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 | 2. Given vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $|\boldsymbol{a}|=1,|\boldsymbol{b}|=\sqrt{3}$, and $(3 a-2 b) \perp a$. Then the angle between $a 、 b$ is $\qquad$ | \frac{\pi}{6} | 69 | 7 |
math | 52. Find the angle between the vectors: a) $\vec{a}=(4 ; 0)$ and $\vec{b}=(2 ;-2)$; b) $\vec{a}=(5 ;-3)$ and $\vec{b}=(3 ; 5)$. | \frac{\pi}{4} | 60 | 7 |
math | 2. Let $a$ be a real number, and the two parabolas
$$
y=x^{2}+x+a \text { and } x=4 y^{2}+3 y+a
$$
have four intersection points.
(1) Find the range of values for $a$;
(2) Prove that these four intersection points are concyclic, and find the coordinates of the center of the circle. | (-\frac{3}{8},\frac{1}{8}) | 90 | 15 |
math | Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ with the f... | \left\lfloor\frac{n^{2}}{4}\right\rfloor | 128 | 18 |
math | Berezin V.N.
Find the sums
a) $1 \cdot n+2(n-1)+3(n-2)+\ldots+n \cdot 1$.
b) $S_{n, k}=(1 \cdot 2 \cdot \ldots \cdot k) \cdot(n(n-1) \ldots(n-k+1))+(2 \cdot 3 \cdot \ldots \cdot(k+1)) \cdot((n-1)(n-2) \ldots(n-k))+\ldots+((n-k+1)(n-k+$
2)... $\cdot n) \cdot(k(k-1) \cdot \ldots \cdot 1)$ | (k!)^2\cdotC_{n+k+1}^{2k+1} | 148 | 19 |
math | (1) Given the function $f(x)=x^{3}+a x^{2}+x+1(a \in \mathbf{R})$ is decreasing in the interval $\left(-\frac{2}{3},-\frac{1}{3}\right)$ and increasing in the interval $\left(-\frac{1}{3},+\infty\right)$, then $a=$ | 2 | 84 | 1 |
math | In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$.
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$.
If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$. | 48 | 96 | 2 |
math | 2. Given positive real numbers $a, b, a<b$. Let $x_{1}, x_{2}, \cdots, x_{2022} \in[a, b]$, find the maximum value of $\frac{\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{2021}-x_{2022}\right|+\left|x_{2022}-x_{1}\right|}{x_{1}+x_{2}+\cdots+x_{2022}}$. | \frac{2(b-)}{+b} | 130 | 11 |
math | Example 10 Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$, satisfying the condition
$$
f[x+f(y)]=f(x) \cdot f(y)(x, y \in \mathbf{Q}) .
$$ | f(x)=0orf(x)=1 | 58 | 8 |
math | EXAMPLE 2. Calculate the line integral
$$
\int_{L}(x-y) d l
$$
where $L$ is the line segment from point $A(0,0)$ to point $B(4,3)$. | \frac{5}{2} | 51 | 7 |
math | Example 1. Derive the equations of the tangent and normal lines to the curve $y=x \ln x$, drawn at the point with abscissa $x=e$. | ():2x-e,(n):-\frac{1}{2}x+\frac{3}{2}e | 36 | 23 |
math | 2. Find the minimum value of the expression $\frac{3 f(1)+6 f(0)-f(-1)}{f(0)-f(-2)}$, if $f(x)=a x^{2}+b x+c$ is an arbitrary quadratic function satisfying the condition $b>2 a$ and taking non-negative values for all real $x$.
(12 points) | 12 | 82 | 2 |
math | 1. [3] Suppose that $x$ and $y$ are positive reals such that
$$
x-y^{2}=3, \quad x^{2}+y^{4}=13
$$
Find $x$. | \frac{3+\sqrt{17}}{2} | 50 | 13 |
math | Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle? | \frac{1}{4} | 57 | 7 |
math | $15(1,0)$ is a fixed point, $P(0, b)$ is a moving point on the $y$-axis, and point $M(a, 0)$ satisfies $\overrightarrow{P M} \cdot \overrightarrow{P F}=0$. If point $N$ satisfies $2 \overrightarrow{P N}+\overrightarrow{N M}=\overrightarrow{0}$.
