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 | 5. The solution to the system of equations $\left\{\begin{array}{l}x=\mathrm{e} \ln y, \\ y=\mathrm{e} \ln z, \\ z=\mathrm{e} \ln x\end{array}\right.$ is | \mathrm{e} | 58 | 5 |
math | 3.
On the sides $AB, BC, CD$, and $AD$ of the convex quadrilateral $ABCD$, points $M, N, K$, and $L$ are located respectively, such that $AM: MB=3: 2, CN: NB=2: 3, CK=KD$, and $AL: LD=1: 2$. Find the ratio of the area of the hexagon $MBN KDL$ to the area of the quadrilateral $ABCD$. | \frac{4}{5} | 104 | 7 |
math | 4. Given a positive integer $n \geqslant 2$. Let integers $a_{0}, a_{1}, \cdots$, $a_{n}$ satisfy $0=a_{0}<a_{1}<\cdots<a_{n}=2 n-1$. Find the minimum possible number of elements in the set $\left\{a_{i}+a_{j} \mid 0 \leqslant i \leqslant j \leqslant n\right\}$. | 3n | 109 | 2 |
math | 29.8. Compute $\int \frac{d x}{1+\sqrt{x}}$. | 2\sqrt{x}-2\ln(1+\sqrt{x})+C | 20 | 16 |
math | 8. For positive integer $a$ and integers $b$, $c$, the quadratic equation $a x^{2}+b x+c=0$ has two roots $\alpha, \beta$. And it satisfies $0<\alpha<\beta<$ 1. Find the minimum value of $a$. | 5 | 64 | 1 |
math | 4. In the vertices of a regular 300-gon, numbers from 1 to 300 are placed once each in some order. It turns out that for each number a, among the 15 nearest numbers to it in the clockwise direction, there are as many numbers less than a as there are among the 15 nearest numbers to it in the counterclockwise direction. ... | 10 | 117 | 2 |
math | Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$. | \frac{7}{18} | 77 | 8 |
math | Find all integers $n \geq 1$ such that $2^{n}-1$ has exactly $n$ positive integer divisors. | n\in{1,2,4,6,8,16,32} | 30 | 20 |
math | 1. Given $\frac{1949}{y^{2}}=\frac{2007}{x^{2}}$, and $\frac{1}{x}+\frac{1}{y}=1(x>$ $0, y>0)$. Then $\sqrt{1949 x+2007 y}=$ $\qquad$ | \sqrt{1949}+\sqrt{2007} | 75 | 16 |
math | 20. (12 points) Given the function $f(x)=\frac{2 x+3}{3 x}$, the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n+1}=f\left(\frac{1}{a_{n}}\right)\left(n \in \mathbf{N}_{+}\right) .
$$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $T_{n}=\sum_{i=1}^{2 n}(-1)^{i+1} a_{i} a_{... | -\frac{4}{9}(2n^2 + 3n) | 157 | 16 |
math | ## Task A-3.3.
Given is a triangle $A B C$ with an area of 5. If for the lengths of the sides of this triangle the equality $|A B|^{2}+|A C|^{2}=17+|B C|^{2}$ holds, determine the tangent of the angle $\varangle C A B$. | \frac{20}{17} | 77 | 9 |
math | I1.3 Given that $p$ and $q$ are real numbers with $p q=b$ and $p^{2} q+q^{2} p+p+q=70$. If $c=p^{2}+q^{2}$, find the value of $c$. | 31 | 62 | 2 |
math | 8.326. $(\cos x-\sin x)^{2}+\cos ^{4} x-\sin ^{4} x=0.5 \sin 4 x$. | x_{1}=\frac{\pi}{2}(2n+1);x_{2}=\frac{\pi}{4}(4k+1) | 41 | 32 |
math | Find all triplets of three strictly positive integers such that:
$$
x+\frac{1}{y+\frac{1}{z}}=\frac{10}{7}
$$ | (1,2,3) | 37 | 7 |
math | ## [ Sums of numerical sequences and difference series ]
In the class, there are $a_{1}$ students who received at least one two during the year, $a_{2}$ students who received at least two twos, ..., $a_{k}$ students who received at least $k$ twos. How many twos are there in total in this class?
