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 | Solve the system of equations
$$
\begin{gathered}
\frac{\sqrt{x+z}+\sqrt{x+y}}{\sqrt{y+z}}+\frac{\sqrt{y+z}+\sqrt{x+y}}{\sqrt{x+z}}=14-4 \sqrt{x+z}-4 \sqrt{y+z} \\
\sqrt{x+z}+\sqrt{x+y}+\sqrt{z+y}=4
\end{gathered}
$$ | 2,2,-1 | 94 | 5 |
math | 4. Let $n$ be a positive integer. Given a real number $x$, let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$. For example, $\lfloor 2.4\rfloor=2,\lfloor 3\rfloor=3$ and $\lfloor\pi\rfloor=3$. Define a sequence $a_{1}, a_{2}, a_{3}, \ldots$ where $a_{1}=n$ and
$$
a_{m}=\left\lfloor\frac{a_{m-1}}{3... | 126 | 291 | 3 |
math | Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:
$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.
$a)$ Prove that the sequence is converging.
$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$. | a^{\frac{m+1}{m+2}} | 103 | 13 |
math | 1. Given are the parabolas $y=x^{2}+3 x+6$ and $y=x^{2}+5 x+3$. Find the equation of their common tangent, as well as the points of tangency of this tangent with these parabolas. | 7x+2 | 58 | 4 |
math | Balkan Olympiads 2009
Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that for all positive integers $m$ and $n$,
$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$ | f(n)=n | 83 | 4 |
math | 3・17 (1) Simplify $\frac{1-a^{2}}{(1+a x)^{2}-(a+x)^{2}}$;
(2) When $x=0.44$, find the value of $\sqrt{1-x-x^{2}+x^{3}}$. | 0.672 | 65 | 5 |
math | 454. Given the function (rational fractional) $f(x)=\frac{2 x-3}{3 x^{2}-1}$. Find $f(-2) ; f(0) ; f(1)$. | f(-2)=-\frac{7}{11};f(0)=3;f(1)=-\frac{1}{2} | 49 | 31 |
math | 【Question 2】
If $E, U, L, S, R, T$ represent $1, 2, 3, 4, 5, 6$ (different letters represent different numbers), and satisfy:
(1) $E+U+L=6$;
(2) $S+R+U+T=18$;
(3) $U \times T=15$;
(4) $S \times L=8$.
Then the six-digit number $\overline{E U L S R T}=$ | 132465 | 120 | 6 |
math | 94. Even or odd number
$$
1-2+3-4+5-6+\ldots+993 ?
$$ | odd | 30 | 1 |
math | 71*. a) Banach's problem ${ }^{1}$. A person simultaneously bought two boxes of matches and put them in his pocket. After that, every time he needed to light a match, he randomly took one or the other box. After some time, upon emptying one of the boxes, the person discovered that it was empty. What is the probability ... | 2^{2n} | 190 | 5 |
math | 1.017. $\frac{0.128: 3.2+0.86}{\frac{5}{6} \cdot 1.2+0.8} \cdot \frac{\left(1 \frac{32}{63}-\frac{13}{21}\right) \cdot 3.6}{0.505 \cdot \frac{2}{5}-0.002}$. | 8 | 99 | 1 |
math | 5. Find the maximum possible value of the expression $x y(x-y)$, given that $x, y \in$ $[0 ; 1]$ | 0.25 | 33 | 4 |
math | 10. Let $a, b, c \in(0,1]$, $\lambda$ be a real number, such that $\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c)$ always holds, find the maximum value of $\lambda$.
Set $a, b, c \in(0,1]$, $\lambda$ as a real number, such that $\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c)$ always... | \frac{64}{27} | 142 | 9 |
math | Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=... | 65 | 126 | 2 |
math | II. (40 points) Let positive real numbers $a, b, c$ satisfy $a+b+c=ab+bc+ca$.
If $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \geqslant k$ always holds, find the maximum value of $k$.
