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 | In a class at school, all students are the same age, except for seven of them who are 1 year younger and two of them who are 2 years older. The sum of the ages of all the students in this class is 330. How many students are there in this class? | 37 | 62 | 2 |
math | 7. The real solution $(x, y, z)=$ $\qquad$ of the equation $2 \sqrt{x-4}+3 \sqrt{y-9}+4 \sqrt{z-16}=\frac{1}{2}(x+y+z)$ | (8,18,32) | 58 | 9 |
math | 5. In $\triangle A B C$, if $\tan A \tan B=\tan A \tan C+\tan C \tan B$, then $\frac{a^{2}+b^{2}}{c^{2}}=$ | 3 | 48 | 1 |
math | What is the probability that at least two individuals in a group of 13 were born in September? (For simplicity, consider each month equally likely.) | 0.296 | 31 | 5 |
math | ## Task 2.
A natural number $n$ is good if we can color each side and diagonal of a regular $n$-gon in some color so that for any two vertices $A$ and $B$, there is exactly one vertex $C$, different from $A$ and $B$, such that the segments $\overline{A B}, \overline{B C}$, and $\overline{C A}$ are colored the same col... | 7 | 126 | 1 |
math | Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
(i) $ p(x)\equal{}p(\minus{}x)$ for all $ x$,
(ii) $ p(x)\ge0$ for all $ x$,
(iii) $ p(0)\equal{}1$
(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1\minus{}x_2|\equal{}2$. | p(x) = a(x^2 - 1)^2 + 1 - a | 106 | 19 |
math | Example 5 It is known that for any $x$,
$$
a \cos x + b \cos 2x \geqslant -1
$$
always holds. Find the minimum value of $a + b$.
(2009, Peking University Independent Admission Examination) | -1 | 63 | 2 |
math | 13th APMO 2001 Problem 2 Find the largest n so that the number of integers less than or equal to n and divisible by 3 equals the number divisible by 5 or 7 (or both). Solution | 65 | 50 | 2 |
math | 8. Let integer $n \geqslant 3, \alpha, \beta, \gamma \in (0,1), a_{k}, b_{k}, c_{k} \geqslant 0 (k=1,2, \cdots, n)$ satisfy
$$
\begin{array}{l}
\sum_{k=1}^{n}(k+\alpha) a_{k} \leqslant \alpha, \\
\sum_{k=1}^{n}(k+\beta) b_{k} \leqslant \beta, \\
\sum_{k=1}^{n}(k+\gamma) c_{k} \leqslant \gamma .
\end{array}
$$
If for a... | \frac{\alpha \beta \gamma}{(1+\alpha)(1+\beta)(1+\gamma)-\alpha \beta \gamma} | 250 | 29 |
math | 9.12. Calculate in the same way the sums $S_{2}(n)=1^{2}+$ $+2^{2}+\ldots+n^{2}$ and $S_{3}(n)=1^{3}+2^{3}+\ldots+n^{3}$. | S_{2}(n)=\frac{n(n+1)(2n+1)}{6},\quadS_{3}(n)=(\frac{n(n+1)}{2})^2 | 62 | 41 |
math | Example 6. Given the equation $x^{2}+(a-6) x+a=0$ ( $a$ $\neq 0$ ) with both roots being integers. Try to find the integer $a$. (1989, Sichuan Province Junior High School Mathematics Competition) | 16 | 63 | 2 |
math | 357. Find \( f^{\prime}(1 / 5) \) if \( f(x)=\operatorname{arctg} 5 x+x^{2} \). | 2.9 | 40 | 3 |
math | A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloo... | 2827 | 98 | 4 |
math | 10. In the garage, there are several two-wheeled motorcycles and four-wheeled sedans. The ratio of the number of vehicles to the number of wheels is $2: 5$. What is the ratio of the number of motorcycles to the number of sedans? | 3:1 | 58 | 3 |
math | 13.7.5 ** Given the parabola $y^{2}=2 p x(p>0)$, two perpendicular chords $F A$ and $F B$ are drawn through the focus $F$ of the parabola. Find the minimum value of the area of $\triangle F A B$. | (3-2\sqrt{2})p^{2} | 65 | 13 |
math | 2. Solve the equation $\left(\frac{3 x}{2}\right)^{\log _{3}(8 x)}=\frac{x^{7}}{8}$. | \frac{729}{8},2 | 36 | 10 |
math | Let's determine $m$ such that
$$
x^{4}-5 x^{2}+4 x-m
$$
is divisible by $(2 x+1)$. | -\frac{51}{16} | 37 | 9 |
math | Example 5. Find $\lim _{x \rightarrow \infty} \frac{\ln x}{x^{\alpha}}$ for $\alpha>0, x>0$. | \lim_{xarrow\infty}\frac{\lnx}{x^{\alpha}}=0\text{for}\alpha>0 | 38 | 29 |
math | Example 11 (2004 China Mathematical Olympiad) In a convex quadrilateral $E F G H$, the vertices $E, F, G, H$ are on the sides $A B, B C, C D, D A$ of another convex quadrilateral $A B C D$, respectively, satisfying: $\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A}=1$; and the points $A... | \lambda | 280 | 2 |
math | 4. Given that the circumradius of $\triangle A B C$ is 1, and $A B, B C, \frac{4}{3} C A$ form a geometric sequence in order, then the maximum value of $B C$ is $\qquad$ . | \frac{4\sqrt{2}}{3} | 58 | 12 |
math | Three. (20 points) Let $a, b$ be integers. How many solutions does the system of equations
$$
\left\{\begin{array}{l}
{[x]+2 y=a,} \\
{[y]+2 x=b}
\end{array}\right.
