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 | 943. A cylindrical tank must hold $V \Omega$ of water. What should its dimensions be so that the surface area (without the lid) is the smallest? | \sqrt[3]{\frac{V}{\pi}} | 36 | 13 |
math | Let $\ell$ be a line and let points $A$, $B$, $C$ lie on $\ell$ so that $AB = 7$ and $BC = 5$. Let $m$ be the line through $A$ perpendicular to $\ell$. Let $P$ lie on $m$. Compute the smallest possible value of $PB + PC$.
[i]Proposed by Ankan Bhattacharya and Brandon Wang[/i] | 19 | 92 | 2 |
math | 26*. The Spartakiad lasted $n$ days; during which $N$ sets of medals were awarded; on the 1st day, 1 set of medals and $1 / 7$ of the remaining quantity were awarded; on the 2nd day - 2 sets of medals and $1 / 7$ of the remaining quantity; ...; on the penultimate, $(n-1)$-th day - $(n-1)$ sets of medals and $1 / 7$ of ... | n=6,N=36 | 147 | 7 |
math | Example 7 Let $M=\{1,2, \cdots, 10\}, A_{1}, A_{2}, \cdots, A_{n}$ be distinct non-empty subsets of $M$, such that when $i \neq j$,
$A_{i} \cap A_{j}$ has at most two elements. Find the maximum value of $n$.
| 175 | 83 | 3 |
math | When the line with equation $y=-2 x+7$ is reflected across the line with equation $x=3$, the equation of the resulting line is $y=a x+b$. What is the value of $2 a+b$ ? | -1 | 49 | 2 |
math | 1. Given two groups of numerical sequences, each consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1,2,3, \ldots, 15$, and their differences are respectively $2,4,6, \ldots, 30$. The second group of progressions has the same first terms $1,2,3... | \frac{160}{151} | 147 | 11 |
math | Problem 9.3. The farmer said: "I have $N$ rabbits. Long ears are exactly on 13 of them. And exactly 17 of them can jump far."
The traveler rightly noted: "Therefore, among your rabbits, there are at least 3 rabbits that simultaneously have long ears and can jump far."
What is the largest value that the number $N$ can... | 27 | 83 | 2 |
math | Three. (50 points) Let $n$ be a given positive integer greater than 1. For any $d_{1}, d_{2}, \cdots, d_{n}>1$, when
$$
\sum_{1 \leqslant i<j \leqslant n} d_{i} d_{j}=(n-1) \sum_{i=1}^{n} d_{i}
$$
find the maximum value of
$$
S=\sum_{i=1}^{n}\left[d_{i} \sum_{j=1}^{n} d_{j}-n d_{i}+n(n-2)\right]^{-1}
$$ | \frac{1}{n} | 148 | 7 |
math | ## Task A-2.2.