保留了源文本的换行和格式。 | \frac{4}{3} | 85 | 7 |
math | 8. There are 10 young men, each with a different weight and height; for any two young men $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or $\mathbf{A}$ is taller than $\mathbf{B}$, then we say “$\mathrm{A}$ is not worse than B”; if a young man is not worse than the other 9 people, he is c... | 10 | 128 | 2 |
math | For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$ | \frac{f_{n+2} - 1}{f_n} | 112 | 17 |
math | 8 (CMO-13 Problem) Find all natural numbers greater than 3, such that $1+C_{n}^{1}+C_{n}^{2}+C_{n}^{3}$ divides $2^{2000}$. | n=7n=23 | 55 | 7 |
math | 248. The equation of the hyperbola $y=\frac{1-3 x}{2 x-1}$ can be transformed using a parallel translation of the coordinate axes to the form $X Y=m$. Plot this hyperbola. | XY=-0.25 | 51 | 6 |
math | 16. [10] Given an angle $\theta$, consider the polynomial
$$
P(x)=\sin (\theta) x^{2}+(\cos (\theta)+\tan (\theta)) x+1 \text {. }
$$
Given that $P$ only has one real root, find all possible values of $\sin (\theta)$. | 0,\frac{\sqrt{5}-1}{2} | 72 | 12 |
math | Task 1 - 250721 Annett, Birgit, and Cornelia performed differently in the last class test; one of these students received a grade of 1, another a grade of 2, and the third a grade of 3.
Kerstin, a classmate, tells at home: "Annett did not get a 1, Birgit did not get a 2, but Cornelia got a 2."
It turns out, however, ... | Birgit2,Annett3,Cornelia1 | 138 | 12 |
math | Four. (50 points) Given a positive integer $n(n \geqslant 2)$. From 1, $2, \cdots, 3 n$, any $m$ numbers are chosen such that there must be four pairwise distinct numbers $a$, $b$, $c$, $d$, such that $a=b+c+d$. Find the minimum value of $m$.
保留源文本的换行和格式，直接输出翻译结果如下：
```
Four. (50 points) Given a positive integer $n(n... | 2n+2 | 184 | 4 |
math | Example 3 If $y=\sqrt{1-x}+\sqrt{x-\frac{1}{2}}$ has a maximum value of $a$ and a minimum value of $b$, then the value of $a^{2}+b^{2}$ is $\qquad$ [2]
(2011, "Mathematics Weekly" Cup National Junior High School Mathematics Competition) | \frac{3}{2} | 80 | 7 |
math | Example 1 (2001 Irish Mathematical Olympiad) Find the smallest positive integer $a$ such that there exists a positive odd integer $n$ satisfying $2001 \mid$
$$55^{n}+a \cdot 32^{n}$$ | 436 | 58 | 3 |
math | Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$ | \angle A = 30^\circ | 79 | 10 |
math | Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations:
\begin{align*} \sqrt{2x-xy} + \sqrt{2y-xy} &= 1 \\ \sqrt{2y-yz} + \sqrt{2z-yz} &= \sqrt2 \\ \sqrt{2z-zx} + \sqrt{2x-zx} &= \sqrt3. \end{align*}
Then $\left[ (1-x)(1-y)(1-z) \right]^2$ can be written as $\frac{m}{n},$ where $m$ and $n$ a... | 33 | 149 | 2 |
math | $7.4 \quad 2.5^{\frac{4+\sqrt{9-x}}{\sqrt{9-x}}} \cdot 0.4^{1-\sqrt{9-x}}=5^{10} \cdot 0.1^{5}$. | x_{1}=8,x_{2}=-7 | 57 | 11 |
math | (5) From the set $M=\{1,2,3, \cdots, 2009\}$, after removing all multiples of 3 and all multiples of 5, the number of elements remaining in $M$ is $\qquad$. | 1072 | 56 | 4 |
math | 3. Among the natural numbers from 1 to 144, the number of ways to pick three numbers that form an increasing geometric progression with an integer common ratio is $\qquad$ . | 78 | 40 | 2 |
math | A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropi... | 2 | 109 | 1 |
math | Let's find a triangle where the lengths of the sides are integers and their product is equal to 600, and the perimeter is as small as possible. Solve the same problem if the product is 144. | 4,6,6 | 46 | 5 |
math | $6 \cdot 11$ 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 $f(4,1981)$. | f(4,1981)=-3+2^{2^{2} \cdot \cdot^{2}} \mid 1984 | 98 | 32 |
math | In $\triangle A B C$, extend $A B$, $A C$, and $B C$ successively, such that $B A^{\prime}=\lambda_{1} A B$, $C B^{\prime}=\lambda_{2} B C$, $A C^{\prime}=\lambda_{3} C A$. Find $S \triangle \triangle^{\prime} \mathrm{B}^{\prime} C^{\prime}: S \triangle \triangle B C$. | 1+\left(\lambda_{1}+\lambda_{2}-\lambda_{3}\right)+\left(\lambda_{1} \lambda_{2}-\lambda_{2} \lambda_{3}-\lambda_{3} \lambda_{1}\right) | 104 | 54 |
math | 11.29 Find the rational roots of the equation
$$
\frac{\sqrt{x+2}}{|x|}+\frac{|x|}{\sqrt{x+2}}=\frac{4}{3} \sqrt{3}
$$
Solve the inequalities (11.30-11.31): | x_{1}=-\frac{2}{3},x_{2}=1 | 70 | 17 |
math | One. (20 points) Let $k, t$ be constants, and the equation about $x$
$$
k x^{2}+(3-3 k) x+2 k-6=0
$$
has only integer roots. The quadratic equation about $y$
$$
(k+3) y^{2}-15 y+t=0
$$
has two positive integer roots $y_{1}, y_{2}$.