(It is assumed that no... | a_{1}+a_{2}+\ldots+a_{k} | 91 | 16 |
math | Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
[asy]
size(8cm);
defaul... | \frac{1}{2} | 735 | 7 |
math | Let's determine those positive integers which are 14 times as large as the sum of their digits. | 126 | 21 | 3 |
math | $14 \cdot 10$ Let $a > 1$ be a positive real number, and $n \geqslant 2$ be a natural number, and the equation $[a x]=x$ has exactly $n$ different solutions. Try to find the range of values for $a$.
(China Sichuan Province High School Mathematics Competition, 1992). | 1+\frac{1}{n}\leqslant<1+\frac{1}{n-1} | 85 | 23 |
math | 7. Given
$$
f(x)=x^{2}-53 x+196+\left|x^{2}-53 x+196\right| \text {. }
$$
Then $f(20)+f(14)=$ $\qquad$ . | 0 | 60 | 1 |
math | 1. For each pair of distinct numbers in the set $\{1,2, \cdots, 19\}$, the sum of all such products is | 16815 | 34 | 5 |
math | Consider all pairs of positive integers $(a,b)$, with $a<b$, such that
$\sqrt{a} +\sqrt{b} = \sqrt{2,160}$
Determine all possible values of $a$. | 15, 60, 135, 240, 375 | 50 | 21 |
math | 6. Given the set
$$
A=\left\{x \mid x=a_{0}+a_{1} \times 7+a_{2} \times 7^{2}+a_{3} \times 7^{3}\right\} \text {, }
$$
where, $a_{i} \in\{0,1, \cdots, 6\}(i=0,1,2,3)$, and $a_{3} \neq 0$.
If positive integers $m 、 n \in A$, and $m+n=2010(m>n)$, then the number of positive integers $m$ that satisfy the condition is $\q... | 662 | 154 | 3 |
math | Anna and Zsófi take turns rolling a die, and they always add the rolled number to the sum of the numbers rolled so far. The one who, after their roll, makes the sum first divisible by 4 wins. If Anna starts, what is the probability that she wins? | \frac{52}{99} | 59 | 9 |
math | Example 1 Find the range of the function $y=\frac{x^{2}-x}{x^{2}-x+1}$. | [-\frac{1}{3},1) | 28 | 10 |
math | 884. Find the length of the cardioid $x=2 a \cos t-a \cos 2 t, y=$ $-2 a \sin t-a \sin 2 t$. | 16a | 42 | 3 |
math | 9.22 Thirty people are divided into three groups (I, II, and III) with 10 people in each. How many different group compositions are possible? | \frac{30!}{(10!)^{3}} | 35 | 14 |
math | 1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i] | 753 | 55 | 3 |
math | 11.1. Find all solutions in natural numbers for the equation: $x!+9=y^{3}$. | 6,9 | 25 | 3 |
math | $4 \cdot 36$ Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers.
untranslated part:
试求 1 的 $n$ 个 $n$ 次方根,并求它们 $n$ 次幕的和.
translated part:
Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers. | n | 106 | 1 |
math | 4. a) Determine the natural numbers $a$ and $b$, if $[a, b]-(a, b)=34$.
b) Write the set $A=\{1,2,3, \ldots, 1000\}$ as a union of 500 disjoint subsets, such that the sum of the elements of each subset is a perfect square.
Subject elaborated by prof. Valentina Blendea
## SOLUTIONS
## Grade VI | (,b)\in{(1,35),(5,7),(7,5),(35,1);(2,36),(4,18),(18,4),(36,2);(17,51),(51,17);(34,68),(68,34)} | 102 | 71 |
math | Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$?
