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 | Find the largest positive real number $k$ such that the inequality$$a^3+b^3+c^3-3\ge k(3-ab-bc-ca)$$holds for all positive real triples $(a;b;c)$ satisfying $a+b+c=3.$ | k = 5 | 55 | 5 |
math | 3. For which positive integers $n$ is $\phi(n)$
a) odd
b) divisible by 4
c) equal to $n / 2$ ? | a) 1,2 \quad b) n \text{ such that } 8|n; 4|n \text{ and } n \text{ has at least one odd prime factor; } n \text{ has at least two odd prime factors; or } n \text{ has a prime factor } p \equiv 1 \pmod{4} \quad c) 2^k, k=1,2,\ldots | 37 | 95 |
math | Problem 8.8. In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that the sum of any two different numbers of the same color is never equal to a power of two? | 256 | 50 | 3 |
math | Let $a<b<c<d<e$ be real numbers. We calculate all possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of $e$.
## High School Statements | \frac{55}{2} | 72 | 8 |
math | 6. It is known that all positive integers are in $n$ sets, satisfying that when $|i-j|$ is a prime number, $i, j$ belong to two different sets. Then the minimum value of $n$ is $\qquad$ . | 4 | 54 | 1 |
math | Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations
$ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
$ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$
Find all possible values of the product $ p_1p_2p_3p_4$ | 570 | 132 | 3 |
math | 11. Given the sequence $\left\{a_{n}\right\}(n \geqslant 0)$ satisfies $a_{0}=0, a_{1}=1$, for all positive integers $n$, there is $a_{n+1}=2 a_{n}+$ $2007 a_{n-1}$, find the smallest positive integer $n$ such that $2008 \mid a_{n}$. | 2008 | 96 | 4 |
math | 2. Four years ago, the father was three times older than his daughters Elena and Natasha combined. Natasha was then twice as old as Elena. How old is Natasha, and how old is Elena if the father is now twice as old as both of them combined? | Natashais12old,Elenais8old | 53 | 11 |
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 | Three, (50 points) Try to find the smallest positive integer $M$, such that the sum of all positive divisors of $M$ is 4896. | 2010 | 37 | 4 |
math | ## Task B-2.2.
Solve the equation $\left(16^{-x}-2\right)^{3}+\left(4^{-x}-4\right)^{3}=\left(16^{-x}+4^{-x}-6\right)^{3}$. | {-1,-\frac{1}{2},-\frac{1}{4}} | 62 | 17 |
math | Let the tangent of the circle with diameter $AB$ be $AT$. Determine the point $M$ on the circle for which the sum of the distances from the lines $AB$ and $AT$ is $l$, a given value. Investigate for which values of $l$ there is a solution and how many there are. | \leqR(\sqrt{2}+1) | 68 | 12 |
math | 6. Find the greatest possible value of $\gcd(x+2015 y, y+2015 x)$, given that $x$ and $y$ are coprime numbers. | 4060224 | 42 | 7 |
math | Question 13. The general term of the sequence $101, 104, 109, 116, \cdots$ is $a_{n}=100+n^{2}$. Here $n=1,2,3$, $\cdots$. For each $n$, let $d_{n}$ denote the greatest common divisor of $a_{n}$ and $a_{n+1}$. Find the maximum value of $\mathrm{d}_{\mathrm{n}}$, where $n$ takes all positive integers. | 401 | 118 | 3 |
math | 22.11. (Jury, NRB, 79). Find all non-zero polynomials $P(x)$ with real coefficients, satisfying the identity
$$
P(x) P\left(2 x^{2}\right) \equiv P\left(2 x^{3}+x\right), \quad x \in \mathbf{R}
$$ | P(x)=(x^2+1)^k,wherek\in{Z}^+ | 79 | 20 |
math | 4. Given $\frac{1}{3} \leqslant a \leqslant 1$, if $f(x)=a x^{2}-2 x+1$ has a maximum value $M(a)$ and a minimum value $N(a)$ on the interval $[1,3]$, and let $g(a)=M(a)-N(a)$, then the minimum value of $g(a)$ is $\qquad$. | \frac{1}{2} | 92 | 7 |
math | 9.3. Find five different numbers if all possible sums of triples of these numbers are equal to $3,4,6$, $7,9,10,11,14,15$ and 17. The numbers do not have to be integers. | -3,2,4,5,8 | 58 | 10 |
math | Example 2 (2002 National High School Competition Question) The range of negative values of $a$ for which the inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ holds for all $x \in \mathbf{R}$ is $\qquad$ . | \leqslant-2 | 72 | 7 |
math | Determine the smallest and the greatest possible values of the expression
$$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.