Determine all triples of natural numbers ( $p, m, n$ ) such that $p$ is a prime number and
$$
p^{m} - n^{3} = 8
$$ | (2,4,2),(3,2,1) | 51 | 13 |
math | 7. If $a$ is the positive root of the equation $x^{2}+3 x-2=0$, and $b$ is the root of the equation $x+\sqrt{x+1}=3$, then $a+b=$ $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 2 | 80 | 1 |
math | Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$. | 21 | 57 | 2 |
math | 1. In a row, the numbers $1,2,3, \ldots, 2014,2015$ are written. We will call a number from this row good if, after its removal, the sum of all the remaining 2014 numbers is divisible by 2016. Find all the good numbers. | 1008 | 75 | 4 |
math | Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Kat... | \frac{25}{72} | 88 | 9 |
math | 1. In the set of natural numbers, solve the equation
$$
3^{x}-5^{y}=z^{2}
$$ | (x,y,z)=(2,1,2) | 28 | 10 |
math | 7. A, B, and C are playing a game. A thinks of a two-digit number, then multiplies this two-digit number by 100. B thinks of a number, then multiplies this one-digit number by 10. C thinks of a one-digit number, then multiplies this number by 7. Finally, the products of the three people are added together, and the resu... | 23 | 113 | 2 |
math | Task B-1.1. How many zeros does the number $N=x^{3}+3 x^{2}-9 x-27$ end with, if $x=999997$? | 12 | 46 | 2 |
math | 10. Koschei is counting his gold coins. When he counts them by tens, there are 7 coins left, and he is 3 coins short of a whole number of dozens. Koschei's wealth is estimated at $300-400$ coins. How many coins does Koschei have? | 357 | 68 | 3 |
math | Find the largest natural number, all digits in the decimal representation of which are different and which
is reduced by 5 times if the first digit is erased. | 3750 | 32 | 4 |
math | 3. For the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, the right focus is $F$, and the right directrix is $l$. Points $P_{1}, P_{2}, \cdots, P_{24}$ are 24 points arranged in a counterclockwise order on the ellipse, where $P_{1}$ is the right vertex of the ellipse, and $\angle P_{1} F P_{2}=\angle P_{2} F P_{3}=\cdots... | 6\sqrt{5} | 170 | 6 |
math | Arrange the following number from smallest to largest: $2^{1000}, 3^{750}, 5^{500}$. | 2^{1000}<5^{500}<3^{750} | 32 | 19 |
math | Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$ | 2011 | 41 | 4 |
math | Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+$ $P(d)$, where $P(X)=X^{6}-X^{5}-X^{4}-X^{3}-X$. | -2 | 70 | 2 |
math | 7. (3 points) In a class of 30 people participating in a jump rope competition, at the beginning, 4 people were late and did not participate in the competition, at which point the average score was 20. Later, these 4 students arrived at the competition venue and jumped $26, 27, 28, 29$ times respectively. At this point... | 21 | 99 | 2 |