(1) Find the value of $k$;
(2) For what value of $t$ is $y_{1}^{2}+y_{2}^{2}$ minimize... | 15 | 133 | 2 |
math | 3. (4 points) Solve the equation of the form $f(f(x))=x$, given that $f(x)=x^{2}+2 x-5$
# | \frac{1}{2}(-1\\sqrt{21}),\frac{1}{2}(-3\\sqrt{17}) | 37 | 30 |
math | 11.1. $\quad$ Find the smallest period of the function $y=\cos ^{10} x+\sin ^{10} x$. | \frac{\pi}{2} | 34 | 7 |
math | 7.3. There are three automatic coin exchange machines. Among them, the first machine can only exchange 1 coin for 2 other coins; the second machine can only exchange 1 coin for 4 other coins; the third machine can exchange 1 coin for 10 other coins. A person made a total of 12 exchanges, turning 1 coin into 81 coins. T... | A=B=2, C=8 | 101 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{2+\cos x \cdot \sin \frac{2}{2 x-\pi}}{3+2 x \sin x}$ | \frac{2}{3+\pi} | 55 | 9 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{2}+5\right)\left(n^{4}+2\right)}-\sqrt{n^{6}-3 n^{3}+5}}{n}
$$ | \frac{5}{2} | 68 | 7 |
math | Find all real numbers $a, b, c$ that satisfy
$$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$ | (-1, -1, -1), (0, 0, 0), (1, 1, 1) | 52 | 28 |
math | 4. The distance between the foci of the conic section
$$
(3 x+4 y-13)(7 x-24 y+3)=200
$$
is $\qquad$ . | 2 \sqrt{10} | 47 | 7 |
math | ## Task B-4.2.
Determine the smallest natural number $m$ for which the product $3^{\frac{1}{55}} \cdot 3^{\frac{4}{55}} \cdot 3^{\frac{7}{55}} \cdot \ldots \cdot 3^{\frac{3 m-2}{55}}$ is greater than 729? | 16 | 88 | 2 |
math | 3-3. The height of a truncated cone is equal to the radius of its larger base; the perimeter of a regular hexagon circumscribed around the smaller base is equal to the perimeter of an equilateral triangle inscribed in the larger base. Determine the angle of inclination of the cone's generatrix to the base plane. | \varphi=\arctan(4) | 67 | 10 |
math | ## 9. Diagonal Squares
Vlado covered the diagonal of a large square with a side length of $2020 \mathrm{~cm}$ using a row of smaller squares with a side length of $4 \mathrm{~cm}$ cut from green collage paper. The diagonals of the green squares align with the diagonal of the large square, and the intersection of any t... | 101 | 134 | 3 |
math | A median of a triangle is $k=5 \mathrm{dm}$; this median forms angles of $\varphi=47^{\circ} 16^{\prime}$ and $\phi=25^{\circ} 38^{\prime}$ with the sides; what is the area of the triangle? | 16.623\mathrm{}^{2} | 67 | 12 |
math | 4. The equation
$$
x y z+x y+y z+z x+x+y+z=2014
$$
has $\qquad$ groups of non-negative integer solutions $(x, y, z)$. | 27 | 46 | 2 |
math | Solve the following equation:
$$
\frac{\frac{5-4 x}{5+4 x}+3}{3+\frac{5+4 x}{5-4 x}}-\frac{\frac{5-4 x}{5+4 x}+2}{2+\frac{5+4 x}{5-4 x}}=\frac{\frac{5-4 x}{5+4 x}+1}{1+\frac{5+4 x}{5-4 x}}
$$ | \frac{15}{2} | 105 | 8 |
math | Example 1 Let $x, y \geqslant 0$, and $x^{3}+y^{3}=1$, find the range of $x+y$.