[i]Proposed by Walther Janous, Au... | \frac{27}{16} | 135 | 9 |
math | SI. 1 Let $a, b, c$ and $d$ be the roots of the equation $x^{4}-15 x^{2}+56=0$. If $P=a^{2}+b^{2}+c^{2}+d^{2}$, find the value of $P$. | 30 | 69 | 2 |
math | 14.59 Find all natural numbers $n$ such that $\min _{k \in N}\left(k^{2}+\left[\frac{n}{k^{2}}\right]\right)=1991$, where $N$ is the set of natural numbers.
(6th China High School Mathematics Winter Camp, 1991) | 1024\cdot967\leqslantn<1024\cdot968 | 75 | 25 |
math | 25. 2. 3 * Sum: $S=19 \cdot 20 \cdot 21+20 \cdot 21 \cdot 22+\cdots+1999 \cdot 2000 \cdot 2001$. | 6\cdot(\mathrm{C}_{2002}^{4}-\mathrm{C}_{21}^{4}) | 62 | 27 |
math | Find the minimum of the function
$$
f(x, y)=\sqrt{(x+1)^{2}+(2 y+1)^{2}}+\sqrt{(2 x+1)^{2}+(3 y+1)^{2}}+\sqrt{(3 x-4)^{2}+(5 y-6)^{2}} \text {, }
$$
defined for all real $x, y>0$. | 10 | 91 | 2 |
math | 10. Let the eccentricities of an ellipse and a hyperbola with common foci be $e_{1}$ and $e_{2}$, respectively. If the length of the minor axis of the ellipse is twice the length of the imaginary axis of the hyperbola, then the maximum value of $\frac{1}{e_{1}}+\frac{1}{e_{2}}$ is $\qquad$. | \frac{5}{2} | 87 | 7 |
math | 5. Given $f(x)=\left(x^{2}+3 x+2\right)^{\cos \pi x}$. Then the sum of all $n$ that satisfy the equation
$$
\left|\sum_{k=1}^{n} \log _{10} f(k)\right|=1
$$
is | 21 | 72 | 2 |
math | Vertices $A$ and $B$ of the prism $A B C A 1 B 1 C 1$ lie on the axis of the cylinder, while the other vertices lie on the lateral surface of the cylinder. Find the dihedral angle in this prism with edge $A B$. | 120 | 61 | 3 |
math | 1. If $S_{\triangle A B C}=\frac{1}{4} b^{2} \tan B$, where $b$ is the side opposite to $\angle B$, then $\frac{\cot B}{\cot A+\cot C}=$ $\qquad$ . | \frac{1}{2} | 60 | 7 |
math | 11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism. | Q\sqrt{Q/3} | 52 | 8 |
math | 1. (51st Czech and Slovak Mathematical Olympiad (Final) Question) Find $a, b$ in the real numbers such that the equation $\frac{a x^{2}-24 x+b}{x^{2}-1}=x$
has two roots, and the sum of these roots equals 12, where a repeated root counts as one root. | (11,-35),(35,-5819) | 76 | 15 |
math | 12. Given that $x, y$ are real numbers, and $x+y=1$, find the maximum value of $\left(x^{3}+1\right)\left(y^{3}+1\right)$. | 4 | 48 | 1 |
math | 81. A straight line intersects the $O x$ axis at some point $M$ and passes through points $A(-2 ; 5)$ and $B(3 ;-3)$. Find the coordinates of point $M$. | (\frac{9}{8};0) | 48 | 9 |
math | 8th Swedish 1968 Problem 2 How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ... , 6? | 30 | 43 | 2 |
math | 3. Now, subtract the same positive integer $a$ from the numerator and denominator of the fractions $\frac{2018}{2011}$ and $\frac{2054}{2019}$, respectively, to get two new fractions that are equal. What is the positive integer $a$? | 2009 | 68 | 4 |
math | \section*{Problem 1 - 241041}
Determine all ordered pairs \((x ; y)\) of integers for which \(\sqrt{x}+\sqrt{y}=\sqrt{1985}\) holds! | (0,1985)(1985,0) | 53 | 15 |
math | The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