math | Tobias downloads $m$ apps. Each app costs $\$ 2.00$ plus $10 \%$ tax. He spends $\$ 52.80$ in total on these $m$ apps. What is the value of $m$ ?
(A) 20
(B) 22
(C) 18
(D) 24
(E) 26 | 24 | 86 | 2 |
math | Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors. | 162000 | 28 | 6 |
math | 8. The number of non-empty subsets of the set $\{1,2, \cdots, 2009\}$ whose elements sum to an odd number is $\qquad$. | 2^{2008} | 40 | 7 |
math | Define a function $f$ on the real numbers by
$$
f(x)= \begin{cases}2 x & \text { if } x<1 / 2 \\ 2 x-1 & \text { if } x \geq 1 / 2\end{cases}
$$
Determine all values $x$ satisfying $f(f(f(f(f)))))=x$. | 0,\frac{1}{31},\frac{2}{31},\ldots,\frac{30}{31},1 | 83 | 30 |
math | 155. Find $\lim _{x \rightarrow 0}\left(\frac{3+x}{3}\right)^{1 / x}$. | \sqrt[3]{e} | 32 | 7 |
math | 3.252. $\frac{\cos ^{2}(4 \alpha-3 \pi)-4 \cos ^{2}(2 \alpha-\pi)+3}{\cos ^{2}(4 \alpha+3 \pi)+4 \cos ^{2}(2 \alpha+\pi)-1}$. | \operatorname{tg}^{4}2\alpha | 67 | 12 |
math | A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$. | 1 \text{ or } 11 | 87 | 9 |
math | Example 2. In Rt $\triangle A B C$, $\angle C=90^{\circ}, \angle A B C$ $=66^{\circ}, \triangle A B C$ is rotated around $C$ to the position of $\triangle A^{\prime} B^{\prime} C^{\prime}$, with vertex $B$ on the hypotenuse $A^{\prime} B^{\prime}$, and $A^{\prime} C$ intersects $A B$ at $D$. Find $\angle B D C$. (1993,... | 72^{\circ} | 137 | 6 |
math | Which is the three-digit (integer) number that, when increased or decreased by the sum of its digits, results in a number consisting of the same digit repeated? | 105 | 33 | 3 |
math | 4. A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex with one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How m... | a_{n}=3^{n}+2+(-1)^{n} | 100 | 17 |
math | 11. Let the three interior angles $A, B, C$ of $\triangle ABC$ form an arithmetic sequence, and the reciprocals of the three side lengths $a, b, c$ also form an arithmetic sequence. Try to find $A, B, C$. | A=B=C=60 | 58 | 6 |
math | 8. (8 points) For a natural number $N$, if at least eight of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an "Eight Immortals Number". The smallest "Eight Immortals Number" greater than 2000 is $\qquad$ . | 2016 | 71 | 4 |
math | Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remai... | 100 | 138 | 3 |
math | 74. The cathetus of a right-angled triangle is a perfect cube, the other cathetus represents the difference between this cube and its side (i.e., the first power), and the hypotenuse is the sum of the cube and its side. Find the sides.
