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 | 28 Remove one number from the set $1,2, \cdots, n+1$, and arrange the remaining numbers into a sequence $a_{1}, a_{2}, a_{3}, \cdots, a_{n}$, such that the $n$ numbers $\left|a_{1}-a_{2}\right|,\left|a_{2}-a_{3}\right|, \cdots,\left|a_{n}-a_{1}\right|$ are all different. For what values of $n(n \geqslant 3)$ can the ab... | n=4korn=4k-1,(k\in{N}) | 127 | 17 |
math | 2.2. Find all possible values of
$$
\left\lfloor\frac{x-p}{p}\right\rfloor+\left\lfloor\frac{-x-1}{p}\right\rfloor,
$$
where $x$ is a real number and $p$ is a nonzero integer.
Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. | -3,-2,-1,0 | 87 | 8 |
math | Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D,E$, and $F$ are chosen on $AB, BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1... | 67 | 148 | 2 |
math | Example. Find the work of the force
\[
\vec{F}=(x-y) \vec{i}+\vec{j}
\]
when moving along the curve \( L \)
\[
x^{2}+y^{2}=4 \quad(y \geq 0)
\]
from point \( M(2,0) \) to point \( N(-2,0) \). | 2\pi | 84 | 3 |
math | G10.3 If $k=\frac{3 \sin \theta+5 \cos \theta}{2 \sin \theta+\cos \theta}$ and $\tan \theta=3$, find $k$. | 2 | 45 | 1 |
math | 42. Find the algebraic complements of the elements $a_{13}, a_{21}, a_{31}$ of the determinant
$$
D=\left|\begin{array}{rrr}
-1 & 2 & 3 \\
2 & 0 & -3 \\
3 & 2 & 5
\end{array}\right|
$$ | A_{13}=4,A_{21}=-4,A_{31}=-6 | 78 | 20 |
math | 9. Inside the tetrahedron $ABCD$ there is a point $O$, such that the lines $AO, BO, CO, DO$ intersect the faces $BCD, ACD, ABD, ABC$ at points $A_1, B_1, C_1, D_1$ respectively, and $\frac{AO}{A_1O}=\frac{BO}{B_1O}=\frac{CO}{C_1O}=\frac{DO}{D_1O}=k$. Find all possible values of $k$.
(1968 Bulgarian Competition Problem) | 3 | 129 | 1 |
math | 1. Find all values of $p$, for each of which the numbers $4 p+5, 2 p$ and $|p-3|$ are respectively the first, second, and third terms of some geometric progression. | p=-1,p=\frac{15}{8} | 47 | 12 |
math | 10. A. Xiaoming volunteered to sell pens at a stationery store one day. Pencils were sold at 4 yuan each, and ballpoint pens at 7 yuan each. At the beginning, it was known that he had a total of 350 pencils and ballpoint pens. Although he did not sell them all that day, his sales revenue was 2013 yuan. Then he sold at ... | 207 | 97 | 3 |
math | 4. Find all pairs of natural numbers $a$ and $b$, for which out of the four statements
5) $a^{2}+6 a+8$ is divisible by $b$;
6) $\quad a^{2}+a b-6 b^{2}-15 b-9=0$
7) $a+2 b+2$ is divisible by 4;
8) $a+6 b+2$ is a prime number
three are true, and one is false. | =5,b=1;=17,b=7 | 108 | 12 |
math | 1. Let the set $A=\{1,2,3, \cdots, 99\}, B=\{2 x \mid x \in A\}, C=\{x \mid 2 x \in A\}$, then the number of elements in $B \cap C$ is $\qquad$ . | 24 | 70 | 2 |
math | Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions?
[i]Propose... | 90 | 107 | 2 |
math | Find all positive integers $A,B$ satisfying the following properties:
(i) $A$ and $B$ has same digit in decimal.
(ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$) | (3, 6), (13, 52) | 99 | 14 |
math | Starting with a natural number, Márcio replaces this number with the sum of its digits, obtaining a new number, with which he repeats the process, until he finally arrives at a number with only one digit. For example, Márcio replaces 1784102 with 23 and then with 8. He also applies this process to lists of $N$ natural ... | 9,8,4,2232445 | 225 | 13 |
math | 7. In the Magic Land, there is a magic stone that grows uniformly upwards. To prevent it from piercing the sky, the Elders of the Immortal World decided to send plant warriors to consume the magic stone and suppress its growth. Each plant warrior consumes the same amount every day. If 14 plant warriors are dispatched, ... | 17 | 124 | 2 |
math | 766. On the sphere $x^{2}+y^{2}+z^{2}=676$ find the points where the tangent plane is parallel to the plane $3 x-12 y+4 z=0$. | (6,-24,8)(-6,24,-8) | 51 | 16 |
math | Fifteen identical sheets of paper placed on top of each other, I folded them all at once. This resulted in a "booklet" whose pages I numbered in sequence from 1 to 60. Which three other numbers are written on the same sheet of paper as the number $25?$
(L. Šimünek) | 26,35,36 | 71 | 8 |
math | \section*{Problem 3 - 071243}
Give all functions \(y=f(x)\) that satisfy the equation
\[
a \cdot f\left(x^{n}\right)+f\left(-x^{n}\right)=b x
\]
in the largest possible domain (within the range of real numbers), where \(b\) is any real number, \(n\) is any odd natural number, and \(a\) is a real number with \(|a| \n... | f(x)=\frac{b}{-1}\cdot\sqrt[n]{x} | 110 | 18 |
math | 8. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=2, a_{2}=1, \\
a_{n+2}=\frac{n(n+1) a_{n+1}+n^{2} a_{n}+5}{n+2}-2\left(n \in \mathbf{N}_{+}\right) .
\end{array}
$$
Then the general term formula of $\left\{a_{n}\right\}$ is $a_{n}=$ | \frac{(n-1)!+1}{n} | 123 | 12 |
math | Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by... | \binom{2n-1}{n-1} n! | 129 | 15 |
math | ## Task 1 - 080731
We are looking for natural numbers that, when divided by 7, leave a remainder of 4, when divided by 4, leave a remainder of 3, and when divided by 3, leave a remainder of 1.
a) Determine the smallest such natural number!
b) How can one obtain further natural numbers from the number found in a) tha... | 67 | 92 | 2 |
math | $3 \cdot 1$ If the numbers $p, p+10, p+14$ are all prime, find $p$.
(Kiev Mathematical Olympiad, 1981) | 3 | 44 | 1 |
math | Task 12. (16 points)
The budget of the Petrovs consists of the following income items:
- parents' salary after income tax deduction - 56000 rubles;
- grandmother's pension - 14300 rubles;
- son's scholarship - 2500 rubles
Average monthly expenses of the family include:
- utility payments - 9800 rubles;
- food - 210... | 16740 | 185 | 5 |
math | 24. A positive integer is called frierdly if it is divisible by the sum of its digits. For example, 111 is friendly but 123 is not. Find the number of all two-digit friendly numbers. | 23 | 49 | 2 |
math | There are the following two number sequences: (1) $1,3,5,7, \cdots, 2013,2015,2017$; (2) 1, 4, 7, 10, , , 2011, 2014, 2017. The numbers that appear in both sequences are $\qquad$ in total. | 337 | 92 | 3 |
math | 【Question 22】
A reservoir has five water pipes. To fill the reservoir, the first four pipes together need 6 hours; the last four pipes together need 8 hours. If only the first and fifth pipes are opened, it takes 12 hours to fill the reservoir. How many hours will it take to fill the reservoir if only the fifth pipe is... | 48 | 78 | 2 |
math | 1-36 Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square.
Find all such four-digit numbers, when 400 is written to their left, the result is a perfect square. | 4001 \text{ or } 8004 | 54 | 14 |
math | 10. (20 points) Let $\lambda$ be a positive real number. For any pairwise distinct positive real numbers $a, b, c$, we have
$$
\frac{a^{3}}{(b-c)^{2}}+\frac{b^{3}}{(c-a)^{2}}+\frac{c^{3}}{(a-b)^{2}} \geqslant \lambda(a+b+c) \text {. }
$$
Find the maximum value of $\lambda$. | 1 | 104 | 1 |
math | 6. Given the quadratic function $f(x)=a x^{2}+b x+c, a$ $\in \mathbf{N}_{+}, c \geqslant 1, a+b+c \geqslant 1$, the equation $a x^{2}+b x+c$ $=0$ has two distinct positive roots less than 1. Then the minimum value of $a$ is | 5 | 88 | 1 |
math | (1) Does there exist a natural number $n$, such that $n+S(n)=1980$?