[i]2018 CCA Math Bonanza Lightning Round #3.1[/i] | 264 | 51 | 3 |
math | (1) (20 points) There is a type of notebook originally priced at 6 yuan per book. Store A uses the following promotional method: for each purchase of 1 to 8 books, a 10% discount is applied; for 9 to 16 books, a 15% discount is applied; for 17 to 25 books, a 20% discount is applied; and for more than 25 books, a 25% di... | y=\left\{\begin{array}{ll}
4.9 x, & 11 \leqslant x \leqslant 16 ; \\
4.8 x, & 17 \leqslant x \leqslant 20 ; \\
4.5 x, & 21 \leqslant x \leqslant 40 .
\end{array}\right.} | 291 | 93 |
math | 4. Solve the system $\left\{\begin{array}{l}x^{2}+y^{2} \leq 1, \\ 16 x^{4}-8 x^{2} y^{2}+y^{4}-40 x^{2}-10 y^{2}+25=0 .\end{array}\right.$ | (-\frac{2}{\sqrt{5}};\frac{1}{\sqrt{5}}),(-\frac{2}{\sqrt{5}};-\frac{1}{\sqrt{5}}),(\frac{2}{\sqrt{5}};-\frac{1}{\sqrt{5}}),(\frac{2}{\sqrt{5}};\frac{1}{} | 77 | 83 |
math | 1. a) Simplify the expression
$$
A=\frac{a^{2}-1}{n^{2}+a n}\left(\frac{1}{1-\frac{1}{n}}-1\right) \frac{a-a n^{3}-n^{4}+n}{1-a^{2}}
$$
b) Is there a natural number $n$ for which the number $A$ is a natural number | n+1+\frac{1}{n} | 92 | 10 |
math | 2.14. Calculate the dihedral angle cosine of a regular tetrahedron. | \frac{1}{3} | 20 | 7 |
math | 15. The equation $(\arccos x)^{2}+(2-t) \arccos x+4$ $=0$ has real solutions. Then the range of values for $t$ is $\qquad$. | [6,+\infty) | 49 | 7 |
math | 2. If the polynomial $P=2 a^{2}-8 a b+17 b^{2}-16 a-4 b$ +2070, then the minimum value of $P$ is | 2002 | 45 | 4 |
math | ## Zadatak B-2.5.
Od žice duljine 4.5 metra treba napraviti šest ukrasa, tri u obliku pravilnog šesterokuta i tri u obliku jednakostraničnog trokuta. Svi šesterokuti, odnosno trokuti su međusobno sukladni. Odredite duljine stranica šesterokuta i trokuta tako da zbroj površina svih likova bude minimalan, a cijela žica ... | =10\mathrm{~},b=30\mathrm{~} | 137 | 17 |
math | Find all strictly positive integers $a$ and $n$ such that
$$
3^{n}+1=a^{2}
$$ | =2,n=1 | 28 | 5 |
math | 7th Putnam 1947 Problem B3 Let O be the origin (0, 0) and C the line segment { (x, y) : x ∈ [1, 3], y = 1 }. Let K be the curve { P : for some Q ∈ C, P lies on OQ and PQ = 0.01 }. Let k be the length of the curve K. Is k greater or less than 2? Solution | k<2 | 96 | 3 |
math | Example 5 Find all natural numbers $a, b$, such that $\left[\frac{a^{2}}{b}\right]+$
$$
\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b
$$ | ^2+1,\in{N}^*orb^2+1,b\in{N}^* | 69 | 24 |
math | 14. For a certain young adult reading material, if sold at the original price, a profit of 0.24 yuan is made per book; now it is sold at a reduced price, resulting in the sales volume doubling, and the profit increasing by 0.5 times. Question: How much has the price of each book been reduced? | 0.06 | 72 | 4 |
math | 14. (15 points) Teacher Li leads the students to visit the science museum. The number of students is a multiple of 5. According to the rules, teachers and students pay half the ticket price, and each person pays an integer number of yuan. In total, they paid 1599 yuan. Ask:
(1) How many students are in this class?