## Problems of Iamblichus. | 10,6,8 | 65 | 6 |
math | Example 2 Let the positive integer $n>4$ satisfy that the decimal representation of $n!$ ends with exactly $k$ zeros, and the decimal representation of $(2n)!$ ends with exactly $3k$ zeros. Find such an $n$.
(Adapted from the 2015 National High School Mathematics Joint Competition Additional Problem) | 8,9,13,14 | 76 | 9 |
math | Determine the integers $n, n \ge 2$, with the property that the numbers $1! , 2 ! , 3 ! , ..., (n- 1)!$ give different remainders when dividing by $n $.
| n \in \{2, 3\} | 50 | 11 |
math | 9.077. $2^{x+2}-2^{x+3}-2^{x+4}>5^{x+1}-5^{x+2}$. | x\in(0;\infty) | 38 | 9 |
math | 2. The function $f(x)=x^{2}-t x+2$ has an inverse function on $[-1,2]$. Then the range of all possible values of the real number $t$ is $\qquad$ . | (-\infty,-2] \cup[4,+\infty) | 49 | 16 |
math | 3. Let $O$ be the circumcenter of $\triangle A B C$, and
$$
\overrightarrow{A O} \cdot \overrightarrow{B C}+2 \overrightarrow{B O} \cdot \overrightarrow{C A}+3 \overrightarrow{C O} \cdot \overrightarrow{A B}=0 \text {. }
$$
Then the minimum value of $\frac{1}{\tan A}+\frac{1}{\tan C}$ is | \frac{2\sqrt{3}}{3} | 104 | 12 |
math | 6. On domino tiles, there are pairs of numbers from 0 to 6 (including numbers 0 and 6) located on two fields. All numbers, except zero, are represented by the corresponding number of dots. An empty field without dots represents zero. In a complete set of domino tiles, there are also tiles with the same values on both f... | 168 | 103 | 3 |
math | Let $S(m)$ be the sum of the digits of the positive integer $m$. Find all pairs $(a, b)$ of positive integers such that $S\left(a^{b+1}\right)=a^{b}$. | (a, b) \in\left\{(1, b) \mid b \in \mathbb{Z}^{+}\right\} \cup\{(3,2),(9,1)\} | 48 | 44 |
math | 10. On the coordinate plane, depict the set of points $(a, b)$ such that the system of equations $\left\{\begin{array}{c}x^{2}+y^{2}=a^{2}, \\ x+y=b\end{array}\right.$ has at least one solution. | |b|\leq\sqrt{2}|| | 64 | 11 |
math | 10. [5] Let
$$
\begin{array}{l}
A=(1+2 \sqrt{2}+3 \sqrt{3}+6 \sqrt{6})(2+6 \sqrt{2}+\sqrt{3}+3 \sqrt{6})(3+\sqrt{2}+6 \sqrt{3}+2 \sqrt{6})(6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}), \\
B=(1+3 \sqrt{2}+2 \sqrt{3}+6 \sqrt{6})(2+\sqrt{2}+6 \sqrt{3}+3 \sqrt{6})(3+6 \sqrt{2}+\sqrt{3}+2 \sqrt{6})(6+2 \sqrt{2}+3 \sqrt{3}+\sqrt{6})
\e... | 1 | 202 | 1 |
math | Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which:
[b]*[/b] $2a - 1$ is a multiple of $b$;
[b]*[/b] $2b - 1$ is a multiple of $c$;
[b]*[/b] $2c - 1$ is a multiple of $a$. | (13, 25, 7) | 88 | 12 |
math | Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$. | 420 | 30 | 3 |
math | 1. Solve the inequality $\log _{\frac{x^{2}-2}{2 x-3}}\left(\frac{\left(x^{2}-2\right)(2 x-3)}{4}\right) \geq 1$. | x\in[\frac{1}{2};1)\cup(1;\sqrt{2})\cup[\frac{5}{2};+\infty) | 51 | 33 |
math | Four, (20 points) Let $a \in \mathbf{R}, A=\left\{x \mid 2^{1+x}+2^{1-x}\right.$ $=a\}, B=\{\sin \theta \mid \theta \in \mathbf{R}\}$. If $A \cap B$ contains exactly one element, find the range of values for $a$. | a=4 | 86 | 3 |
math | 69. The area of $\triangle A B C$ is $1, D, E$ are points on sides $A B, A C$ respectively, $B E, C D$ intersect at point $P$, and the area of quadrilateral $B C E D$ is twice the area of $\triangle P B C$. Find the maximum value of the area of $\triangle P D E$.
---
The area of $\triangle A B C$ is $1, D, E$ are poi... | 5 \sqrt{2}-7 | 165 | 7 |
math | 1. Oleg has 550 rubles, and he wants to give his mother tulips for March 8, and there must be an odd number of them, and no color shade should be repeated. In the store where Oleg came, one tulip costs 49 rubles, and there are eleven shades of flowers available. How many ways are there for Oleg to give his mother flowe... | 1024 | 108 | 4 |
math | ## Task Condition
Find the derivative.
$$
y=3 e^{\sqrt[3]{x}}\left(\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+2\right)
$$ | e^{\sqrt[3]{x}} | 48 | 9 |
math | ## Task 20/89
Determine all triples $(x ; y ; z)$ of natural numbers $x, y, z$ for which the equation $x+y+z+2=x y z$ holds. | (1;2;5),(1;3;3),(2;2;2) | 46 | 19 |
math | 11. Given real numbers $x, y$ satisfy: $1+\cos ^{2}(x+y-1)=\frac{x^{2}+y^{2}+2(x+1)(1-y)}{x-y+1}$, find the minimum value of $x y$.