[i]Ray Li[/i] | 75^\circ | 76 | 4 |
math | 5. [4 points] Find the number of eight-digit numbers, the product of whose digits equals 7000. The answer should be presented as an integer. | 5600 | 35 | 4 |
math | 4. There is a math competition problem, the probabilities of A, B, and C solving it alone are $\frac{1}{a} 、 \frac{1}{b} 、 \frac{1}{c}$, respectively, where $a 、 b 、 c$ are all positive integers in the range of 1 to 9. Now A, B, and C are solving this problem independently at the same time. If the probability that exac... | \frac{4}{15} | 128 | 8 |
math | ## Task 3 - 340713
Franziska is looking for a four-digit natural number $z$ that satisfies the following statements (1), (2), and (3):
(1) The units digit of $z$ is 1 greater than the tens digit of $z$.
(2) The hundreds digit of $z$ is twice the tens digit of $z$.
(3) The number $z$ is twice a prime number.
Prove ... | 9634 | 113 | 4 |
math | 3. Find all real numbers $x \in\left[0, \frac{\pi}{2}\right]$, such that
$$
(2-\sin 2 x) \sin \left(x+\frac{\pi}{4}\right)=1 \text {, }
$$
and prove your conclusion.
(Li Shenghong) | x=\frac{\pi}{4} | 72 | 8 |
math | 10.090. The distance from the center of the circle to a chord of length 16 cm is 15 cm. Find the area of the triangle circumscribed around the circle, if the perimeter of the triangle is $200 \mathrm{~cm}$. | 1700\mathrm{~}^{2} | 61 | 12 |
math | Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i] | 4 | 82 | 1 |
math | 2nd USAMO 1973 Problem 4 Find all complex numbers x, y, z which satisfy x + y + z = x 2 + y 2 + z 2 = x 3 + y 3 + z 3 = 3. Solution | x=y=z=1 | 58 | 5 |
math | 10.5. After watching the movie, viewers rated it one by one with an integer score from 0 to 10. At any given time, the movie's rating was calculated as the sum of all the given scores divided by their number. At some point in time $T$, the rating became an integer, and then with each new voting viewer, it decreased by ... | 5 | 109 | 1 |
math | ## Subject 3.
The sum of four numbers is 2014. Find the four numbers knowing that if we add 16 to the first number, subtract 16 from the second number, divide the third number by 16, we get the fourth number in each case. | =90,b=122,=1696,=106 | 61 | 19 |
math | Example 4 Given real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=5$, let $S=x^{2}+y^{2}$. Then $\frac{1}{S_{\text {max }}}+\frac{1}{S_{\text {min }}}=$ $\qquad$ ( $\max$ denotes the maximum value, $\min$ denotes the minimum value). | \frac{8}{5} | 89 | 7 |
math | 12. (3 points) Xiao Xi bought a total of 30 exercise books of type A and type B, paying 54 yuan, and got 0.5 yuan back. Type A exercise books cost 1.5 yuan each. Type B exercise books cost 2 yuan each. How many type A exercise books did Xiao Xi buy? $\qquad$ books. | 13 | 79 | 2 |
math | Example 14 Let non-negative real numbers $a, b, c$ satisfy $ab + bc + ca = 1$. Find the minimum value of $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$.
(2003 China National Team Training Problem) | \frac{5}{2} | 70 | 7 |
math | The CMU Kiltie Band is attempting to crash a helicopter via grappling hook. The helicopter starts parallel (angle $0$ degrees) to the ground. Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x... | \frac{269}{32} | 133 | 10 |
math | Example 9. Determine all triples of real numbers $(x, y, z)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
2 x+x^{2} y=y, \\
2 y+y^{2} z=z, \\
2 z+z^{2} x=x .