(2) ... | 40 | 91 | 2 |
math | $$
\begin{array}{l}
\text { Three. (20 points) (1) The quadratic function } \\
f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)
\end{array}
$$
satisfies
(i) For $x \in \mathbf{R}$, there is $4 x \leqslant f(x) \leqslant \frac{1}{2}(x+2)^{2}$
always holds;
(ii) $f(-4+2 \sqrt{3})=0$.
Find $f(x)$.
(2) Let $f_{1}(x)=... | f_{2009}(0)=\frac{3^{2010}+3}{3^{2010}-1} | 196 | 31 |
math | 38th Putnam 1977 Problem A3 R is the reals. f, g, h are functions R → R. f(x) = (h(x + 1) + h(x - 1) )/2, g(x) = (h(x + 4) + h(x - 4) )/2. Express h(x) in terms of f and g. | (x)=(x)-f(x-3)+f(x-1)+f(x+1)-f(x+3) | 85 | 24 |
math | 11. Given $x+y=1$, for what values of real numbers $x, y$ does $\left(x^{3}+1\right)\left(y^{3}+1\right)$ achieve its maximum value. | 4 | 48 | 1 |
math | 5. (6 points) 12 question setters make guesses about the answer to this question, with their guesses being “not less than 1”, “not greater than 2”, “not less than 3”, “not greater than 4”, “not less than 11”, “not greater than 12” (“not less than” followed by an odd number, “not greater than” followed by an even number... | 7 | 105 | 1 |
math | 7-15 In a given regular $(2n+1)$-sided polygon, three different vertices are randomly selected. If all such selections are equally likely. Find the probability that the center of the regular polygon lies inside the triangle formed by the randomly chosen three points. | \frac{n+1}{2(2 n-1)} | 56 | 13 |
math | 322. Express $\operatorname{tg} 6 \alpha$ in terms of $\operatorname{tg} \alpha$. | \frac{6\operatorname{tg}\alpha-20\operatorname{tg}^{3}\alpha+6\operatorname{tg}^{5}\alpha}{1-15\operatorname{tg}^{2}\alpha+15\operatorname{tg}^{4}\alpha-\operatorname{tg}^{6}\alpha} | 28 | 75 |
math | 351*. Find all natural $n>1$, for which $(n-1)!$ is divisible by $n$. | allcomposite,except4 | 26 | 5 |
math | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | 275 | 204 | 3 |
math |
G3 Let $A B C$ be a triangle in which ( $B L$ is the angle bisector of $\widehat{A B C}(L \in A C), A H$ is an altitude of $\triangle A B C(H \in B C)$ and $M$ is the midpoint of the side $[A B]$. It is known that the midpoints of the segments $[B L]$ and $[M H]$ coincides. Determine the internal angles of triangle $\... | 60 | 110 | 2 |
math | 9. Given the function
$$
f(x)=\frac{2 x}{a x+b}, f(1)=1, f\left(\frac{1}{2}\right)=\frac{2}{3} \text {. }
$$
Let $x_{1}=\frac{1}{2}, x_{n+1}=f\left(x_{n}\right)$. Then the general term formula for the sequence $\left\{x_{n}\right\}$ is $x_{n}=$ $\qquad$ . | \frac{2^{n-1}}{2^{n-1}+1} | 112 | 19 |
math | Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\
y^2 - 3z + p = x, \\
z^2 - 3x + p = y \end{cases}$ with real parameter $p$.
a) For $p \ge 4$, solve the considered system in the field of real numbers.
b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$.