untranslated part:
已知实数 $x, y$ 满足: $1+\cos ^{2}(x+y-1)=\frac{x^{2}+y^{2}+2(x+1)(1-y)}{x-y+1}$, 求 $x y$ 的最小值。
translated part:
Given real numbers $x, y$ satisfy: $1+\cos ^{2}(x+y-1)=\f... | \frac{1}{4} | 194 | 7 |
math | Find all functions $f$ from $\mathbb{Z}$ to $\mathbb{Z}$ such that:
$$
f(2a) + 2f(b) = f(f(a+b))
$$
## Bonus: Replace $\mathbb{Z}$ with $\mathbb{Q}$ | f(x)=2xorf(x)=0 | 61 | 9 |
math | Example. Compute the limit
$$
\lim _{x \rightarrow 3} \frac{x^{3}-4 x^{2}-3 x+18}{x^{3}-5 x^{2}+3 x+9}
$$ | \frac{5}{4} | 51 | 7 |
math | 3. The real number $x$ satisfies
$$
\sqrt{x^{2}+3 \sqrt{2} x+2}-\sqrt{x^{2}-\sqrt{2} x+2}=2 \sqrt{2} x \text {. }
$$
Then the value of the algebraic expression $x^{4}+x^{-4}$ is $\qquad$ | 14 | 80 | 2 |
math | Example 5 Given that $f(x)$ is a function defined on the set of real numbers, and $f(x+2)[1-f(x)]=1+f(x)$, if $f(5)$ $=2+\sqrt{3}$, find $f(2009)$. | \sqrt{3}-2 | 62 | 6 |
math | # Assignment 1. (10 points)
Each of the 2017 middle school students studies English or German. English is studied by $70 \%$ to $85 \%$ of the total number of students, and both languages are studied by $5 \%$ to $8 \%$. What is the maximum number of students who can study German. | 766 | 75 | 3 |
math | Task B-3.8. In a basketball tournament, 8 teams participated and each played one match against each other. For a win, 2 points are awarded, and for a loss, 0 points (no match ended in a draw). The teams accumulated 14, 12, 8, 8, 6, 4, 2, 2 points respectively. How many matches did the last 4 teams lose to the first 4 t... | 15 | 99 | 2 |
math | 7. If the equation about $x$
$$
x^{3}+a x^{2}+b x-4=0\left(a 、 b \in \mathbf{N}_{+}\right)
$$
has a positive integer solution, then $|a-b|=$ | 1 | 61 | 1 |
math | Let $A B C D$ be a unit square and let $P$ and $Q$ be points in the plane such that $Q$ is the circumcentre of triangle $B P C$ and $D$ is the circumcentre of triangle $P Q A$. Find all possible values of the length of segment $P Q$. | \sqrt{2-\sqrt{3}} | 69 | 9 |
math | 8. (10th "Hope Cup" Invitational Competition Question) Let $x_{1}, x_{2}$ be the roots of the equation $x^{2}+x+1=0$, then the value of the sum $\frac{x_{1}}{x_{2}}+\left(\frac{x_{1}}{x_{2}}\right)^{2}+\left(\frac{x_{1}}{x_{2}}\right)^{3}+\cdots+\left(\frac{x_{1}}{x_{2}}\right)^{1998}$ is $\qquad$ $ـ$. | 0 | 131 | 1 |
math | 6. Let $A B C D$ be a parallelogram, with three vertices $A, B, C$ having coordinates $(-1,0),(2,1),(0,3)$, respectively. Then the coordinates of vertex $D$ are $\qquad$. | (-3,2) | 57 | 5 |
math | 10. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy
$$
\begin{array}{l}
a_{1}=-1, b_{1}=2, \\
\quad a_{n+1}=-b_{n}, b_{n+1}=2 a_{n}-3 b_{n}\left(n \in \mathbf{Z}_{+}\right) .
\end{array}
$$
Then $b_{2015}+b_{2016}=$ . $\qquad$ | -3\times2^{2015} | 128 | 11 |
math | We are laying 15 m long railway tracks at a temperature of $t=-8^{\circ} \mathrm{C}$. What gap must be left between the individual tracks if the maximum temperature we are planning for is $t=60^{\circ} \mathrm{C}$. The thermal expansion coefficient of the track is $\lambda=0,000012$. | 12.24 | 81 | 5 |
math | Let's find such numbers that when multiplied by 12345679, we get numbers consisting of 9 identical digits. | 9,18,27,36,45,54,63,72,81 | 29 | 25 |
math | There is a tunnel between locations $A$ and $B$. A car departs from $B$ towards $A$ at 08:16, and a truck departs from $A$ towards $B$ at 09:00. It is known that the truck and the car arrive at the two ends of the tunnel simultaneously, but the truck leaves the tunnel 2 minutes later than the car. If the car arrives at... | 10:00 | 137 | 5 |
math | 2. There are 20 teams participating in the national football championship finals. To ensure that in any group of three teams, at least two teams have played against each other, what is the minimum number of matches that need to be played? | 90 | 49 | 2 |
