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 | 6. Given that $n, k$ are positive integers, $n>k$. Given real numbers $a_{1}, a_{2}, \cdots, a_{n} \in(k-1, k)$. Let positive real numbers $x_{1}, x_{2}$, $\cdots, x_{n}$ satisfy that for any $k$-element subset $I$ of $\{1,2, \cdots, n\}$, we have $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$. Find the maximum ... | a_{1} a_{2} \cdots a_{n} | 159 | 15 |
math |
Let $a$, $b$, $c$, $d$, $e$, $f$ be integers selected from the set $\{1,2,\dots,100\}$, uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when $M$ is divided by $64$? | \frac{63}{2} | 99 | 8 |
math | Determine $x^2+y^2+z^2+w^2$ if
$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{... | 36 | 254 | 2 |
math | 1. Find all $x$ for which $2[x]+\{3 x\}=\frac{7}{3}$, where $[x]$ is the integer part of $x$, and $\{x\}$ is the fractional part of $x$, that is, $\{x\}=x-[x]$. | {1\frac{1}{9};1\frac{4}{9};1\frac{7}{9}} | 66 | 25 |
math | 1. Given that $a$ and $b$ are integers, $\frac{127}{a}-\frac{16}{b}=1$. Then the maximum value of $b$ is $\qquad$ . | 2016 | 47 | 4 |
math | Problem 4. Determine the functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ with the property that $f(x+3 f(y))=f(x)+f(y)+2 y$, for all $x, y \in \mathbb{Q}$. | f(x)=x | 61 | 4 |
math | ## 7. Natural Fraction
What is the sum of all natural numbers $n$ for which the fraction
$$
\frac{6 n+900}{2 n-3}
$$
is a natural number?
Result: $\quad 672$ | 672 | 56 | 3 |
math | 47.1. Find the sum of all three-digit numbers divisible by 7.
$(9-10$ grade $)$
6 | 70336 | 29 | 5 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{3 ; 3 ;-1\}$
$p=\{3 ; 1 ; 0\}$
$q=\{-1 ; 2 ; 1\}$
$r=\{-1 ; 0 ; 2\}$ | p+q-r | 74 | 4 |
math | 1. (3 points) The sum of the digits of the result of $100$ ones $+50$ twos $+25$ fours $+2010$ is $\qquad$
Translating the text as requested, while retaining the original formatting and structure. | 303 | 62 | 3 |
math | SUBIECTUL III
(7p)Consider the triangle $A B C, A E$ the bisector of $\angle B A C$, such that [AE] $\equiv[E C]$. Find the measure of $\angle A B C$ if $A C=2 A B$. | 90 | 61 | 2 |
math | 9. (16 points) Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=1,\left|a_{n+1}-a_{n}\right|=p^{n}(n=1,2, \cdots) \text {, }
$$
and $\left\{a_{2 n-1}\right\}$ is a monotonically increasing sequence, $\left\{a_{2 n}\right\}$ is a strictly monotonically decreasing sequence. Find the general term formula of ... | a_{n}={\begin{pmatrix}1,&p=0\\\frac{1+(-1)^{n-1}}{2},&p=1\\\frac{1-(-p)^{n}}{1+p},&0<p<1 | 130 | 56 |
math | Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, ag... | \frac{130}{3} | 149 | 9 |
math | 2. Dima, Sasha, Kolya, and Gleb participated in an olympiad and took the first four places. A year later, their classmates managed to recall only three facts: "Dima took first place or Gleb took third," "Gleb took second place or Kolya took first," "Sasha took third place or Dima took second." Who performed better - Di... | Dima | 113 | 2 |
math | Determine the largest 6 -digit positive integer that is divisible by 5 . | 999995 | 17 | 6 |
math | 1. Let $p$ be a prime, $\left(a, p^{2}\right)=p,\left(b, p^{3}\right)=p^{2}$. Find $\left(a b, p^{4}\right), \quad\left(a+b, p^{4}\right)$. | p^3, p | 62 | 5 |
math | Task B-3.1. Solve the equation
$$
\log _{5 x-2} 2+2 \cdot \log _{5 x-2} x=\log _{5 x-2}(x+1)
$$ | 1 | 52 | 1 |
math | Example 9 (APMO Problem) Let $f$ be a function from $\mathbf{R} \rightarrow \mathbf{R}$, and
(1) For any $x, y \in \mathbf{R}, f(x)+f(y)+1 \geqslant f(x+y) \geqslant f(x)+f(y)$;
(2) For any $x \in[0,1)$, $f(0) \geqslant f(x)$;
(3) $-f(-1)=f(1)=1$.