(2) Prove that among any two consecutive natural numbers, at least one can be expressed in the form $n+S(n)$ (where $n$ is another natural number). | 1962+S(1962)=1980 | 62 | 15 |
math | Calculate the minimum of $\sqrt{(x-1)^{2}+y^{2}}+\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \in \mathbb{R}$. | 2+\sqrt{3} | 53 | 6 |
math | 11. Find $\varphi(\pi / 4)$, if $\varphi(t)=\frac{2 t}{1+\sin ^{2} t}$. | \frac{\pi}{3} | 36 | 7 |
math | \section*{Problem 1}
A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
| 23 | 62 | 2 |
math | 8.275. $\frac{\cot 4 t}{\sin ^{2} t}+\frac{\cot t}{\sin ^{2} 4 t}=0$. | t_{1}=\frac{\pik}{5},k\neq5;t_{2}=\frac{\pi}{6}(2n+1),n\neq3+1,k,n,\inZ | 40 | 45 |
math | 4. Try to determine all positive integer pairs $(a, b)$ such that $a b^{2}+b+7$ divides $a^{2} b+a+b$. | (,b)=(11,1),(49,1),(7k^{2},7k)(k\in{N}^{*}) | 38 | 31 |
math | 237. $\left\{\begin{array}{l}x+y+\sqrt{x^{2}+y^{2}}=\frac{x y}{2} \\ x y=48 .\end{array}\right.$ | 8,6 | 48 | 3 |
math | 18. Find the largest integer $n$ such that $n$ is a divisor of $a^{5}-a$ for all integers $a$. | 30 | 32 | 2 |
math | 10.316. In a right-angled triangle, the distance from the midpoint of the hypotenuse to one of the legs is 5 cm, and the distance from the midpoint of this leg to the hypotenuse is 4 cm. Calculate the area of the triangle. | \frac{200}{3} | 60 | 9 |
math | 1. $[x]$ represents the greatest integer not exceeding the real number $x$. Suppose the real number $x$ is not an integer, and $x+\frac{99}{x}=[x]+\frac{99}{[x]}$. Then, the value of $x$ is | -9.9 | 62 | 4 |
math | 5. There are six thin rods, with lengths sequentially being $3$, $2 \sqrt{2}$, $2$, $2$, $2$, $2$. Use them to form a triangular pyramid. Then, the cosine value of the angle formed by the two longer edges is $\qquad$. | \frac{13\sqrt{2}}{24} | 62 | 14 |
math | $3 \cdot 68$ For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+3 \quad(i=1$, $2, \cdots, n)$. If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime. Find the largest possible value of $n$.
（Hungarian Mathematical Olympiad, 1990) | 2 | 127 | 1 |
math |
Problem 2. The quadratic function $f(x)=-x^{2}+4 p x-p+1$ is given. Let $S$ be the area of the triangle with vertices at the intersection points of the parabola $y=f(x)$ with the $x$-axis and the vertex of the same parabola. Find all rational $p$, for which $S$ is an integer.
| 0,1,\frac{1}{4},-\frac{3}{4} | 86 | 17 |
math | In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$. | 36 | 93 | 2 |
math | 3. Let $x, y$ be real numbers. Then
$$
f(x, y)=x^{2}+x y+y^{2}-x-y
$$
the minimum value is $\qquad$ . | -\frac{1}{3} | 46 | 7 |
math | 3. Find the sum of all four-digit numbers in which the digits $0,4,5,9$ do not appear. | 6479352 | 27 | 7 |
math | 8.13 Does a perfect square exist whose sum of digits is (1)1983; (2)1984?
(Kiev Mathematical Olympiad, 1983) | 1984 | 42 | 4 |
math | In a classroom, there are 50 students, including boys and girls. At least one of the students is a boy. Taking any pair of students, at least one of the two is a girl. How many girls are there in this classroom?
# | 49 | 52 | 2 |
math | 14. Find all positive numbers $a$ such that the quadratic equation $\left(a^{2}+1\right) x^{2}+2 a x+\left(a^{2}-1\right)=0$ has both roots as integers. | a=1 | 52 | 3 |
math | 14.28 The sequence $\left\{a_{n}\right\}$ is defined as $a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{a_{n}} \quad(n \geqslant 1)$. Find $\left[a_{100}\right]$.