math | 9. Given the sequence $\left\{a_{n}\right\}$ satisfies the conditions: $a_{1}=1, a_{n+1}=2 a_{n}+1, n \in \mathbf{N}_{+}$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Let $c_{n}=\frac{2^{n}}{a_{n} \cdot a_{n+1}}, T_{n}=\sum_{k=1}^{n} c_{k}$, prove: $T_{n}<1$. | 1-\frac{1}{2^{n+1}-1}<1 | 133 | 15 |
math | Two bikers, Bill and Sal, simultaneously set off from one end of a straight road. Neither biker moves at a constant rate, but each continues biking until he reaches one end of the road, at which he instantaneously turns around. When they meet at the opposite end from where they started, Bill has traveled the length of ... | 8 | 92 | 1 |
math | 4. Given real numbers $x, y$ satisfy
$$
2 x=\ln (x+y-1)+\ln (x-y-1)+4 \text {. }
$$
Then the value of $2015 x^{2}+2016 y^{3}$ is $\qquad$ | 8060 | 66 | 4 |
math | Three players: $A, B$, and $C$ play the following game: on each of three cards, an integer is written. For these three numbers $(p, q$, and $r)$, it holds that $0<p<q<r$.
The cards are shuffled and then distributed so that each player gets one. Then, the players are given as many marbles as the number on their card. A... | C | 191 | 1 |
math | Determine all pairs $(x, y)$ of positive integers for which the equation \[x + y + xy = 2006\] holds. | (2, 668), (668, 2), (8, 222), (222, 8) | 33 | 32 |
math | ## Task Condition
Find the derivative.
$y=x+\frac{1}{1+e^{x}}-\ln \left(1+e^{x}\right)$ | \frac{1}{(1+e^{x})^{2}} | 35 | 15 |
math | Suppose that $0^{\circ}<A<90^{\circ}$ and $0^{\circ}<B<90^{\circ}$ and
$$
\left(4+\tan ^{2} A\right)\left(5+\tan ^{2} B\right)=\sqrt{320} \tan A \tan B
$$
Determine all possible values of $\cos A \sin B$.
## PART B
For each question in Part B, your solution must be well-organized and contain words of explanation or... | \frac{1}{\sqrt{6}} | 145 | 10 |
math |
2. Find the least possible value of $a+b$, where $a, b$ are positive integers such that 11 divides $a+13 b$ and 13 divides $a+11 b$.
| 28 | 48 | 2 |
math | (9) Given, in $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, the circumradius of $\triangle A B C$ is $R=\sqrt{3}$, and it satisfies $\tan B+\tan C=\frac{2 \sin A}{\cos C}$.
(1) Find the size of angle $B$ and side $b$;
(2) Find the maximum area of $\triangle A B C$. | \frac{9}{4}\sqrt{3} | 106 | 11 |
math |
4. Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.
| 2,3,6 | 40 | 5 |
math | A polyhedron has 20 triangular faces. Determine its number of vertices and edges. | 12 | 19 | 2 |
math | Example 7 (2006 Beijing College Entrance Examination Final Question) In the sequence $\left\{a_{n}\right\}$, if $a_{1}, a_{2}$ are positive integers, and $a_{n}=\left|a_{n-1}-a_{n-2}\right|$, $n=3,4,5, \cdots$, then $\left\{a_{n}\right\}$ is called an "absolute difference sequence".
(1) Give an example of an "absolute ... | 6 | 277 | 1 |
math | Problem 2. A group of adventurers is showing off their loot. It is known that exactly 4 adventurers have rubies; exactly 10 have emeralds; exactly 6 have sapphires; exactly 14 have diamonds. Moreover, it is known that
- if an adventurer has rubies, then they have either emeralds or diamonds (but not both at the same t... | 18 | 128 | 2 |
math | There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get? | 127 | 82 | 3 |
math | $\mathrm{Na}$ our planned cottage, we brought the cat Vilda. On Monday, she caught $\frac{1}{2}$ of all the mice, on Tuesday $\frac{1}{3}$ of the remaining, on Wednesday $\frac{1}{4}$ of those left after Tuesday's hunt, and on Thursday only $\frac{1}{5}$ of the remainder. On Friday, the remaining mice preferred to move... | 60 | 144 | 2 |
math | 7. Given a positive integer $n(n \geqslant 2)$, find the largest real number $\lambda$ such that the inequality $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+$ $2 a_{n}$ holds for any positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $a_{1}<a_{2}<\cdots<a_{n}$.