Find all functions $f$ that satisfy the conditions. | f(x)=[x] | 134 | 6 |
math | ## Task A-4.1.
Determine all complex numbers $z$ that satisfy the equations
$$
|z+1|=1 \quad \text{and} \quad\left|z^{2}+1\right|=1
$$ | z_{1}=0,\;z_{2,3}=\frac{-1\\sqrt{3}i}{2} | 53 | 26 |
math | Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions. | 168 | 34 | 3 |
math | 1. Determine all positive integers $x$, such that $2x+1$ is a perfect square, but among the numbers $2x+2, 2x+3, \cdots, 3x+2$, there is no perfect square. | 4 | 54 | 1 |
math | 741*. Which years of the $XX$ century can be represented in the form $2^{n}-2^{k}$, where $n$ and $k$ are natural numbers? Find all such years. | 1984,1920 | 45 | 9 |
math | There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maxima... | 50 | 95 | 2 |
math | Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.
How many games did father play with uncle?
(M. Volfová) | 16 | 69 | 2 |
math | 2. A truck traveling at $60 \mathrm{~km} / \mathrm{h}$ started from city $A$ to city $B$. After some time, a car traveling at $90 \mathrm{~km} / \mathrm{h}$ also started from city $A$ to city $B$. It was planned for the car to catch up with the truck in city $B$. However, after covering $\frac{2}{3}$ of the distance, t... | 200 | 168 | 3 |
math | [ equations in integers $]$
Solve the equation $9 x+2=(y+1) y$ in integers.
# | (k(k+1),3k+1),k\in{Z} | 27 | 16 |
math | [ [equations in integers $]$ ]
In the room, there are three-legged stools and four-legged chairs. When people sat on all these seats, there were 39 legs in the room.
How many stools are in the room? | 3 | 49 | 1 |
math | 3. Some integers, when divided by $\frac{5}{7}, \frac{7}{9}, \frac{9}{11}, \frac{11}{13}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{5}, \frac{2}{7}, \frac{2}{9}, \frac{2}{11}$ respectively. The smallest integer greater than 1 that satisfies these conditions ... | 3466 | 104 | 4 |
math | In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played? | 1535 | 39 | 4 |
math | Three, let $S_{\mathrm{n}}$ denote the sum of the elements of all subsets of the set of natural numbers $\{1,2,3, \cdots, n\}$ (the sum of elements of the empty set is defined as 0). Find $S_{1980}$. | 2^{1978} \cdot 1980 \cdot 1981 | 66 | 21 |
math | Example 12 Given $f(\theta)=\sin ^{2} \theta+\sin ^{2}(\theta+\alpha)$ $+\sin ^{2}(\theta+\beta)$, where $\alpha, \beta$ are constants satisfying $0 \leqslant \alpha \leqslant \beta \leqslant \pi$. For what values of $\alpha, \beta$ is $f(\theta)$ a constant value? | \alpha=\frac{\pi}{3}, \beta=\frac{2 \pi}{3} | 96 | 20 |
math | 16. Let $x_{1}, x_{2}, x_{3}, x_{4}$ be non-negative real numbers satisfying
$$
x_{1}+x_{2}+x_{3}+x_{4}=1
$$
Find the maximum value of $\sum_{1 \leqslant i<j \leqslant 4}\left(x_{i}+x_{j}\right) \sqrt{x_{i} x_{j}}$, and determine the values of $x_{1}, x_{2}, x_{3}, x_{4}$ when the maximum value is achieved. | \frac{3}{4} | 129 | 7 |
math | A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks. | 7 | 43 | 1 |
math | 1. Given an integer $n \geq 2$, let $f(n)$ be the second largest positive divisor of $n$. For example, $f(12)=6$ and $f(13)=1$. Determine the largest positive integer $n$ such that $f(n)=35$. | 175 | 65 | 3 |
math | 2. On the day when Dima's brother and sister were congratulating him on his birthday, Dima said: "Look how interesting, I am now twice as old as my brother and three times as old as my sister!" - "And your average age is 11 years," - added Dad. How old did Dima turn? | 18 | 70 | 2 |