\end{array}\right.
$$ | x = \tan \frac{k\pi}{7}, \quad y = \tan \frac{2k\pi}{7}, \quad z = \tan \frac{4k\pi}{7}. \quad (k = 0, \pm 1, \pm 2, \pm 3) | 73 | 67 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\ln \frac{1-t}{1+t} \\
y=\sqrt{1-t^{2}}
\end{array}\right.
$$ | \frac{\cdot\sqrt{1-^{2}}}{2} | 59 | 15 |
math | 4. Points $A, B, C, D, E$ are sequentially located on a line, such that $A B=C D=1, B C=D E=2$. Circles $\Omega$ and $\omega$, touching each other, are such that $\Omega$ passes through points $A$ and $E$, and $\omega$ passes through points $B$ and $C$. Find the radii of circles $\Omega$ and $\omega$, given that their ... | R=\frac{27}{2\sqrt{19}},r=\frac{8}{\sqrt{19}} | 109 | 26 |
math | 11. From the three-digit numbers $100, 101, 102, \ldots, 699, 700$, if $n$ different numbers are taken, such that there are always three numbers among them with the same sum of digits. Then the minimum value of $n$ is $\qquad$ | 47 | 75 | 2 |
math | 10.1. New smartwatches cost 2019 rubles. Namzil has $\left(500^{2}+4 \cdot 500+3\right) \cdot 498^{2}-500^{2}$. $503 \cdot 497$ rubles. Will he have enough money to buy the smartwatches? | 2012<2019 | 86 | 9 |
math | ## SUBIECTUL I
(4p) a) Calculaţi: $x=\sqrt{\left(\frac{12}{11}+\frac{13}{22}+\frac{14}{33}+\ldots+\frac{110}{1089}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{99}\right)}$;
(3p)b)Calculaţi: $y=\left(\frac{1}{1 \bullet 4}+\frac{1}{2 \bullet 6}+\frac{1}{3 \bullet 8}+\ldots+\frac{1}{49 \bullet 100}\right)$;
| 3 | 166 | 1 |
math | 8. Given that the three sides and the area of a triangle are all integers, and the perimeter is a perfect square, then the minimum value of such a triangle's perimeter is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly... | 16 | 65 | 2 |
math | 29.38. Calculate the length of the cycloid branch
$$
x=a(t-\sin t), \quad y=a(1-\cos t), \quad \text { where } \quad 0 \leqslant t \leqslant 2 \pi
$$
## 29.7. Surface Area
Let $f(x)$ be a continuous positive function on the interval $[a, b], x_{0}=a<x_{1}<\ldots<x_{n-1}<b=x_{n}$ - some points on the interval $[a, b]... | 8a | 240 | 2 |
math | 106. Prime factorization. Find the prime divisors of the number 1000027. | 103\cdot7\cdot73\cdot19 | 25 | 14 |
math | # Task 3. Maximum 20 points
At the conference "Economics of the Present," an intellectual tournament was held, in which more than 198 but fewer than 230 scientists, including doctors and candidates of sciences, participated. Within one match, participants had to ask each other questions and record correct answers with... | 105 | 156 | 3 |
math | Solve the system
$x^{2}+y^{2}=1$
$4 x y\left(2 y^{2}-1\right)=1$ | (\\frac{\sqrt{2-\sqrt{2}}}{2},\\frac{\sqrt{2+\sqrt{2}}}{2}),(\\frac{\sqrt{2+\sqrt{2}}}{2},\\frac{\sqrt{2-\sqrt{2}}}{2}) | 34 | 58 |
math | Example: Let $x, y, z$ be 3 non-zero real numbers. Find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$. | \frac{\sqrt{5}}{2} | 47 | 10 |
math | 4. For which values of the real parameter $m$ does the equation
$$
x^{2}-(m+1) x+2 m-4=0
$$
have real solutions, and at the same time, the sum of their squares is the smallest possible? | 1 | 58 | 1 |
math | 8. In the Cartesian coordinate system, given two points $A(0, a), B(0, b), a>b>0$ on the positive half of the $y$-axis. $C$ is a point on the positive half of the $x$-axis, and makes $\angle A C B$ maximum, then the coordinates of point $C$ are $\qquad$. | (\sqrt{},0) | 82 | 5 |
math | Example 5 Given the quadratic equation in $x$
$$
\left(k^{2}-8 k+15\right) x^{2}-2(13-3 k) x+8=0
$$
both roots are integers. Find the value of the real number $k$.