(1) Find the equation of the trajectory curve $C$ of point $N$;
(2) Find th... | -1 | 130 | 2 |
math | Example 1 On the Cartesian plane, given the parabola $y=1-x^{2}$ and the line $y=x+a(-1<a<1)$ intersect at points $A$ and $B$, and point $C(1,0)$. Question: For what value of $a$ is the area of $\triangle ABC$ maximized? Find the maximum area of $\triangle ABC$. ${ }^{[1]}$
$(2010$, Shanghai $\mathrm{TI}$ Cup High Scho... | \frac{3 \sqrt{3}}{4} | 111 | 12 |
math | Example 2.21. Find the limit $\lim _{x \rightarrow 0}\left(\left(\int_{0}^{x^{2}} \cos x d x\right) / x\right)$. | 0 | 47 | 1 |
math | II. (50 points)
Solve the system of equations: $\left\{\begin{array}{l}a b+a+2 b=78 \\ b c+3 b+c=101 \\ c d+5 c+3 d=232 \\ d e+4 d+5 e=360 \\ e a+2 e+4 a=192\end{array}\right.$ | (,b,,,e)=(8,7,10,14,16)or(-12,-9,-16,-24,-24) | 90 | 35 |
math | 9. Given real numbers $x, y, z$ satisfy: $x \geqslant y \geqslant z, x+y+z=1, x^{2}+y^{2}+z^{2}=3$. Find the range of real number $x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | [1,\frac{5}{3}] | 86 | 9 |
math | Determine all pairs $(a,b)$ of real numbers with $a\leqslant b$ that maximise the integral $$\int_a^b e^{\cos (x)}(380-x-x^2) \mathrm{d} x.$$ | (a, b) = (-20, 19) | 55 | 15 |
math | 6. Let the non-empty set $A \subseteq\{1,2,3,4,5,6,7\}$, when $a \in A$ it must also have $8-a \in A$, there are $\qquad$ such $A$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 15 | 80 | 2 |
math | ## Task B-2.2.
Determine all ordered pairs of integers $(x, y)$ for which
$$
(x y-1)^{2}=(x+1)^{2}+(y+1)^{2}
$$ | (2,3),(3,2),(-1,b),(,-1) | 50 | 16 |
math | ## Task B-3.3.
Determine all natural numbers $n$ for which $n^{3}-10 n^{2}+28 n-19$ is a prime number. | 2,3,6 | 42 | 5 |
math | 11. Let $a, b, c, d$ be four distinct real numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4$, and $a c=$ $b d$. Find the maximum value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$. | -12 | 93 | 3 |
math | 15. Find the area of the rhombus $ABCD$, if the radii of the circles circumscribed around triangles $ABC$ and $ABD$ are $R$ and $r$. | \frac{8R^{3}r^{3}}{(R^{2}+r^{2})^{2}} | 43 | 25 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{2\left(\sqrt{2^{x}-1}-\operatorname{arctg} \sqrt{2^{x}-1}\right)}{\ln 2}
$$ | \sqrt{2^{x}-1} | 51 | 9 |
math | 10.045. The common chord of two intersecting circles is seen from the centers at angles of $90^{\circ}$ and $60^{\circ}$. Find the radii of the circles if the distance between their centers is $\sqrt{3}+1$. | 2 | 61 | 1 |
math | [Integer and fractional parts. Archimedes' principle]
How many solutions in natural numbers does the equation $\left[{ }^{x} / 10\right]=\left[{ }^{x} / 11\right]+1$ have? | 110 | 53 | 3 |
math | If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
| 1 | 49 | 1 |
math | 7.5. Find the largest natural number $n$ for which $3^{n}$ divides the number $a=1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2020$. | 1005 | 47 | 4 |
math | 2. Solve the equation
$$
\left(\sqrt{\sqrt{x^{2}-5 x+8}+\sqrt{x^{2}-5 x+6}}\right)^{x}+\left(\sqrt{\sqrt{x^{2}-5 x+8}}-\sqrt{x^{2}-5 x+6}\right)^{x}=2^{\frac{x+4}{4}}
$$ | 0,2,3 | 82 | 5 |
math | 11. Find the value of $\tan \left(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{2 \times 2^{2}}+\tan ^{-1} \frac{1}{2 \times 3^{2}}+\cdots+\tan ^{-1} \frac{1}{2 \times 2009^{2}}\right)$.
(2 marks)求 $\tan \left(\tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{2 \times 2^{2}}+\tan ^{-1} \frac{1}{2 \times 3^{2}}+\cdots+\tan ^{-1} ... | \frac{2009}{2010} | 182 | 13 |
math | # Problem 5.