(Jarosl... | x = y = z = 2 | 120 | 9 |
math | ## Task 1 - 261211
Determine all triples $(x ; y ; z)$ of numbers with the following properties (1), (2):
(1) The numbers $x, y, z$ are consecutive integers in this order.
(2) It holds: $x \cdot(x+y+z)=x \cdot y \cdot z$. | (-1;0;1),(0;1;2),(1;2;3) | 77 | 19 |
math | Shaovalov A.v.
Two players take turns coloring the sides of an $n$-gon. The first player can color a side that borders with zero or two colored sides, the second player - a side that borders with one colored side. The player who cannot make a move loses. For which $n$ can the second player win, regardless of how the f... | 4 | 79 | 1 |
math | 5. A small ball was released without initial speed from a height of $h=45$ m. The impact with the horizontal surface of the Earth is perfectly elastic. Determine at what moment of time after the start of the fall the average path speed of the ball will be equal to its instantaneous speed. The acceleration due to gravit... | 4.24 | 95 | 4 |
math | 2. The sequence $\left(a_{n}\right)_{n \geqslant 1}$ is defined as
$$
a_{1}=20, \quad a_{2}=30, \quad a_{n+2}=3 a_{n+1}-a_{n} \quad \text { for } n \geqslant 1 .
$$
Determine all natural $n$ for which $1+5 a_{n} a_{n+1}$ is a perfect square.
(Bulgarian) | 3 | 113 | 1 |
math | 9.2. A merchant bought several bags of salt in Tver and sold them in Moscow with a profit of 100 rubles. With all the money earned, he again bought salt in Tver (at the Tver price) and sold it in Moscow (at the Moscow price). This time the profit was 120 rubles. How much money did he spend on the first purchase? | 500 | 84 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{5^{2 x}-2^{3 x}}{\sin x+\sin x^{2}}$ | \ln\frac{25}{8} | 44 | 10 |
math | Example 11 Given that $\alpha$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Find the value of $\frac{\alpha^{3}-1}{\alpha^{5}+\alpha^{4}-\alpha^{3}-\alpha^{2}}$.
(1995, National Junior High School Mathematics League) | 20 | 79 | 2 |
math | ## Problem II - 4
Calculate the numbers $p$ and $q$ such that the roots of the equation
$$
x^{2} + p x + q = 0
$$
are $D$ and 1 - $D$, where $D$ is the discriminant of this quadratic equation. | (p,q)=(-1,0)(p,q)=(-1,\frac{3}{16}) | 66 | 21 |
math | ## Condition of the problem
Find the derivative.
$$
y=\frac{1+8 \operatorname{ch}^{2} x \cdot \ln (\operatorname{ch} x)}{2 \operatorname{ch}^{2} x}
$$ | \frac{\sinhx\cdot(4\cosh^{2}x-1)}{\cosh^{3}x} | 55 | 27 |
math | Martina came up with a method for generating a numerical sequence. She started with the number 52. From it, she derived the next term in the sequence as follows: $2^{2}+2 \cdot 5=4+10=14$. Then she continued in the same way and from the number 14, she got $4^{2}+2 \cdot 1=16+2=18$. Each time, she would take a number, d... | 18 | 155 | 2 |
math | Shapovalov A.V.