math | 8.2. Two cyclists, Andrey and Boris, are riding at a constant and identical speed along a straight highway in the same direction, so that the distance between them remains constant. There is a turnoff to a village ahead. At some point in time, the distance from Andrey to the turnoff was equal to the square of the dista... | 2 | 119 | 1 |
math | 7. Let $n$ be a given integer greater than 2. There are $n$ indistinguishable bags, the $k$-th bag contains $k$ red balls and $n-k$ white balls $(k=1,2, \cdots, n)$. From each bag, 3 balls are drawn consecutively (without replacement), and the third ball drawn from each bag is placed into a new bag. The mathematical ex... | \frac{n-1}{2} | 119 | 8 |
math | 5. On a circle, 25 points are marked, painted either red or blue. Some of the points are connected by segments, with one end of each segment being blue and the other end red. It is known that there do not exist two red points that belong to the same number of segments. What is the maximum possible number of red points? | 13 | 72 | 2 |
math | 5. For a positive integer $n$, denote by $\varphi(n)$ the number of positive integers $k \leq n$ relatively prime to $n$. How many positive integers $n$ less than or equal to 100 are divisible by $\varphi(n)$ ? | 16 | 59 | 2 |
math | Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC =60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
[i]Proposed by Kaan Dokmeci[/i] | 408 | 89 | 3 |
math | 5. Given $\cos \left(\theta+\frac{\pi}{4}\right)<3\left(\sin ^{5} \theta-\cos ^{5} \theta\right)$, where $\theta \in[-\pi, \pi]$. Then the range of $\theta$ is | \left[-\pi,-\frac{3 \pi}{4}\right) \cup\left(\frac{\pi}{4}, \pi\right] | 62 | 33 |
math | Example 4 Given $u+v=96$, and the quadratic equation $x^{2} + u x + v = 0$ has integer roots, then its largest root is $\qquad$
(1996, Anhui Province Partial Areas Junior High School Mathematics League) | 98 | 60 | 2 |
math | Example 1 (Sichuan Preliminary Contest 2004) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n}=2 a_{n-1}+n-2(n \geqslant 2)$, find the general term $a_{n}$. | a_{n}=2^{n}-n(n\geqslant1) | 73 | 17 |
math | N19 (19-5, German Democratic Republic) Let $a, b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b$, the quotient is $q$, and the remainder is $r$. Find all pairs $(a, b)$ such that $q^{2}+r=1977$.
When $a^{2}+b^{2}$ is divided by $a+b$, the quotient is $q$, and the remainder is $r$, find all pairs $(a, b)$... | (50,7),(50,37),(7,50),(37,50) | 132 | 23 |
math | 4.28 Let \( m \) and \( n \) be natural numbers. Determine the minimum number of distinct prime divisors of the natural number
\[
m(n+9)\left(m+2 n^{2}+3\right)
\]
(53rd Moscow Mathematical Olympiad, 1990) | 2 | 69 | 1 |
math | 【Example 3】How many four-digit numbers without repeated digits can be formed from the digits $1,2,3,4,5,6,7,8,9$? | 3024 | 39 | 4 |
math | Problem 5. Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}=(y-z)^{2}-8 \\
y^{2}=(z-x)^{2}-16 \\
z^{2}=(x-y)^{2}+32
\end{array}\right.
$$ | (-1,-3,-6),(1,3,6) | 72 | 13 |
math | How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$? | 42 | 49 | 2 |
math | 430. A man, on his death, left a will according to which the eldest son receives 100 francs and one-tenth of the remainder, the second son - 200 francs and one-tenth of the new remainder, the third - 300 francs and one-tenth of the new remainder, and so on until the last. In this way, the shares of all the sons should ... | 8100, | 118 | 5 |
math | Example 7 Find the real solutions of the system of equations:
$$
\left\{\begin{array}{l}
(x-2)^{2}+\left(y+\frac{3}{2}\right)^{2}+(z-6)^{2}=64, \\
(x+2)^{2}+\left(y-\frac{3}{2}\right)^{2}+(z+6)^{2}=25 .