(2003 National Girls' Olymp... | \frac{2(n-2)}{n-1} | 125 | 13 |
math | 2. Solve the system of equations
$$
\left\{\begin{array}{r}
\log _{2}(x+y)|+| \log _{2}(x-y) \mid=3 \\
x y=3
\end{array}\right.
$$ | 3,1 | 58 | 3 |
math | Determine all integer numbers $x$ and $y$ such that:
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{19}
$$ | (38,38);(380,20);(20,380);(-342,18);(18,-342) | 40 | 39 |
math | 4. How many sets of three consecutive three-digit natural numbers have a product that is divisible by 120? | 336 | 24 | 3 |
math | 7. (6 points) Xiao Qing and Xiao Xia start from locations A and B respectively at the same time, and meet for the first time 60 meters from location A. After meeting, they continue to move forward at their original speeds, reach locations A and B respectively, and then immediately return. They meet for the second time ... | 165 | 90 | 3 |
math | 2. Solve the inequality $\frac{\sqrt{x^{2}-5}}{x}-\frac{x}{\sqrt{x^{2}-5}}<\frac{5}{6}$. (8 points) | x\in(-\infty;-3)\cup(\sqrt{5};+\infty) | 43 | 20 |
math | 14. The sum of the digits of some natural number $A$ is $B$, the sum of the digits of number $B$ is $C$. It is known that the sum of the numbers $A$, $B$, and $C$ is 60. What is the number $A$? Are you sure you have found all solutions? | 44,47,50 | 74 | 8 |
math | 11. If $n \in \mathbf{N}^{+}$, and $[\sqrt{n}] \mid n$, then such $n$ is | k^{2},k^{2}+k,k^{2}+2k(k\in{N}^{*}) | 34 | 26 |
math | 2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \left(-\frac{3}{7}\right)$, and the numbers $\frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z}$ also form an arithmetic progression in the given order. Find $\cos ^{2} y$. | \frac{10}{13} | 95 | 9 |
math | 3. In a right triangle $\triangle A B C\left(\measuredangle C=90^{\circ}\right)$, the line passing through the midpoint of the hypotenuse and the center of the inscribed circle intersects the leg $A C$ at point $N$ at an angle of $75^{\circ}$. Determine the acute angles in $\triangle A B C$. | \alpha=60,\beta=30 | 82 | 10 |
math | . Pierre says: «The day before yesterday I was 10 years old. Next year, I will celebrate my 13th birthday.» What day is it? | January1 | 36 | 2 |
math | The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 12 | 85 | 2 |
math | 116. Find the minimum value of the fraction $\frac{x^{2}-3 x+3}{1-x}$, if $x<1$. | 3 | 32 | 1 |
math | 13.129. At 9 AM, a self-propelled barge left $A$ upstream and arrived at point $B$; 2 hours after arriving at $B$, the barge set off on the return journey and arrived back at $A$ at 7:20 PM on the same day. Assuming the average speed of the river current is 3 km/h and the barge's own speed is constant throughout, deter... | 14 | 119 | 2 |
math | 4. If $\cos ^{5} \theta-\sin ^{5} \theta<7\left(\sin ^{3} \theta-\cos ^{3} \theta\right), \theta \in[0,2 \pi)$, then the range of values for $\theta$ is | \theta\in(\frac{\pi}{4},\frac{5\pi}{4}) | 64 | 20 |
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