f(1990)+f(5)+f(13)]=$ $\qquad$. | 1 | 150 | 1 |
math | ## Task A-2.1.
Determine, if it exists, the real parameter $k$ such that the maximum value of the function
$$
f_{1}(x)=(k-8) x^{2}-2(k-5) x+k-9
$$
is equal to the minimum value of the function
$$
f_{2}(x)=(k-4) x^{2}-2(k-1) x+k+7
$$ | k=5,k=7 | 95 | 6 |
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