Given numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{2018}$. It is known that $x_{1}=1 / 2$ and
$$
x_{n}=\frac{x_{n-1}}{2 n x_{n-1}+1} \text { for } n=2, \ldots, 2018
$$
Find the sum $x_{1}+x_{2}+\cdots+x_{2018}$.
# | \frac{2018}{2019} | 120 | 13 |
math | 4. A natural number $n$ is such that the number $100 n^{2}$ has exactly 55 different natural divisors. How many natural divisors does the number 10n have? | 18 | 45 | 2 |
math | For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$ | 48 | 56 | 2 |
math | 2. Find all three-digit numbers $\overline{abc}$ with distinct digits for which $a^{2}-b^{2}-c^{2}=a-b-c$. | 746,764,967,976 | 35 | 15 |
math | Example 4. Solve the equation $2 x y \ln y d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0$. | x^{2}\lny+\frac{1}{3}(y^{2}+1)^{3/2}=C | 44 | 26 |
math | ## Task 5 - 070935
For a triangle $A B C$, let the side lengths be $a, b$, and $c$.
Calculate the length $s_{c}$ of the median to side $A B$! | s_{}^{2}=\frac{1}{4}(2^{2}+2b^{2}-^{2}) | 54 | 26 |
math | 3. A natural number greater than 0, if the sum of all its divisors equals twice itself, is called a perfect number or a complete number. For example, the divisors of 6 are $1,2,3,6,1+2+3+6=12$, so 6 is the smallest perfect number. The question of whether there are infinitely many perfect numbers remains one of the chal... | 16256 | 127 | 5 |
math | Determine with as few calculations as possible which of the following two fractions is larger:
$\frac{33333333331}{33333333334}, \quad \frac{22222222221}{22222222223}$. | \frac{22222222221}{22222222223}>\frac{33333333331}{33333333334} | 74 | 54 |
math | 5. The clock shows 00:00, at which the hour and minute hands of the clock coincide. Counting this coincidence as number 0, determine how much time (in minutes) will pass before they coincide for the 21st time. Round your answer to the nearest hundredth. | 1374.55 | 63 | 7 |
math | 7.287. $\left\{\begin{array}{l}4^{x}-7 \cdot 2^{x-0.5 y}=2^{3-y}, \\ y-x=3 .\end{array}\right.$ | (1;4) | 51 | 5 |
math | # Problem 8.
Inside a convex $n$-gon, 100 points are placed such that no three of these $n+100$ points lie on the same line. The polygon is divided into triangles, each of whose vertices are 3 of the given $n+100$ points. For what maximum value of $n$ can there not be more than 300 triangles? | 102 | 87 | 3 |
math | 7.1. Two wheels of radii $r_{1}$ and $r_{2}$ roll along a straight line $l$. Find the set of points of intersection $M$ of their common internal tangents. | \frac{2r_{1}r_{2}}{r_{1}+r_{2}} | 45 | 22 |
math | G7.3 Find the largest value of $c$, if $c=2-x+2 \sqrt{x-1}$ and $x>1$. | 2 | 32 | 1 |
math | 1. What is the probability that in a random sequence of 8 ones and two zeros, there are exactly three ones between the two zeros? | \frac{2}{15} | 29 | 8 |
math | ## Task B-2.1.