$$
(here $[x]$ denotes the greatest integer less than or equal to $x$) have? | 2 \text{ or } 1 | 85 | 8 |
math | 12. Using 6 different colors to color $\mathrm{n}(\mathrm{n} \geq 2)$ interconnected regions $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{A}_{\mathrm{n}}$, such that any two adjacent regions are colored differently, the number of all different coloring schemes $a_{\mathrm{n}}=$ $\qquad$ | 5(-1)^{\mathrm{n}}+5^{\mathrm{n}} | 84 | 15 |
math | Example 8 Given a positive integer $n \geqslant 2, x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+}$ and $x_{1}+x_{2}+\cdots+x_{n}=\pi$, find the minimum value of $\left(\sin x_{1}+\frac{1}{\sin x_{1}}\right)\left(\sin x_{2}+\frac{1}{\sin x_{2}}\right) \cdots\left(\sin x_{n}+\frac{1}{\sin x_{n}}\right)$. | (\sin\frac{\pi}{n}+\frac{1}{\sin\frac{\pi}{n}})^{n} | 139 | 27 |
math | 14. (15 points) A rectangular container without a lid, with a length of 40, width of 25, and height of 60 (neglecting the thickness), is filled with water to a depth of $a$, where $0<a \leqslant 60$. Now, a cubic iron block with an edge length of 10 is placed on the bottom of the container. What is the water depth afte... | \frac{10}{9}when0<<9;+1when9\leqslant<59;60when59\leqslant\leqslant60 | 104 | 43 |
math | 4. [6 points] Solve the inequality $4 x^{4}+x^{2}+4 x-5 x^{2}|x+2|+4 \geqslant 0$. | (-\infty;-1]\cup[\frac{1-\sqrt{33}}{8};\frac{1+\sqrt{33}}{8}]\cup[2;+\infty) | 43 | 43 |
math | Example 7 A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex with one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the same... | a_{n}=\left\{\begin{array}{ll}3^{n}+1, & n=2 k+1 \text {; } \\ 3^{n}+3, & n=2 k .\end{array}\right.} | 101 | 56 |
math | nine (not necessarily distinct) nine-digit numbers have been formed; each digit has been used in each number exactly once. What is the maximum number of zeros that the sum of these nine numbers can end with
# | 8 | 43 | 1 |
math | $$
\underbrace{2 \times 2 \times \ldots \times 2}_{20 \uparrow 2}-1
$$
The result's units digit is $\qquad$ | 5 | 43 | 1 |
math | 5. Given that $x, y, z$ are 3 real numbers greater than or equal to 1, then
$$
\left(\frac{\sqrt{x^{2}(y-1)^{2}+y^{2}}}{x y}+\frac{\sqrt{y^{2}(z-1)^{2}+z^{2}}}{y z}+\frac{\sqrt{z^{2}(x-1)^{2}+x^{2}}}{z x}\right)^{2}
$$
the sum of the numerator and denominator of the minimum value written as a simplified fraction is $\... | 11 | 133 | 2 |
math | 2. If the real number $\alpha$ satisfies $\cos \alpha=\tan \alpha$, then $\frac{1}{\sin \alpha}+\cos ^{4} \alpha=$ $\qquad$ | 2 | 43 | 1 |
math | 15. (MEX 2) Determine for which positive integers $k$ the set
$$
X=\{1990,1990+1,1990+2, \ldots, 1990+k\}
$$
can be partitioned into two disjoint subsets $A$ and $B$ such that the sum of the elements of $A$ is equal to the sum of the elements of $B$. | k=4r+3,r\geq0,k=4r,r\geq23 | 96 | 21 |
math | 12. There are 900 three-digit numbers (100, 101, 999). If these three-digit numbers are printed on cards, with one number per card, some cards, when flipped, still show a three-digit number, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when fl... | 34 | 126 | 2 |
math | 1. The sum of all real numbers $x$ that satisfy the equation $\sqrt{3 x-4}+\sqrt[3]{5-3 x}=1$ is $\qquad$ . | \frac{22}{3} | 41 | 8 |