\end{array}\right.
$$ | \left(-\frac{6}{13}, \frac{9}{26},-\frac{18}{13}\right) | 100 | 30 |
math | 37. Compute
$$
\sum_{n=1}^{\infty} \frac{2 n+5}{2^{n} \cdot\left(n^{3}+7 n^{2}+14 n+8\right)}
$$ | \frac{137}{24}-8\ln2 | 55 | 14 |
math | 1. After pulling out the turnip, a lot of earth was left. The Grandpa removed half of the pile, Grandma removed a third of the remaining earth, the Granddaughter removed a quarter of the remainder, the Dog removed a fifth of the remainder, the Cat removed a sixth, and the Mouse removed a seventh. What part of the pile ... | \frac{1}{7} | 74 | 7 |
math | Alex, Pei-Hsin, and Edward got together before the contest to send a mailing to all the invited schools. Pei-Hsin usually just stuffs the envelopes, but if Alex leaves the room she has to lick them as well and has a $25 \%$ chance of dying from an allergic reaction before he gets back. Licking the glue makes Edward a b... | \frac{1}{2} | 132 | 7 |
math | Example 4 Given that $a, b, c$ satisfy $a+b+c=0, abc=8$. Then the range of values for $c$ is $\qquad$ | c<0 \text{ or } c \geqslant 2 \sqrt[3]{4} | 38 | 23 |
math | Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed ... | 31 | 160 | 2 |
math | Solve the following system of equations on the set of real numbers:
$$
\begin{aligned}
4 a+b c & =32 \\
2 a-2 c-b^{2} & =0 \\
a+12 b-c-a b & =6
\end{aligned}
$$ | \begin{pmatrix}a_{1}=11.5,&b_{1}=-4,&c_{1}=3.5\\a_{2}=6.5,&b_{2}=1,&c_{2}=6\\a_{3}=14,&b_{3}=6,&c_{3}=-4\\a_{4}=14,&b_{4}=-2,&c_{4} | 61 | 88 |
math | 7.3. A trading organization wholesale purchased exotic fruits, the moisture content of which was $99\%$ of their mass. After delivering the fruits to the market, the moisture content dropped to $98\%$. By what percentage should the trading organization increase the retail price of the fruits (the price at which it will... | 100 | 90 | 3 |
math | 298. Solve the equation:
$$
x^{3}-2 x^{2}-\left(a^{2}-a-1\right) x+a^{2}-a=0
$$ | 1,,1- | 41 | 4 |
math | Three. (25 points) A positive integer $M$ has 8 positive divisors. Xiao Feng, while calculating the sum of the 8 positive divisors of $M$, forgot to add one of the divisors, resulting in a sum of 2776. Find all possible values of the positive integer $M$.
| 2008 | 70 | 4 |
math | 1. Calculate $\sqrt{(31)(30)(29)(28)+1}$. | 869 | 21 | 3 |
math | 6. Let the vectors $\boldsymbol{\alpha}=(x+3, x), \boldsymbol{\beta}=(2 \sin \theta \cos \theta, a \sin \theta+a \cos \theta)$ satisfy that for any $x \in \mathbb{R}$ and $\theta \in\left[0, \frac{\pi}{2}\right]$, we have $|\boldsymbol{\alpha}+\boldsymbol{\beta}| \geq \sqrt{2}$ always holds. Then the range of the real ... | \leq1or\geq5 | 121 | 9 |
math | Tímár Mihály found himself in a difficult situation because the red crescent fell off the bag where the treasure was hidden. All he knows is that the treasure is hidden in the heaviest of the four bags, buried in the wheat. It was revealed during three measurements that the first bag together with the second is lighter... | D | 98 | 1 |
math | 10. (40 points) Let $P(n)$ denote the product of the digits of a natural number $n$. For what largest natural number $k$ does there exist a natural number $n>10$ such that
$$
P(n)<P(2 n)<\ldots<P(k n) ?
$$ | 9 | 68 | 1 |
math | Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it.
[i]Dan Brânzei[/i] | n+1 | 38 | 3 |
math | Example 1. Integrate the system of equations
$$
\left\{\begin{array}{l}
\frac{d x}{d t}=y+1 \\
\frac{d y}{d t}=x+1
\end{array}\right.