| 1 \leqslant x+y \leqslant 2^{\frac{2}{3}} | 38 | 23 |
math | 5. Given $f(x)=\frac{1}{1+x^{2}}$. Then
$$
\begin{array}{l}
f(1)+f(2)+\cdots+f(2011)+ \\
f\left(\frac{1}{2}\right)+f\left(\frac{1}{3}\right)+\cdots+f\left(\frac{1}{2011}\right) \\
=
\end{array}
$$ | 2010.5 | 98 | 6 |
math | Let $ n \geq 3$ be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where $ x_{i}$ are positive real numbers from the interval $ [0,1]$. | n - 2 + \sqrt{2} | 111 | 10 |
math | # Task No. 5.1
## Condition:
A Dog, a Cat, and a Mouse are running around a circular lake. They all started simultaneously in the same direction from the same point and finished at the same time, each running at a constant speed. The Dog ran 12 laps, the Cat ran 6 laps, and the Mouse ran 4 laps. How many total overta... | 13 | 119 | 2 |
math | 1. Let $a<b<c<d$. If variables $x, y, z, t$ are a permutation of $a, b, c, d$, then the expression
$$
\begin{array}{l}
n(x, y, z, t)=(x-y)^{2}+(y-z)^{2} \\
\quad+(z-t)^{2}+(t-x)^{2}
\end{array}
$$
can take different values. | 3 | 98 | 1 |
math | Consider a paper punch that can be centered at any point
of the plane and that, when operated, removes from the
plane precisely those points whose distance from the
center is irrational. How many punches are needed to
remove every point? | 3 | 49 | 1 |
math | We shuffle a deck of French cards, then draw the cards one by one. On which turn is it most likely to draw the second ace? | 18 | 29 | 2 |
math | Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$ | 35 | 43 | 2 |
math | Mike bought two books from the Canadian Super Mathematics Company. He paid full price for a $\$ 33$ book and he received $50 \%$ off the full price of a second book. In total, he saved $20 \%$ on his purchase. In dollars, how much did he save? | 11 | 64 | 2 |
math | 2. We consider the sequence of numbers $0,1,2,4,6,9,12, \ldots$ which we generate by starting with 0, then adding 1, and adding 1 again, then adding 2 and adding 2 again, then adding 3 and adding 3 again, and so on. If we denote the terms of this sequence by $a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, \ldots$, then $a_{0}=0$ a... | 0alloddk | 195 | 4 |
math | 3. [4 points] Solve the equation $\frac{1}{2}(x+5) \sqrt{x^{3}-16 x+25}=x^{2}+3 x-10$. | 3;\frac{\sqrt{13}+1}{2} | 44 | 14 |
math | ## Problem 2.
Determine the smallest 100 consecutive natural numbers whose sum is divisible by 105. | 3,4,5,\ldots100,101,102 | 26 | 19 |
math | Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 | 58 | 3 |
math | Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$?
[i]Proposed by Evan Chen... | 2 | 155 | 1 |
math | 24 Find the value of $\frac{\frac{1}{2}-\frac{1}{3}}{\frac{1}{3}-\frac{1}{4}} \times \frac{\frac{1}{4}-\frac{1}{5}}{\frac{1}{5}-\frac{1}{6}} \times \frac{\frac{1}{6}-\frac{1}{7}}{\frac{1}{7}-\frac{1}{8}} \times \ldots \times \frac{\frac{1}{2004}-\frac{1}{2005}}{\frac{1}{2005}-\frac{1}{2006}} \times \frac{\frac{1}{2006}... | 1004 | 194 | 4 |
math | Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits. | 227 | 68 | 3 |
math | 13.383 Several workers complete a job in 14 days. If there were 4 more people and each worked 1 hour longer per day, then the same job would be done in 10 days. If there were still 6 more people and each worked 1 hour longer per day, then this
job would be completed in 7 days. How many workers were there and how many h... | 20 | 92 | 2 |
math | G8.2 If the area of the equilateral triangle $P Q R$ is $6+b \sqrt{3}$, find the value of $b$. | 4 | 34 | 1 |
math | If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$ | 16 | 36 | 2 |
math | Five. (20 points) Given that the side lengths of $\triangle A B C$ are $a, b, c$, and they satisfy
$$
a b c=2(a-1)(b-1)(c-1) .