math | 10. Let $S_{n}=1+2+\cdots+n$. Then among $S_{1}, S_{2}$, $\cdots, S_{2015}$, there are $\qquad$ numbers that are multiples of 2015. | 8 | 58 | 1 |
math | 5. Given the number $800 \ldots 008$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 198. In how many ways can this be done? | 14080 | 60 | 5 |
math | Gardener Mr. Malina was selling strawberries. In the last nine crates, he had 28, 51, 135, 67, 123, 29, 56, 38, and 79 strawberry plants, respectively. He sold the crates whole, never removing any plants from the crates. The gardener wanted to sell the crates to three customers so that nothing was left and each of thes... | 202 | 121 | 3 |
math | A cell can divide into 42 or 44 smaller cells. How many divisions are needed to obtain, starting from one cell, exactly 1993 cells?
## - Solutions - | 48 | 40 | 2 |
math | [Cubic Polynomials]
One of the roots of the equation $x^{3}-6 x^{2}+a x-6=0$ is 3. Solve the equation.
# | x_{1}=1,x_{2}=2,x_{3}=3 | 39 | 15 |
math | ## Task A-2.5.
Ivica made a large cube with a side length of $n$ from $n^3$ unit cubes and then painted some of the six faces of the large cube, while leaving others unpainted. When he disassembled the large cube, he found that exactly 1000 unit cubes had no painted faces. Show that this is indeed possible and determi... | 3 | 96 | 1 |
math | A game works as follows: the player pays $2$ tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive $3$ tokens; if the coin lands on tails, they receive nothing. A player starts with $2$ tokens and keeps playing this game until they do not have enough tokens to play again. What... | \frac{3 - \sqrt{5}}{2} | 110 | 13 |
math | Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$. | 2^{1990} - 1 | 66 | 10 |
math | 7. Given odd primes $x, y, z$ satisfying
$$
x \mid \left(y^{5}+1\right), y \mid \left(z^{5}+1\right), z \mid \left(x^{5}+1\right) \text {. }
$$
Find the minimum value of the product $x y z$. (Cheng Chuanping) | 2013 | 83 | 4 |
math | 7. Let $P$ be a moving point on the ellipse $\frac{y^{2}}{4}+\frac{x^{2}}{3}=1$, and let points $A(1,1), B(0,-1)$. Then the maximum value of $|P A|+|P B|$ is $\qquad$ . | 5 | 72 | 1 |
math | Example 4: From the numbers $1, 2, \cdots, 2012$, select a set of numbers such that the sum of any two numbers cannot be divisible by their difference. How many such numbers can be selected at most?
(2012, Joint Autonomous Admission Examination of Peking University and Other Universities) | 671 | 71 | 3 |
math | 3. (6 points) Using the digits $3, 0, 8$, the number of three-digit numbers that can be formed without repeating digits is $\qquad$. | 4 | 36 | 1 |
math | [ Arithmetic. Mental calculation, etc. ] [ Problems on percentages and ratios ]
Author: Raskina I.V.
Children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms, and if instead Anya gives all her mushrooms to Sasha, Sasha will ha... | 6 | 89 | 1 |
math | 5・17 Let $a<b<c<d$. If the variables $x, y, z, t$ are some permutation of the numbers $a, b, c, d$, how many different values can the expression
$$
n=(x-y)^{2}+(y-z)^{2}+(z-t)^{2}+(t-x)^{2}
$$
take? | 3 | 81 | 1 |
math | 8. Given real numbers $x, y, z$ satisfy
$$
\begin{array}{l}
\left(2 x^{2}+8 x+11\right)\left(y^{2}-10 y+29\right)\left(3 z^{2}-18 z+32\right) \\
\leqslant 60 .