In the equation $x^{2}+m-3 x=m x-2$, determine the positive real number $m$ so that the total sum of all solutions of the equation and their squares is 44. | 4 | 53 | 1 |
math | 3. Given $f(x)=\log _{\frac{1}{3}}\left(3^{x}+1\right)+\frac{1}{2} a b x$ is an even function, $g(x)=2^{x}+\frac{a+b}{2^{x}}$ is an odd function, where $a 、 b \in \mathbf{C}$. Then $a+a^{2}+a^{3}+\cdots+a^{2000}+b+b^{2}$ $+b^{3}+\cdots+b^{2000}=$ | -2 | 129 | 2 |
math | 3. In the expansion of $(\sqrt{3}+i)^{10}$, the sum of all odd terms is $\qquad$ . | 512 | 32 | 3 |
math | 1. Two circles $\omega$ and $\gamma$ have radii 3 and 4 respectively, and their centers are 10 units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$, and let $y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$. Find the product $x... | 51 | 85 | 2 |
math | Task B-4.5. If $\cos 4 \alpha=\frac{3}{\sqrt{5 \sqrt{5 \sqrt{5 \sqrt{5 \cdots}}}}}$, what is $(\operatorname{tg} \alpha+\operatorname{ctg} \alpha)^{2}$? | 20 | 67 | 2 |
math | 2. In the Cartesian coordinate system $x O y$, it is known that points $A$ and $B$ lie on the parabola $y^{2}=4 x$, and satisfy $\overrightarrow{O A} \cdot \overrightarrow{O B}=-4, F$ is the focus of the parabola. Then $S_{\triangle O F A} \cdot S_{\triangle O F B}=$ $\qquad$ . | 2 | 96 | 1 |
math | Example 5 Find all positive integers $n$ such that the indeterminate equation
$$n=x_{1}^{2}+x_{2}^{2}+\cdots+x_{5}^{2}$$
has a unique solution in integers satisfying $0 \leqslant x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{5}$. | 1,2,3,6,7,15 | 91 | 12 |
math | 4. A divisor of a natural number is called proper if it is greater than 1 and less than the number itself. A natural number is called elegant if it has at least two proper divisors and is divisible by the difference of any two of them. Find all elegant numbers. | 6,8,12 | 57 | 6 |
math | II. (16 points) Given that $a$ and $b$ are real numbers, and $\mathrm{i}$ is the imaginary unit, the quadratic equation in $z$ is
$$
4 z^{2}+(2 a+\mathrm{i}) z-8 b(9 a+4)-2(a+2 b) \mathrm{i}=0
$$
has at least one real root. Find the maximum value of this real root. | \frac{3 \sqrt{5}}{5} + 1 | 93 | 15 |
math | Example 5 Let $n \in \mathbf{N}_{+}, a_{1}, a_{2}, \cdots, a_{n}$ and $b_{1}, b_{2}, \cdots, b_{n}$ be positive real numbers, and $\sum_{i=1}^{n} a_{i}=1, \sum_{i=1}^{n} b_{i}=1$. Find the minimum value of $\sum_{i=1}^{n} \frac{a_{i}^{2}}{a_{i}+b_{i}}$.
(2004, French Team Selection Exam) | \frac{1}{2} | 136 | 7 |
math | Example 2 If the ellipse $x^{2}+4(y-a)^{2}=4$ and the parabola $x^{2}=2 y$ have common points, then the range of the real number $a$ is $\qquad$
(1998, National High School Mathematics Competition) | -1 \leqslant a \leqslant \frac{17}{8} | 65 | 21 |
math | ## Condition of the problem
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\frac{1}{\ln t} \\
y=\ln \frac{1+\sqrt{1-t^{2}}}{t}
\end{array}\right.
$$ | \frac{\ln^{2}}{\sqrt{1-^{2}}} | 68 | 15 |
math | From the conversation of the sailors, we noted the following detail this year:
Cook: When I was as old as the sailor is now, I was twice as old as he is.
Engineer: I am only 4 years older than the sailor.
Sailor: Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's b... | 1938 | 87 | 4 |
math | How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)? | 16 | 41 | 2 |
math | 10. From any point $P$ on the parabola $y^{2}=2 x$, draw a perpendicular to its directrix $l$, with the foot of the perpendicular being $Q$. The line connecting the vertex $O$ and $P$ and the line connecting the focus $F$ and $Q$ intersect at point $R$. Then the equation of the locus of point $R$ is | y^{2}=-2x^{2}+x | 85 | 12 |
math | 3. It is possible to write $15129=123^{2}$ as the sum of three distinct squares: $15129=$ $27^{2}+72^{2}+96^{2}$
(i) By using the identity $(a+b)^{2} \equiv a^{2}+2 a b+b^{2}$, or otherwise, find another way to write 15129 as the sum of three distinct squares.
(ii) Hence, or otherwise, show that $378225=615^{2}$ can be... | 6^{2}+66^{2}+363^{2}+8^{2}+88^{2}+484^{2} | 138 | 35 |
math | $[$ P $[\quad$ Case Analysis $\quad]$
Find all three-digit numbers that are 12 times the sum of their digits.
# | 108 | 31 | 3 |
math | ## Task 36/79
Solve the equation
$$
\binom{4 k}{k} \cdot k^{2}=\binom{4 k}{k+1} \cdot(k+1)
$$
with $k \in \mathbb{N}, k>0$, and calculate the value of both terms for the found solution(s). | 1980 | 78 | 4 |
math | 2. Problem: Find the smallest positive integer $N$ satisfying the following three properties.
- $N$ leaves a remainder of 5 when divided by 7 .
- $N$ leaves a remainder of 6 when divided by 8 .