Two players take turns coloring the sides of an $n$-gon. The first player can color a side that borders with zero or two colored sides, the second player - a side that borders with one colored side. The player who cannot make a move loses. For which $n$ can the second player win, regardless of how the ... | 4 | 79 | 1 |
math | Example 3 (2nd IM() problem) For which values of $x$ is the inequality $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ satisfied? | -\frac{1}{2}\leqslantx<0 | 49 | 14 |
math | 1. Find all polynomials $f(x)$ of degree not higher than two such that for any real $x$ and $y$, the difference of which is rational, the difference $f(x)-f(y)$ is also rational. | f(x)=bx+ | 48 | 5 |
math | 9. (NET 2) ${ }^{\text {IMO4 }}$ Find all solutions in positive real numbers $x_{i}(i=$ $1,2,3,4,5)$ of the following system of inequalities:
$$
\begin{array}{l}
\left(x_{1}^{2}-x_{3} x_{5}\right)\left(x_{2}^{2}-x_{3} x_{5}\right) \leq 0, \\
\left(x_{2}^{2}-x_{4} x_{1}\right)\left(x_{3}^{2}-x_{4} x_{1}\right) \leq 0, \... | x_{1}=x_{2}=x_{3}=x_{4}=x_{5} | 281 | 20 |
math | 4. In the Cartesian coordinate system $x O y$, points $A, B$ are two moving points on the right branch of the hyperbola $x^{2}-y^{2}=1$. Then the minimum value of $\overrightarrow{O A} \cdot \overrightarrow{O B}$ is
(Wang Huixing provided the problem) | 1 | 74 | 1 |
math | 7. Let
$$
f(x)=\ln x-\frac{1}{2} a x^{2}-2 x(a \in[-1,0)) \text {, }
$$
and $f(x)<b$ holds for all $x$ in the interval $(0,1]$. Then the range of the real number $b$ is $\qquad$ . | (-\frac{3}{2},+\infty) | 79 | 12 |
math | Example 6. Find the general solution of the equation
$$
y^{\prime \prime}-3 y^{\prime}=x^{2}
$$ | C_{1}+C_{2}e^{3x}-\frac{1}{9}x^{3}-\frac{1}{9}x^{2}-\frac{2}{27}x | 32 | 45 |
math | 2. (17 points) Point $M$ lies inside segment $A B$, the length of which is 60 cm. Points are chosen: $N$ at the midpoint of $A M, P$ at the midpoint of $M B, C$ at the midpoint of $N M, D$ at the midpoint of $M P$. Find the length of segment $C D$. | 15 | 82 | 2 |
math | Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$. | y = \pm x^{1/2} + o(x^{1/2}) | 42 | 19 |
math | The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs? | 14400 | 52 | 5 |
math | 2. Dan is holding one end of a 26 inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along straight lines). If he starts with his hands together at the start and leaves his hands at the same height, how far does he need to pull his hands apart so that the bead moves upwa... | 24 | 78 | 2 |
math | 1. Given the sum of $n$ positive integers is 2017. Then the maximum value of the product of these $n$ positive integers is $\qquad$ . | 2^{2} \times 3^{671} | 38 | 13 |
math | 7.062. $8^{\frac{x-3}{3 x-7}} \sqrt[3]{\sqrt{0.25^{\frac{3 x-1}{x-1}}}}=1$. | \frac{5}{3} | 49 | 7 |
math | Task 1. On a plate, there are various candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Svetlana sequentially ate all of them, choosing each subsequent candy at random. Find the probability that the first and last candies eaten were of the same type. | \frac{14}{45} | 67 | 9 |
math | 3. In tetrahedron $ABCD$,
\[
\begin{array}{l}
\angle ABC=\angle BAD=60^{\circ}, \\
BC=AD=3, AB=5, AD \perp BC,
\end{array}
\]
$M, N$ are the midpoints of $BD, AC$ respectively. Then the size of the acute angle formed by lines $AM$ and $BN$ is \qquad (express in radians or inverse trigonometric functions). | \arccos \frac{40}{49} | 107 | 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 | 12. $k$ is a real number, let $f(x)=\frac{x^{4}+k x^{2}+1}{x^{4}+x^{2}+1}$. If for any three real numbers $a, b, c$ (which can be the same), there exists a triangle with side lengths $f(a), f(b), f(c)$, then the range of $k$ is $\qquad$. | -\frac{1}{2}<k<4 | 94 | 10 |
math | 4. Given arithmetic sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, the sums of the first $n$ terms are $S_{n}, T_{n}$ respectively, and $\frac{S_{n}}{T_{n}}=\frac{3 n+2}{2 n+1}$. Then $\frac{a_{7}}{b_{5}}=$ $\qquad$ | \frac{41}{19} | 94 | 9 |
math | 3. Let $x+y+z=1$. Then the minimum value of the function $u=2 x^{2}+3 y^{2}$ $+z^{2}$ is . $\qquad$ | \frac{6}{11} | 43 | 8 |
math | ### 3.493 Find the sum $1+\cos 4 \alpha+\cos 8 \alpha+\ldots+\cos 4 n \alpha$. | \frac{\sin2\alpha(n+1)\cdot\cos2n\alpha}{\sin} | 35 | 22 |
math | 【Question 2】Using 1 one, 2 twos, 2 threes to form some four-digit numbers, the total number of different four-digit numbers that can be formed is $\qquad$.