\end{array}
$$
Then $x+y-z=$ $\qquad$ | 0 | 97 | 1 |
math | 3.358. $\frac{\cos 68^{\circ} \cos 8^{\circ}-\cos 82^{\circ} \cos 22^{\circ}}{\cos 53^{\circ} \cos 23^{\circ}-\cos 67^{\circ} \cos 37^{\circ}}$. | 1 | 80 | 1 |
math | Example 1 The sum of several positive integers is 1976, find the maximum value of their product. | 3^{658} \times 2 | 24 | 10 |
math | Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$. | \sin \angle PBO = \frac{1}{6} | 72 | 15 |
math | 6. The ball invited 2018 couples, who were arranged in areas numbered $1,2, \cdots, 2018$. The ball organizers stipulated: when the ball reaches the $i$-th minute, the couple standing in area $s_{i}$ (if there is someone in this area) will move to area $r_{i}$, and the couple originally in area $r_{i}$ (if there is som... | 505 | 239 | 3 |
math | 10. The number of integer solutions to the inequality $\log _{6}(1+\sqrt{x})>\log _{25} x$ is $\qquad$ . | 24 | 37 | 2 |
math | 13.044. Sugar and granulated sugar were delivered to the store in 63 bags, totaling 4.8 tons, with the number of bags of granulated sugar being $25 \%$ more than those of sugar. The mass of each bag of sugar was $3 / 4$ of the mass of a bag of granulated sugar. How much sugar and how much granulated sugar were delivere... | 1.8 | 88 | 3 |
math | 2.3. Find all values of $x$ for which there exists at least one $a, -1 \leq a \leq 2$, such that the inequality holds: $(2-a) x^{3}+(1-2 a) x^{2}-6 x+\left(5+4 a-a^{2}\right)<0$. | x\in(-\infty,-2)\cup(0,1)\cup(1,\infty) | 74 | 23 |
math | 9.1. For what values of the parameter $a$ do the equations $a x+a=7$ and $3 x-a=17$ have a common integer root? | 1 | 38 | 1 |
math | 1. Find the largest real number $\theta(\theta<\pi)$ such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0 \text {, and } \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1
$$
(2015, Harvard-MIT Mathematics Tournament) | \frac{2046 \pi}{2047} | 98 | 15 |
math | Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$. | \frac{2017}{2016} | 35 | 13 |
math | How to measure 15 minutes using sandglasses of 7 minutes and 11 minutes?
# | 15 | 22 | 2 |
math | 5. The elements of set $M$ are consecutive positive integers, and $|M| \geqslant 2, M$ the sum of the elements in $M$ is 2002, the number of such sets $M$ is $\qquad$. | 7 | 58 | 1 |
math | A trapez has parallel sides $A B$ and $C D$, and the intersection point of its diagonals is $M$. The area of triangle $A B M$ is 2, and the area of triangle $C D M$ is 8. What is the area of the trapezoid? | 18 | 66 | 2 |
math | Example 4 Let real numbers $a \geqslant b \geqslant c \geqslant d>0$. Find the minimum value of the function
$$
\begin{array}{l}
f(a, b, c, d) \\
=\left(1+\frac{c}{a+b}\right)\left(1+\frac{d}{b+c}\right)\left(1+\frac{a}{c+d}\right)\left(1+\frac{b}{d+a}\right)
\end{array}
$$ | (\frac{3}{2})^{4} | 115 | 10 |
math | 11.4. (7 points)
Solve the equation $(\sqrt[5]{7+4 \sqrt{3}})^{x}+(\sqrt[5]{7-4 \sqrt{3}})^{x}=194$. | -10;10 | 52 | 6 |
math | 5. Given a fixed point $A(1,1)$, and $F$ is the left focus of the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$. A moving point $P$ is on the ellipse. Try to find the maximum and minimum values of $|P F|+|P A|$, and the coordinates of point $P$ when these values are achieved. | 5\sqrt{2} | 91 | 6 |