$$
Does there exist a $\triangle A B C$ with all side lengths being integers? If so, find the three side lengths; if not, explain the reason. | 3,7,8or4,5,6 | 86 | 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 | Find all natural integers $a$ such that $a^{2}+a+1$ divides $a^{7}+3 a^{6}+3 a^{5}+3 a^{4}+a^{3}+a^{2}+3$ | =0or=1 | 56 | 5 |
math | Problem 4. Let $n \geq 2$ be a natural number. Determine the set of values that the sum
$$
S=\left[x_{2}-x_{1}\right]+\left[x_{3}-x_{2}\right]+\ldots+\left[x_{n}-x_{n-1}\right]
$$
can take, where $x_{1}, x_{2}, \ldots, x_{n}$ are real numbers with integer parts $1,2, \ldots, n$. | {0,1,2,\ldots,n-1} | 109 | 13 |
math | 4. If $m$ and $n$ are both positive integers, find the minimum value of $\left|2^{m}-181^{n}\right|$.
The above text translated into English, please retain the original text's line breaks and format, and output the translation result directly. | 7 | 61 | 1 |
math | Two spheres of one radius and two of another are arranged so that each sphere touches three others and one plane. Find the ratio of the radius of the larger sphere to the radius of the smaller one.
# | 2+\sqrt{3} | 41 | 6 |
math | Carla wrote on the blackboard the integers from 1 to 21. Diana wants to erase some of them so that the product of the remaining numbers is a perfect square.
a) Show that Diana must necessarily erase the numbers 11, 13, 17, and 19 to achieve her goal.
b) What is the minimum number of numbers that Diana must erase to a... | 5 | 87 | 1 |
math | 2. If $a-2$ is a positive integer and a divisor of $3 a^{2}-2 a+10$, then the sum of all possible values of $a$ is $\qquad$ . | 51 | 45 | 2 |
math | 1 In an exam, there are 30 multiple-choice questions. Correct answers earn 5 points each, incorrect answers earn 0 points, and unanswered questions earn 1 point each. If person A scores more than 80 points, and tells B the score, B can deduce how many questions A answered correctly. If A's score is slightly lower but s... | 119 | 110 | 3 |
math | 19. Find the remainder when $(x-1)^{100}+(x-2)^{200}$ is divided by $x^{2}-3 x+2$. | 1 | 40 | 1 |
math | 6.169. $\sqrt{x}+\frac{2 x+1}{x+2}=2$.
6.169. $\sqrt{x}+\frac{2 x+1}{x+2}=2$. | 1 | 49 | 1 |
math | Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangl... | (0,0) | 91 | 5 |
math | 3. $\log _{\sin 1} \cos 1, \log _{\sin 1} \tan 1, \log _{\operatorname{cos1}} \sin 1$, from largest to smallest is | \log_{\sin1}\cos1>\log_{\cos1}\sin1>\log_{\sin1}\tan1 | 48 | 27 |
math | 3. (8 points) The teacher distributed 9 cards, each with a number from $1 \sim 9$, to three students, Jia, Yi, and Bing, giving each of them 3 cards.
Jia said: The numbers on my three cards form an arithmetic sequence;
Yi said: Mine do too;
Bing said: Only mine do not form an arithmetic sequence.
If what they said is a... | 9 | 105 | 1 |
math | 4. Given the complex sequence $\left\{a_{n}\right\}$ satisfies: $a_{n+1}^{2}-a_{n} a_{n+1}+$ $a_{n}^{2}=0(n \geqslant 0)$, and $a_{n+1} \neq a_{n-1}(n \geqslant 1), a_{1}=1$, then $\sum_{n=0}^{2006} a_{n}$ is $\qquad$. | 2 | 114 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\cos 5 x-\cos 3 x}{\sin ^{2} x}$ | 8 | 42 | 1 |
math | |
| $[\underline{\text { Evenness and Oddness }}]$ | | |
Authors: Akopyan E., Kalinin D. $\underline{\text {. }}$.