Translating the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 30 | 71 | 2 |
math | Example 10 (45th IMO Problem) Find all real-coefficient polynomials $P(x)$ such that for all real numbers $a, b, c$ satisfying $a b+b c+c a=0$, we have
$$
P(a-b)+P(b-c)+P(c-a)=2 P(a+b+c) .
$$ | P(x)=\alphax^{4}+\betax^{2} | 71 | 15 |
math | 2. Given a set of data $x_{1}, x_{2}, \cdots, x_{6}$ with the variance
$$
S^{2}=\frac{1}{6}\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{6}^{2}-24\right) \text {. }
$$
then the mean of the data $x_{1}+1, x_{2}+1, \cdots, x_{6}+1$ is | -1 \text{ or } 3 | 111 | 9 |
math | Four. (30 points) Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) If the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=1+\frac{1}{a_{n... | 13 | 222 | 2 |
math | 3-ча 1. Solve the system of equations:
\[
\left\{\begin{aligned}
1-x_{1} x_{2} & =0 \\
1-x_{2} x_{3} & =0 \\
1-x_{3} x_{4} & =0 \\
\cdots & \cdots \\
1-x_{n-1} x_{n} & =0 \\
1-x_{n} x_{1} & =0
\end{aligned}\right.
\] | x_{1}=x_{2}=\cdots=x_{n}=\1foroddn,\,x_{1}=x_{3}=\cdots=x_{n-1}=x_{2}=x_{4}=\cdots=x_{n}=\frac{1}{}(\neq0)forevenn | 106 | 68 |
math | \section*{Problem \(3-340943=341042\)}
On the side \(AB\) of the square \(ABCD\), a point \(X \neq A\) is chosen. The square is then divided into four regions by the segments \(AC\) and \(X D\).
Determine all possibilities for choosing \(X\) such that there exist natural numbers \(p, q\), and \(r\) for which the area... | |AX|=\frac{1}{r}\cdot|AB| | 118 | 14 |
math | 4. When the real number $a \in$ $\qquad$, there does not exist a real number $x$ such that $|x+a+1|+\left|x+a^{2}-2\right|<3$. | (-\infty,-2] \cup[0,1] \cup[3, \infty) | 48 | 23 |
math | Several natural numbers were written on the board, and the difference between any two adjacent numbers is the same. Kolya replaced different digits with different letters, and the same digits with the same letters in this sequence. Restore the original numbers if the sequence on the board is Т, ЕЛ, ЕК, ЛА, СС.
# | 5,12,19,26,33 | 69 | 13 |
math | Example 17 (1998 "Hope Cup" National Invitational Competition for Senior High School Training Question) Given $x, y, z \in \mathbf{R}^{+}$, find the maximum value of $\frac{x y + y z}{x^{2} + y^{2} + z^{2}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output th... | \frac{\sqrt{2}}{2} | 97 | 10 |
math | Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$ | \frac{3}{\sqrt{3} + 1} | 73 | 15 |
math | (Example 4 Given $0<x<\frac{3}{2}$, find the maximum value of the algebraic expression $x^{2}(3-2 x)$. | 1 | 37 | 1 |
math | 1. A mowing crew mowed the entire meadow in two days. On the first day, half of the meadow and another 3 hectares were mowed, and on the second day, a third of the remaining area and another 6 hectares were mowed. What is the area of the meadow? | 24 | 65 | 2 |
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