math | 15. (6 points) The poetry lecture lasted for 2 hours $m$ minutes, and at the end, the positions of the hour and minute hands on the clock were exactly swapped compared to when it started. If $[x]$ represents the integer part of the decimal number $x$, then $[m]=$ $\qquad$ . | 46 | 71 | 2 |
math | 4. Let $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, where $a, b, c, d$ are constants. If $f(1)=$ $1, f(2)=2, f(3)=3$, then the value of $f(0)+f(4)$ is $\qquad$ . | 28 | 82 | 2 |
math | Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively p... | 6509 | 130 | 4 |
math | 12. (18th Korean Mathematical Olympiad) Find all positive integers $n$ that can be uniquely expressed as the sum of the squares of 5 or fewer positive integers (here, two expressions with different orders of summation are considered the same, such as $3^{2}+4^{2}$ and $4^{2}+3^{2}$ being considered the same expression ... | 1,2,3,6,7,15 | 86 | 12 |
math | Example 6. Integrate the equation $y y^{\prime \prime}-y^{\prime 2}=0$. | C_{2}e^{C_{1}x} | 26 | 12 |
math | The hundreds digit of 2025 is 0, and after removing the 0, it becomes 225 (only the 0 in the hundreds place is removed), $225 \times 9=2025$, such a 4-digit number is called a "zero-clever number". Therefore, please list all the "zero-clever numbers" are $\qquad$. | 2025,4050,6075 | 84 | 14 |
math | Example 12. Find $\lim _{n \rightarrow \infty} \frac{1^{2}+2^{2}+3^{2}+\ldots+n^{2}}{n^{3}}$. | \frac{1}{3} | 47 | 7 |
math | Determine all pairs $(p, q)$, $p, q \in \mathbb {N}$, such that
$(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$. | (p, q) = (1, 1); (2, 2) | 56 | 19 |
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 | (50 points) 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 $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3}{2}\sqrt{3} | 136 | 11 |
math | 1. 10 Start from 1, write natural numbers in sequence, and ask what the number at the one millionth position is? | 1 | 29 | 1 |
math | 12. Given that $F$ is the right focus of the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the ratio of the distance from any point $P$ on the ellipse $C$ to the point $F$ to the distance from point $P$ to the line $l: x=m$ is $\frac{1}{2}$.
(1) Find the equation of the line $l$.
(2) Let $A$ be the left vertex of... | (1,0),(7,0) | 205 | 9 |
math | 1. Given points $A(1,1)、B(3,2)、C(2,3)$ and line $l: y=k x(k \in \mathbf{R})$. Then the maximum value of the sum of the squares of the distances from points $A、B、C$ to line $l$ is $\qquad$ | 27 | 75 | 2 |
math | 505. Solve the system of equations:
$$
\begin{gathered}
x+y+z=9 \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1 \\
x y+x z+y z=27
\end{gathered}
$$
Problem from "Mathesis". | x=y=z=3 | 71 | 5 |
math | Given a circle with center $O$ and radius 1. From point $A$, tangents $A B$ and $A C$ are drawn to the circle. Point $M$, lying on the circle, is such that the quadrilaterals $O B M C$ and $A B M C$ have equal areas. Find $M A$. | 1 | 73 | 1 |
math |
Problem 8.1. Let three numbers $a, b$ and $c$ be chosen so that $\frac{a}{b}=\frac{b}{c}=\frac{c}{a}$.
a.) Prove that $a=b=c$.
b.) Find the $\operatorname{sum} x+y$ if $\frac{x}{3 y}=\frac{y}{2 x-5 y}=\frac{6 x-15 y}{x}$ and the expression $-4 x^{2}+36 y-8$ has its maximum value.
| 2 | 122 | 1 |
math | 2. (1988 US Mathcounts Math Competition) In a game, answering an easy question scores 3 points, and answering a hard question scores 7 points. Among the set of integers that cannot be the total score of a contestant, find the maximum value.