- $N$ leaves a remainder of 7 when divided by 9 . | 502 | 66 | 3 |
math | Example 19. Solve the equation: $\cos ^{2} x+\cos ^{2} 3 x=2 \cos ^{2} 2 x$. | x=k \pi \text{ or } x=\frac{k \pi}{4}+\frac{\pi}{8}(k \in \mathbb{Z}) | 37 | 34 |
math | Let $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$ | 28^\circ | 67 | 4 |
math | # 3. Option 1.
Café "Buratino" operates 6 days a week with a day off on Mondays. Kolya said that from April 1 to April 20, the café was open for 17 days, and from April 10 to April 30, it was open for 18 days. It is known that he made a mistake once. What was the date of the last Tuesday in April? | 29 | 96 | 2 |
math | Consider a regular polygon with $2^n$ sides, for $n \ge 2$, inscribed in a circle of radius $1$. Denote the area of this polygon by $A_n$. Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | \frac{2}{\pi} | 66 | 8 |
math | 3. Given an arithmetic sequence $a_{1}, a_{2}, \cdots, a_{1000}$, the sum of the first 100 terms is 100, and the sum of the last 100 terms is 1000, then $a_{1}=$ | \frac{101}{200} | 68 | 11 |
math | (9) (20 points) The angles of triangle $ABC$ satisfy: $\frac{A}{B}=\frac{B}{C}=\frac{1}{3}$; Find the value of $T=\cos A+\cos B+\cos C$. | \frac{1+\sqrt{13}}{4} | 55 | 13 |
math | 1. Arrange the numbers $1,2, \cdots, 13$ in a row $a_{1}, a_{2}$, $\cdots, a_{13}$, where $a_{1}=13, a_{2}=1$, and ensure that $a_{1}+a_{2}+$ $\cdots+a_{k}$ is divisible by $a_{k+1}(k=1,2, \cdots, 12)$. Then the value of $a_{4}$ $+a_{5}+\cdots+a_{12}$ is $\qquad$ . | 68 | 131 | 2 |
math | Senderov V.A.
Find all pairs $(a, b)$ of natural numbers such that for any natural $n$, the number $a^{n}+b^{n}$ is an exact $(n+1)$-th power. | (2,2) | 48 | 5 |
math | 2. Find any pair of natural numbers $a$ and $b$, both greater than 1, that satisfy the equation $a^{13} \cdot b^{31}=6^{2015}$. | =2^{155},b=3^{65} | 46 | 14 |
math | 2. Find all pairs of positive integers $a, b$, for which the number $b$ is divisible by the number $a$ and at the same time the number $3a+4$ is divisible by the number $b+1$.
| (1,6),(2,4),(3,12) | 52 | 14 |
math | 7. (3 points) A deck of playing cards has 54 cards. How many cards must be drawn to ensure that: there are cards of all four suits.
At least $\qquad$ cards must be drawn to ensure that there are cards of all four suits. | 42 | 56 | 2 |
math | $ABC$ is triangle. $l_1$- line passes through $A$ and parallel to $BC$, $l_2$ - line passes through $C$ and parallel to $AB$. Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$. $XY=AC$. What value can take $\angle A- \angle C$ ? | 60^\circ | 86 | 4 |
math | Problem 3. The product of two numbers is 2250. If one of them is reduced by 6, and the other remains the same, then the new product is 1800. What are these numbers? | =30,b=75 | 49 | 7 |
math | Example 6 A farmer contracts 2 hectares of farmland. According to his experience: if he plants rice, the yield per hectare per season is 6000 kilograms; if he plants peanuts, the yield per hectare per season is 1500 kilograms. However, the cost of rice is higher, requiring 3600 yuan per hectare per season, while peanut... | x=\frac{3}{2},y=\frac{1}{2} | 144 | 16 |
math | Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition
\[y^2 \equiv x^3 - x \pmod p\]
is exactly $p.$ | p \equiv -1 \pmod{4} | 62 | 12 |
math | The equation
$$ (x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016) $$
is written on the board. One tries to erase some linear factors from both sides so that each side still has at least one factor, and the resulting equation has no real roots. Find the least number of linear factors one needs to erase to achieve ... | 2016 | 102 | 4 |
math | 9. (5 points) There are 4 distinct natural numbers, their average is 10. The largest number is at least
保留源文本的换行和格式，翻译结果如下：
9. (5 points) There are 4 distinct natural numbers, their average is 10. The largest number is at least | 12 | 68 | 2 |
math | Tokaeva I.