$$ | C_{1}e^{}+C_{2}e^{-}-1,\quadC_{1}e^{}-C_{2}e^{-}-1 | 57 | 34 |
math | 5. Let the sequences $\left\{x_{n}\right\},\left\{y_{n}\right\}$ satisfy $x_{n}+i y_{n}=\left(\frac{\sqrt{3} i-1}{2}\right)^{n}$, find the sum of the first 1994 terms of the sequence $\left\{x_{n}\right\}$, $S_{1994}$. | -1 | 96 | 2 |
math | A courier departs from $A$ towards $B$; at the same time, another courier departs from $B$ towards $A$. They meet after 8 hours. How many $\mathrm{km}$ does each courier travel per hour, if the distance from $A$ to $B$ is $176 \mathrm{~km}$ and it takes one courier 1 hour longer to travel $60 \mathrm{~km}$ than it take... | 12 | 100 | 2 |
math | On the board, only the numbers $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$, and $\frac{1}{6}$ are written. We can write on the board the sum or the product of any two numbers from the board. Is it possible to achieve by such writing that the following numbers appear on the board: a) $\frac{1}{60}$, b) $\fra... | )\frac{1}{60},b)\frac{2011}{375},)proof | 133 | 23 |
math | 1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additi... | \frac{5\sqrt{13}}{3\sqrt{2}} | 140 | 17 |
math | 1.2. In triangle $A B C$, the altitude $A H$ and the bisector $A L$ were drawn. It turned out that $B H: H L: L C=1: 1: 4$ and side $A B$ is equal to 1. Find the lengths of the other sides of triangle $A B C$. | AC=2,BC=3/\sqrt{2} | 76 | 12 |
math | In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled w... | \left\lceil \frac{n}{2} \right\rceil | 180 | 16 |
math | 1. How many ordered (integer) quadruples $(i, j, k, h)$ satisfy the condition $1 \leqslant i<j \leqslant k<h \leqslant n+1$ ? | C_{n+2}^{4} | 48 | 9 |
math | 9. Put 12 small balls marked with numbers $1, 2, 3, \cdots \cdots, 12$ into a paper box. Three people, A, B, and C, each take 4 balls from the box. It is known that the sum of the numbers on the balls they each took is equal. A has two balls marked with numbers 6 and 11, B has two balls marked with numbers $4$ and $8$,... | 3,10,12 | 126 | 7 |
math | $\underline{\text { Tolkpy A.K. }}$
A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^{\circ}$.
Name the largest number $A$ such that in any such division, each of the seven arcs contains at least $A^{\circ}$. | 51 | 70 | 2 |
math | A servant confesses to his master that he stole one liter of wine from a 250-liter barrel every day for 50 days and always replaced the missing amount with water. What percentage of wine was still in the barrel? | 81.846 | 48 | 6 |
math | Find all primes that can be written both as a sum of two primes and as a difference of two primes. | 5 | 22 | 1 |
math | 12.113. The base of the pyramid $ABCD$ is a right triangle $ABC\left(\angle C=90^{\circ}\right)$. The lateral edge $AD$ is perpendicular to the base. Find the acute angles of triangle $ABC$, if $\angle DBA=\alpha$ and $\angle DBC=\beta(\alpha<\beta)$. | \arcsin\frac{\cos\beta}{\cos\alpha};\arccos\frac{\cos\beta}{\cos\alpha} | 80 | 32 |
math | 5.1.1. A particle moves in the plane $O x y$ of a rectangular coordinate system according to the law (here $x, y-$ are the coordinates of the point in meters, $t$ - time in seconds):
$$
x(t)=3+\sin t \cdot \cos t-\sin t-\cos t ; y(t)=1
$$
In the same plane, a light ray constantly emanates from the origin according to... | \in[\frac{2(7-2\sqrt{2})}{41};\frac{1}{2}] | 137 | 26 |
math | An urn contains $k$ balls marked with $k$, for all $k=1,2, \ldots, 2016$. What is the minimum number of balls we must withdraw, without replacement and without looking at the balls, to be sure that we have 12 balls with the same number? | 22122 | 66 | 5 |
math | Example 9. Find $\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}$. | 6 | 30 | 1 |
math | Problem 9.2. During the first half of the year, lazy Pasha forced himself to solve math problems. Every day he solved no more than 10 problems, and if on any day he solved more than 7 problems, then for the next two days he solved no more than 5 problems per day. What is the maximum number of problems Pasha could solve... | 52 | 83 | 2 |
math | (9) (20 points) Let $A+B+C=180^{\circ}$, and satisfy: $\frac{\sin A+\sin B+\sin C}{\cos A+\cos B+\cos C}=$
1, find the value of $\frac{\cos 2 A+\cos 2 B+\cos 2 C}{\cos A+\cos B+\cos C}$. | 2 | 84 | 1 |
math | 9. Let the line $l: y=k x+m$ (where $k, m$ are integers) intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$ at two distinct points $A, B$, and intersect the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{12}=1$ at two distinct points $C, D$. Does there exist a line $l$ such that the vector $\overrightarrow{A C}+\over... | 9 | 143 | 1 |
math | What is the $21^{\varrho}$ term of the sequence
$$
1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \ldots ?