untranslated text remains in its original format and line breaks are preserve... | 11 | 69 | 2 |
math | 13. A and B are two car retailers (hereinafter referred to as A and B) who ordered a batch of cars from a car manufacturer. Initially, the number of cars A ordered was 3 times the number of cars B ordered. Later, due to some reason, A transferred 6 cars from its order to B. When picking up the cars, the manufacturer pr... | 18 | 144 | 2 |
math | 8. Given $x, y>0$, and $x+2 y=2$. Then the minimum value of $\frac{x^{2}}{2 y}+\frac{4 y^{2}}{x}$ is . $\qquad$ | 2 | 51 | 1 |
math | 4. Two classes are planting trees. Class one plants 3 trees per person, and class two plants 5 trees per person, together planting a total of 115 trees. The maximum sum of the number of people in the two classes is $\qquad$ . | 37 | 56 | 2 |
math | Consider a right circular cylinder with radius $r$ and height $m$. There exists another right circular cylinder that is not congruent to the given one, but has the same volume and surface area as the given one. Determine the dimensions of the latter cylinder in general, and specifically, if $r=8 \text{~cm}, m=48 \text{... | \frac{}{2} | 94 | 6 |
math | 11.4. 2011 Warehouses are connected by roads in such a way that from any warehouse you can drive to any other, possibly by driving along several roads. On the warehouses, there are $x_{1}, \ldots, x_{2011}$ kg of cement respectively. In one trip, you can transport an arbitrary amount of cement from any warehouse to ano... | 2010 | 209 | 4 |
math | Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the for... | 254 | 112 | 3 |
math | Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime. | 17 | 31 | 4 |
math | 11.3. Consider the quadratic trinomial $P(x)=a x^{2}+b x+c$, which has distinct positive roots. Vasya wrote four numbers on the board: the roots of $P(x)$, as well as the roots of the trinomial $Q(x)=c x^{2}+b x+a$. What is the smallest integer value that the sum of the four written numbers can have? | 5 | 90 | 1 |
math | 13.008. Seawater contains $5 \%$ salt by mass. How much fresh water needs to be added to 30 kg of seawater to make the salt concentration $1.5 \%$? | 70 | 47 | 2 |
math | a) Find all solutions of the system of equations (1)-(3). - Furthermore, provide a system of equations that differs from the given one by exactly one coefficient and
b) has infinitely many solutions,
c) which is not satisfied by any triple of real numbers $x, y, z$.
$$
\begin{gathered}
2 x+3 y+z=1 \\
4 x-y+2 z=2 \\
... | x=\frac{1}{2},y=0,z=0 | 106 | 14 |
math | 10.4. Ten chess players over nine days played a full round-robin tournament, during which each of them played exactly one game with each other. Each day, exactly five games were played, with each chess player involved in exactly one of them. For what maximum $n \leq 9$ can it be claimed that, regardless of the schedule... | 5 | 112 | 1 |
math | 8. [25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.) | 42 | 56 | 2 |
math | 2. From the numbers $1,2, \cdots, 2017$, select $n$ numbers such that the difference between any two of these $n$ numbers is a composite number. The maximum value of $n$ is $\qquad$ | 505 | 55 | 3 |
math | Let $n$ be a fixed positive integer. How many sequences $1 \leq a_{1}<a_{2}<\cdots<a_{k} \leq n$ are there, in which the terms with odd indices are odd, and the terms with even indices are even integers? | A_{n}=\frac{(\frac{1+\sqrt{5}}{2})^{n+2}-(\frac{1-\sqrt{5}}{2})^{n+2}}{\sqrt{5}}-1 | 61 | 49 |
math | 10. Given that line segment $A B$ is the diameter of sphere $O$ with radius 2, points $C, D$ are on the surface of sphere $O$, $C D=2, A B \perp C D$, $45^{\circ} \leqslant \angle A O C \leqslant 135^{\circ}$, then the range of the volume of tetrahedron $A B C D$ is $\qquad$ . | [\frac{4}{3},\frac{4\sqrt{3}}{3}] | 107 | 19 |
math | $3 \cdot 25$ For each positive integer $n$, let
$$\begin{array}{l}
S_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} \\
T_{n}=S_{1}+S_{2}+S_{3}+\cdots+S_{n} \\
U_{n}=\frac{1}{2} T_{1}+\frac{1}{3} T_{2}+\frac{1}{4} T_{3}+\cdots+\frac{1}{n+1} T_{n}
\end{array}$$
Find integers $0<a, b, c, d<1000000$, such that
$$\begin{... | T_{1988}=1989 S_{1989}-1989, \quad U_{1988}=1990 S_{1989}-3978 | 219 | 47 |
math | 10. The number of positive integers not exceeding 2012 and having exactly three positive divisors is $\qquad$ . | 14 | 28 | 2 |
math | 7.271. $\left\{\begin{array}{l}3^{x} \cdot 2^{y}=972 \\ \log _{\sqrt{3}}(x-y)=2 .\end{array}\right.$ | (5;2) | 52 | 5 |
math | 4. 1 class has 6 boys, 4 girls, and now 3 class leaders need to be selected from them, requiring that there is at least 1 girl among the class leaders, and each person has 1 role, then there are $\qquad$ ways to select. | 600 | 60 | 3 |
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