Analysis: Since $k$ is a real number, we cannot solve it using the discriminant. We can first find the two roots of the equation $x_{1}=$ $\frac{2}{5-k}, x... | k=4,7, \frac{13}{3} | 187 | 14 |
math | 1. Given the function $f(\cos x)=\frac{\cos 3 x}{\cos ^{2} x}, x \in\left[0, \frac{\pi}{3}\right]$, then the minimum value of the function $f(x)$ is | -4 | 57 | 2 |
math | 5 (1246). By what percentage will the area of a rectangle increase if its length is increased by $20 \%$ and its width by $10 \%$? | 32 | 38 | 2 |
math | Find all the primes between 1 and 15. | 2,3,5,7,11,13 | 12 | 13 |
math | Example 6 Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}$, $x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$.
(2002 National High ... | \frac{3\sqrt{3}}{2} | 146 | 12 |
math | 8. In triangle $A B C$, the lengths of the sides are known: $|A B|=12,|B C|=13,|C A|=15$. On side $A C$, a point $M$ is taken such that the radii of the circles inscribed in triangles $A B M$ and $B C M$ are equal. Find the ratio $|A M|:|M C|$. | \frac{22}{23} | 92 | 9 |
math | 277. Find a two-digit number that is equal to the sum of the cube of its tens digit and the square of its units digit. | 24 | 30 | 2 |
math | Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$. | 6 | 37 | 1 |
math | Find the real numbers $x, y$ and $z$ such that
$$
\left\{\begin{array}{l}
x+y+z=3 \\
x^{2}+y^{2}+z^{2}=3 \\
x^{3}+y^{3}+z^{3}=3
\end{array}\right.
$$ | 1 | 74 | 1 |
math | ## 2. Lock
Mislav has forgotten the four-digit code of the bicycle lock, but he remembers some details. The number is divisible by 15, but not by 6, and the digits decrease from the thousands place to the units place. Determine the code of the lock and write down its first three digits (excluding the units digit).
Re... | 976 | 75 | 3 |
math | Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$? | 91 | 45 | 2 |
math | Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression
$$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$ | \sqrt{\frac{3}{8}} | 56 | 9 |
math | Let $d(n)$ denote the number of positive divisors of $n$. Find all triples $(n, k, p)$, where $n$ and $k$ are positive integers and $p$ is a prime number, such that
$$
n^{d(n)}-1=p^{k} .
$$ | (2,1,3) \text{ and } (3,3,2) | 65 | 19 |
math | 6. $\sum_{k=0}^{2022} C_{2022}^{k} \cos \frac{(1011-k) \pi}{2}$ The value is $\qquad$ . | 2^{1011} | 49 | 7 |
math | 2. A fruit store is conducting a promotional sale, with the following combinations: Combination A: 2 kg of fruit $A$, 4 kg of fruit $B$; Combination B: 3 kg of fruit $A$, 8 kg of fruit $B$, 1 kg of fruit $C$; Combination C: 2 kg of fruit $A$, 6 kg of fruit $B$, 1 kg of fruit $C$. It is known that fruit $A$ costs 2 yuan... | 150 | 188 | 3 |
math | The 1996 Canadian Mathematical Olympiad, Problem 2 is: Find all real solutions to the system of equations
$$
\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y, \\
\frac{4 y^{2}}{1+4 y^{2}}=z, \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.
$$
The 1996, Issue 5 provided one solution method. Below, we present two cl... | (x, y, z)=(0,0,0), (x, y, z)=\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) | 129 | 44 |
math | 5. In the sequence $\left\{a_{n}\right\}$, $a_{1}=1, a_{n+1}=1+\frac{2}{a_{n}}(n=1,2, \cdots)$, then $a_{n}=$ $\qquad$ | a_{n}=2+\frac{3}{(-2)^{n}-1}(n\in{N}^{*}) | 63 | 27 |
math | (2) Given that $\triangle A B C$ is an equilateral triangle, $M$ and $N$ are the midpoints of sides $B C$ and $B A$ respectively, $O$ is the circumcenter of $\triangle B M N$, and there is a point $D$ on side $A C$ such that the area of $\triangle A O D$ is $\frac{1}{n}$ of the area of $\triangle A B C$, where $n$ is a... | \frac{3}{2n-3} | 129 | 10 |
math | ## Task Condition
Find the derivative.