$$ | 4641 | 63 | 4 |
math | 383. A random variable $X$ is uniformly distributed in the interval $(-\pi / 2, \pi / 2)$. Find the density function $g(y)$ of the random variable $Y=\sin X$. | (y)=\frac{1}{\pi\sqrt{1-y^{2}}},\quad-1<y<1 | 49 | 25 |
math | $4.$ Harry, Hermione, and Ron go to Diagon Alley to buy chocolate frogs. If Harry and Hermione spent one-fourth of their own money, they would spend $3$ galleons in total. If Harry and Ron spent one-fifth of their own money, they would spend $24$ galleons in total. Everyone has a whole number of galleons, and the numbe... | 6 | 115 | 1 |
math | Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$.
When does the equality occur? | \sqrt[3]{abc} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 2\sqrt{3} | 61 | 42 |
math | Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies:
$\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$.
$\bullet$ $f (1998) = 2$
Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is sati... | 1997003 | 111 | 7 |
math | 13.236. The master is giving a simultaneous chess exhibition on several boards. By the end of the first two hours, he won $10 \%$ of the total number of games played, while 8 opponents drew their games with the master. Over the next two hours, the master won $10 \%$ of the games with the remaining opponents, lost two g... | 20 | 98 | 2 |
math | 8. If the internal radius $R=2+\sqrt{6} \mathrm{~cm}$ of a hollow sphere contains 4 solid spheres of radius $r$, then the maximum value that $r$ can take is
保留了原文的换行和格式。 | 2 | 56 | 1 |
math | Problem 10. A cylinder of volume 9 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a frustum of volume 63 from the original cone. Find the volume of the original cone. | 64 | 50 | 2 |
math | 1. In $\triangle A B C$, it is known that: $a=2, b=2 \sqrt{2}$. Find the range of values for $\angle A$. | A\in(0,\frac{\pi}{4}] | 38 | 12 |
math | 1. Let $A, B, C, D$ be four points on a circle in that order. Also, $A B=3, B C=5, C D=6$, and $D A=4$. Let diagonals $A C$ and $B D$ intersect at $P$. Compute $\frac{A P}{C P}$. | \frac{2}{5} | 75 | 7 |
math | 8. Let $\triangle A B C$ have three interior angles in arithmetic progression, and the reciprocals of the three corresponding sides $a, b, c$ also form an arithmetic progression. Try to find $A, B, C$. | A=B=C=\frac{\pi}{3} | 50 | 10 |
math | Question 150: Four spheres with a radius of 1 are pairwise externally tangent. Then the volume of the circumscribed regular tetrahedron is $\qquad$ . | \frac{\sqrt{3}(6+\sqrt{6})^{3}}{27} | 38 | 20 |
math | 1. The solution to the equation $\frac{1+3^{x}+3^{2 x}+\cdots+3^{2010 x}}{1+3^{-x}+3^{-2 x}+\cdots+3^{-2010 x}}=3$ is $x=$ | x=\frac{1}{2010} | 66 | 11 |
math | $\begin{array}{c}\text { Example 3. Simplify: } \sin \theta \sin \left(\frac{\pi}{2}-\theta\right)+\sin ^{2}\left(\frac{\pi}{4}+\theta\right) \\ -\sin ^{2}\left(\frac{\pi}{6}-\theta\right)-\frac{1}{2} \sin \left(\frac{\pi}{6}+2 \theta\right) .\end{array}$ | \sin 2 \theta | 108 | 6 |
math | 3. (5 points) If $\overline{\mathrm{abc}}$ and $\overline{\operatorname{def}}$ are two three-digit numbers, and $a=b+1, b=c+2, \overline{\mathrm{abc}} \times 3+4=\overline{\operatorname{def}}$, then $\overline{\mathrm{def}}$ $=$ $\qquad$ | 964 | 85 | 3 |
math | Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\] | 6 | 49 | 3 |
math | 13.1. From 22 kg of fresh mushrooms, 2.5 kg of dried mushrooms are obtained, which still contain $12 \%$ water. What percentage of water is contained in fresh mushrooms?
$$
\text { (5-7 grades) }
$$ | 90 | 58 | 2 |
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