Malvina wrote down 2016 common proper fractions in order: $1 / 2,1 / 3,2 / 3,1 / 4,2 / 4,3 / 4, \ldots$ (including reducible ones). Fractions whose value is less than $1 / 2$ she painted red, and the r... | 32 | 141 | 2 |
math | 4. Let's introduce the notations: $B H=2 a, H C=a, B F=y, F C=x$. Since angle $B F H$ is a right angle, by the theorem of relations in a right-angled triangle for the two legs $B H, H C$, we have:
$$
\left\{\begin{array}{l}
a^{2}=x(y+x) \\
4 a^{2}=y(x+y)
\end{array} \quad \Rightarrow \frac{y}{x}=4 \Rightarrow y=4 x\ri... | \frac{1}{10};\frac{1}{2}(\arcsin\frac{4}{5}+\frac{4}{5}) | 522 | 33 |
math | Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double? | 315 | 83 | 3 |
math | $$
\begin{array}{l}
\text { Three. (25 points) Given the equation in } x \text { : } \\
x^{2}+\sqrt{a-2009} x+\frac{a-2061}{2}=0
\end{array}
$$
has two integer roots. Find all real values of $a$ that satisfy the condition. | 2013, 2109 | 85 | 10 |
math | Example 1. Find the singular solutions of the differential equation
$$
x y^{\prime}+\left(y^{\prime}\right)^{2}-y=0
$$ | -\frac{x^{2}}{4} | 38 | 9 |
math | 6. When $s$ and $t$ take all real values, the minimum value that can be reached by $(s+5-3|\cos t|)^{2}$ $+(s-2|\sin t|)^{2}$ is $\qquad$
(1989, National High School Mathematics Competition) | 2 | 67 | 1 |
math | A7. The product of five different integers is 12 . What is the largest of the integers? | 3 | 22 | 1 |
math | [ Combinatorics of orbits ]
[ Product rule ]
$p$ - a prime number. How many ways are there to color the vertices of a regular $p$-gon using $a$ colors?
(Colorings that can be matched by rotation are considered the same.)
# | \frac{^{p}-}{p}+ | 56 | 10 |
math | ## 246. Math Puzzle $11 / 85$
A hollow round ceiling support in a large bakery is loaded with 28 tons. The outer diameter of the support is 145 millimeters, and the clear width, i.e., the inner diameter, is 115 millimeters.
What is the load per square centimeter of the cross-sectional area? | 457\mathrm{~}/\mathrm{}^{2} | 81 | 14 |
math | 7. Xiao Wang walks along the street at a uniform speed and finds that a No. 18 bus passes him from behind every 6 min, and a No. 18 bus comes towards him every $3 \mathrm{~min}$. Assuming that each No. 18 bus travels at the same speed, and the No. 18 bus terminal dispatches a bus at fixed intervals, then, the interval ... | 4 | 99 | 1 |
math | Question 105: If for all $\theta \in R$, the modulus of the complex number $z=(a+\cos \theta)+(2 a-\sin \theta)$ i does not exceed 2, then the range of the real number $a$ is $\qquad$ _. | [-\frac{\sqrt{5}}{5},\frac{\sqrt{5}}{5}] | 60 | 21 |
math | 12.5. Let $I_{n}=\int_{\frac{1}{n}}^{n} \frac{d x}{(x+1)\left(\ln ^{2 n} x+1\right)}, n \geq 2$. Calculate $\lim _{n \rightarrow \infty} I_{n}$. | 1 | 73 | 1 |
math | Consecutive natural numbers are added and subtracted according to the following guide:
$$
1+2-3-4+5+6-7-8+9+10-11-12+\ldots,
$$
that is, two positive and two negative addends always repeat.
Determine the value of such an expression whose last term is 2015.
(L. Hozová) | 0 | 87 | 1 |
math | 39. Find all three-digit numbers $x$, in the notation of which the digits do not repeat, such that the difference between this number and the number written with the same digits but in reverse order is also a three-digit number consisting of the same digits as the number $x$. | 954459 | 58 | 6 |
math | Example 6 Find the largest positive integer $n$ that satisfies the following condition: $n$ is divisible by all positive integers less than $\sqrt[3]{n}$. (1998, Asia Pacific Mathematical Olympiad) | 420 | 48 | 3 |
math | 6. Given the parabola $y^{2}=2 p x(p>0), P$ is a moving point on the negative half-axis of the $x$-axis, $P A 、 P B$ are tangent lines to the parabola, and $A 、 B$ are the points of tangency. Then the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is $\qquad$ . | -\frac{p^{2}}{4} | 96 | 10 |
math | Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take? | \sqrt{2} - 1 | 48 | 8 |
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