$y=x^{\arcsin x}$ | x^{\arcsinx}\cdot(\frac{\lnx}{\sqrt{1-x^{2}}}+\frac{\arcsinx}{x}) | 18 | 31 |
math | 8. If for all $\theta \in \mathbf{R}$, the modulus of the complex number $(a+\cos \theta)+(2 a-\sin \theta) \mathrm{i}$ does not exceed 2, then the range of the real number $a$ is $\qquad$ | \in[-\frac{\sqrt{5}}{5},\frac{\sqrt{5}}{5}] | 61 | 23 |
math | Example 4 Find all non-negative integer solutions $(x, y, z, w)$ of the indeterminate equation
$$
2^{x} \times 3^{y}-5^{x} \times 7^{w}=1
$$
[6] | (1,0,0,0),(3,0,0,1),(1,1,1,0),(2,2,1,1) | 55 | 33 |
math | Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if
[list]
[*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and
[*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and
[*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list... | p^{p-2}(p-1) | 173 | 10 |
math | II. (40 points) Let $n(n \geqslant 3)$ be a given natural number, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i < j \leqslant n)$ as $m$. If $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=1$, find the maximum value of $m$. | \sqrt{\frac{12}{n\left(n^{2}-1\right)}} | 126 | 19 |
math | 2. Let $\left\{a_{n}\right\}_{n \geq 1}$ be an arithmetic sequence and $\left\{g_{n}\right\}_{n \geq 1}$ be a geometric sequence such that the first four terms of $\left\{\bar{a}_{n}+g_{n}\right\}$ are $0,0,1$, and 0 , in that order. What is the 10 th term of $\left\{a_{n}+g_{n}\right\}$ ? | -54 | 115 | 3 |
math | Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).
a) Prove that $H^x=H^y$ if and... | 1 + \frac{m}{n}(n - 1) | 183 | 16 |
math | ## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\sqrt{t} \\
y=\sqrt[3]{t-1}
\end{array}\right.
$$ | -\frac{2(+3)}{9\sqrt[3]{(-1)^{5}}} | 66 | 20 |
math | 17. The numbers $1,2,3, \cdots, 7$ are randomly divided into two non-empty subsets. The probability that the sum of the numbers in the two subsets being equal is $\frac{p}{q}$ expressed in the lowest term. Find $p+q$. | 67 | 62 | 2 |
math | 7.5. Find all three-digit numbers $\mathrm{N}$ such that the sum of the digits of the number $\mathrm{N}$ is 11 times smaller than the number $\mathrm{N}$ itself (do not forget to justify your answer). | 198 | 53 | 3 |
math | # Problem 5
Find all values of \(a\) for which the equation
$$
(\operatorname{tg} x+6)^{2}-\left(a^{2}+2 a+8\right)(\operatorname{tg} x+6)+a^{2}(2 a+8)=0
$$
has exactly two solutions on the interval \(\left[0 ; \frac{3 \pi}{2}\right]\). | (-\sqrt{6};-2)(-2;-1);4 | 94 | 15 |
math | Example 5. Find the integral $\int \sin ^{4} x \cos ^{2} x d x$. | \frac{1}{16}x-\frac{1}{64}\sin4x-\frac{1}{48}\sin^{3}2x+C | 26 | 35 |
math | Problem 4. For what values of $a$ and $b$ do the equations $2 x^{3}+a x-12=0$ and $x^{2}+b x+2=0$ have two common roots? | =-14,b=3 | 52 | 6 |
math | ## problem statement
Calculate the area of the parallelogram constructed on vectors $a_{\text {and }} b$.