Let $F_{1}, F_{2}, F_{3}, \ldots$ be a sequence of convex quadrilaterals, where $F_{k+1}$ (for $k=1,2,3, \ldots$) is obtained by cutting $F_{k}$ along a diagonal, flipping one of the parts, and gluing it back along the cut line to the other part. What is the maximum number of different quadrilaterals that t... | 6 | 120 | 1 |
math | 7. (5 points) A set of Go costs 24 yuan, and a set of Chinese chess costs 18 yuan. With 300 yuan, you can exactly buy a total of 14 sets of the two types of chess, among which there are $\qquad$ sets of Chinese chess. | 6 | 66 | 1 |
math | IMO 1992 Problem A1 Find all integers a, b, c satisfying 1 < a < b < c such that (a - 1)(b -1)(c - 1) is a divisor of abc - 1. Solution | 2,4,8;or3,5,15 | 53 | 13 |
math | 10 Four people, A, B, C, and D, are practicing passing a ball. The ball is initially passed by A. Each person, upon receiving the ball, passes it to one of the other three people with equal probability. Let $p_{n}$ denote the probability that the ball returns to A after $n$ passes. Then $p_{6}=$ $\qquad$ | \frac{61}{243} | 81 | 10 |
math | Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist? | 35 | 70 | 2 |
math |
NT3. Find all pairs of positive integers $(x, y)$ such that $2^{x}+3^{y}$ is a perfect square.
| (4,2) | 32 | 5 |
math | Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ to $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.
| 880 | 84 | 3 |
math | 40.3. From point $A$ along the circumference, two bodies start moving simultaneously in opposite directions: the first with a constant speed $v$, and the second with a constant linear acceleration $a$ and an initial speed of 0. After what time did the bodies meet for the first time, if they met again for the second tim... | t_{1}=\frac{v}{}(\sqrt{5}-1) | 77 | 17 |
math | 12. Let the function $f(x)$ be a differentiable function defined on $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and it satisfies $2 f(x)+x f^{\prime}(x)>x^{2}$. Then the solution set of the inequality $(x+2017)^{2} f(x+2017)-f(-1)>0$ is $\qquad$ . | (-\infty,-2018) | 99 | 10 |
math | ## Subject II. (20 points)
Determine the real numbers $x, y, z, t$ that satisfy the relations:
$$
x-\sqrt{y}=y-\sqrt{z}=z-\sqrt{t}=t-\sqrt{x}=2 .
$$
Prof. Gheorghe Lobonț, National College "Mihai Viteazul" Turda | 4 | 80 | 1 |
math | Example 4 Find
$$
M=(x+1)(x+2) \cdots(x+n)
$$
the coefficient of $x^{n-2}$ in the expansion. | \frac{1}{24}(n-1) n(n+1)(3 n+2) | 39 | 22 |
math | Let ABCD be a trapezoid with $AB \parallel CD, AB = 5, BC = 9, CD = 10,$ and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \tfrac ab$ for relatively prime pos... | 90011 | 126 | 5 |
math | 5. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with radius 13, such that $AC \| B_{1}D_{1}, BD \| A_{1}C_{1}$. Find the ratio of the areas of $ABCD$ and $A_{1}B_{1}C_{1}D_{1}$, given that $AD=24, BC=10$. | \frac{289}{338} | 117 | 11 |
math | 4. Let the natural number $n$ have the following property: from $1,2, \cdots \cdots, n$ any 50 different numbers are taken, among these 50 numbers there must be two numbers whose difference is 7. The largest such $n$ is $\qquad$ | 98 | 66 | 2 |
math | Find all finite sets $S$ of positive integers with at least $2$ elements, such that if $m>n$ are two elements of $S$, then
$$ \frac{n^2}{m-n} $$
is also an element of $S$. | \{n, 2n\} | 55 | 10 |
math | ## Task 3 - 060723
Someone writes down all natural numbers from 1 to 5555, each exactly once. Calculate the total number of digit 9s written! | 1605 | 44 | 4 |
math | 11. We define: $a @ b=a \times(a+1) \times \ldots \times(a+b-1)$. Given that $x @ y @ 2=420$, then $y @ x=(\quad)$ | 20or120 | 53 | 6 |
math | Example 10 If the equation
$$
\left(x^{2}-1\right)\left(x^{2}-4\right)=k
$$
has four non-zero real roots, and the four points corresponding to them on the number line are equally spaced, find the value of $k$. | \frac{7}{4} | 62 | 7 |
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