$a=3 p+2 q$
$$
\begin{aligned}
& b=p-q \\
& |p|=10 \\
& |q|=1 \\
& (\widehat{p, q})=\frac{\pi}{2}
\end{aligned}
$$ | 50 | 81 | 2 |
math | Example 3. Solve the system of equations
$$
\left\{\begin{array}{l}
t \frac{d x}{d t}=-x+y t \\
t^{2} \frac{d y}{d t}=-2 x+y t
\end{array}\right.
$$ | C_{1}+C_{2},\quad\frac{C_{1}}{}+2C_{2},\quad\neq0 | 64 | 31 |
math | 4. Xiao Pang, Xiao Dingding, Xiao Ya, and Xiao Qiao's four families, a total of 8 parents and 4 children, went to the amusement park together. The amusement park's ticket pricing is: 100 yuan per adult; 50 yuan per child; for 10 people or more, a group ticket is available at 70 yuan per person. They need to spend at le... | 800 | 94 | 3 |
math | 24.3. Solve the equation $x^{3}+p x+q=0$, using the identity
$x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x+\omega y+\omega^{2} z\right)\left(x+\omega^{2} y+\omega z\right)$, where $\omega^{2}+\omega+1=0$. (Choose $y$ and $z$ such that $-3 y z=p$ and $\left.y^{3}+z^{3}=q.\right)$ | x_{1}=-(y+z),x_{2}=-(\omegay+\omega^{2}z),x_{3}=-(\omega^{2}y+\omegaz) | 124 | 39 |
math | 60th Putnam 1999 Problem B3 Let R be the reals. Define f : [0, 1) x [0, 1) → R by f(x, y) = ∑ x m y n , where the sum is taken over all pairs of positive integers (m, n) satisfying m ≥ n/2, n ≥ m/2. Find lim (x, y)→(1, 1) (1 - xy 2 )(1 - x 2 y)f(x, y). | 3 | 113 | 1 |
math | ## Task 22/69
If the first digit of the birth and death years of a famous German scholar is omitted, two numbers $a$ and $b$ are obtained, for which the following applies:
1. Both numbers can be factored into a product of three different factors (all greater than 1), and among the 6 factors, 5 are prime numbers.
2. T... | 17771855 | 181 | 8 |
math | 5. Define the operation $a * b=a b-5\left[\frac{a b}{5}\right]$, where $[x]$ denotes the greatest integer not exceeding the real number $x$, and the set $A=$ $\{0,1,2,3,4\}$. A bijection $f: A \rightarrow A$ satisfies $f(a * b)=$ $f(a) * f(b)$. Then the number of such $f$ is . | 2 | 101 | 1 |
math | [ Ratio of areas of similar triangles ]
In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line perpendicular to the hypotenuse divides the triangle into two equal areas. In what ratio does this line divide the hypotenuse? | 2:1 | 56 | 3 |
math | 1. Let $a, b, c, d$ be positive numbers such that $\frac{1}{a^{3}}=\frac{512}{b^{3}}=\frac{125}{c^{3}}=\frac{d}{(a+b+c)^{3}}$. Find $d$.
(1 mark)設 $a 、 b 、 c 、 d$ 為正數, 使得 $\frac{1}{a^{3}}=\frac{512}{b^{3}}=\frac{125}{c^{3}}=\frac{d}{(a+b+c)^{3}}$ 。求 $d$ 。 | 2744 | 143 | 4 |
math | Consider the sequence $1,3,4,9,10,12,13 \ldots$ consisting of integers greater than or equal to 1, in ascending order, which are powers of 3 or sums of distinct powers of 3 (for example: $4=3^{1}+3^{0}, 10=3^{2}+3^{0}, 13=3^{2}+3^{1}+3^{0} \ldots$ ).
What is the integer that appears at the hundredth position? | 981 | 117 | 3 |
math | 174. Find the derivative of the function $y=x^{2}-3 x+5$. | 2x-3 | 21 | 4 |
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