(Japan Mathematical Olympiad, 1990) | 14 | 85 | 2 |
math | 3. If the quadratic function $y=a x^{2}+(1-4 a) x-2$ intersects the reciprocal function $y=\frac{8}{x}(x>0)$ at only one point, then this intersection point is $\qquad$ | (4,2) | 54 | 5 |
math | Example 2.77. Calculate the work $A$ performed during the isothermal ( $T=$ const) expansion of an ideal gas (pressure $P$ is related to the absolute temperature $T$ and volume $V$, mass $m$ and molar mass $\mu$ and the gas constant $R$, by the law $\left.P V=\frac{m}{\mu} R T\right)$. | \frac{}{\mu}RT\ln(\frac{V_{2}}{V_{1}}) | 87 | 22 |
math | 5. Let integer $n \geqslant 2$,
$$
A_{n}=\sum_{k=1}^{n} \frac{3 k}{1+k^{2}+k^{4}}, B_{n}=\prod_{k=2}^{n} \frac{k^{3}+1}{k^{3}-1} \text {. }
$$
Then the size relationship between $A_{n}$ and $B_{n}$ is | A_{n}=B_{n} | 99 | 8 |
math | Given an integer $n\ge 2$, compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$, where all $n$-element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$. | n! | 68 | 3 |
math | 5. On the line $2 x-y-4=0$, there is a point $P$, which has the maximum difference in distance to two fixed points $A(4,-1), B(3,4)$. Then the coordinates of $P$ are $\qquad$. | (5,6) | 58 | 5 |
math | 12. Given that $A$ and $B$ are two moving points on the parabola $C: y^{2}=4 x$, point $A$ is in the first quadrant, and point $B$ is in the fourth quadrant. Lines $l_{1}$ and $l_{2}$ pass through points $A$ and $B$ respectively and are tangent to the parabola $C$. $P$ is the intersection point of $l_{1}$ and $l_{2}$.
... | x=-1 | 202 | 3 |
math | 6. If positive integers $x, y, z$ satisfy $x+2 x y+3 x y z=115$, then $x+y+z=$ | 10or16 | 35 | 5 |
math | 1. The sequence $\left\{a_{n}\right\}$ has nine terms, $a_{1}=a_{9}=1$, and for each $i \in\{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in\left\{2,1,-\frac{1}{2}\right\}$. The number of such sequences is $\qquad$ .
(2013, National High School Mathematics League Competition) | 491 | 113 | 3 |
math | Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$,
with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. | 677 | 97 | 3 |
math | 9. A chemistry student conducted an experiment: from a bottle filled with syrup solution, he poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again and refilled the bottle with water. As a result, the percentage of syrup decreased from 9 to 4 percent. Determine the vol... | 3 | 74 | 1 |
math | In a fruit shop, Jaime noticed that an orange costs the same as half an apple plus half a real, and he also noticed that a third of an apple costs the same as a quarter of an orange plus half a real. With the value of 5 oranges plus 5 reals, how many apples can Jaime buy?
# | 5 | 67 | 1 |
math | 7. For a certain game activity,
the rewards are divided into first, second, and third prizes (all participants in the game activity will receive a prize), and the corresponding winning probabilities form a geometric sequence with the first term $a$ and a common ratio of 2. The corresponding prize money forms an arithm... | 500 | 102 | 3 |
math | 2. (2 points) Find the minimum value of the expression $x^{2}-6 x \sin y-9 \cos ^{2} y$. | -9 | 33 | 2 |
math | 2. (8 points) Shuaishuai finished memorizing English words in five days. It is known that in the first three days, he memorized $\frac{1}{2}$ of all the words, and in the last three days, he memorized $\frac{2}{3}$ of all the words, and he memorized 120 fewer words in the first three days than in the last three days. T... | 120 | 107 | 3 |
math | 10. Even number of successes. Find the probability that heads will appear an even number of times, in an experiment where:
a) a fair coin is tossed $n$ times;
b) a coin, for which the probability of heads on a single toss is $p(0<p<1)$, is tossed $n$ times. | \frac{1+(1-2p)^{n}}{2} | 70 | 16 |
math | ## Task 4.
Determine all natural numbers $n$ for which there exist distinct divisors $a$ and $b$ of $n$ such that there are no other divisors of $n$ between them and that
$$
n=a^{2}-b
$$ | 8 | 58 | 1 |
math | 6. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with eccentricity $e \in\left[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{2}}\right]$, the line $y=-x+1$ intersects the ellipse at points $M$ and $N$, and $O$ is the origin with $O M \perp O N$. Then the range of values for $a$ is $\qquad$ | \frac{\sqrt{5}}{2}\leqslant\leqslant\frac{\sqrt{6}}{2} | 120 | 29 |
math | Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$, where $m$ and $n$ are nonnegative integers. If $1776 $is one of the numbers that is not expressible, find $a + b$. | 133 | 73 | 3 |
math | 19. Find all prime numbers $p$ such that the numbers $p+4$ and $p+8$ are also prime. | 3 | 29 | 1 |
math | 2・43 Let $p$ be an odd prime. Consider the subsets $A$ of the set $\{1,2, \cdots, 2 p\}$ that satisfy the following two conditions:
（1） $A$ has exactly $p$ elements;
（2） The sum of all elements in $A$ is divisible by $p$.
Find the number of all such subsets $A$. | \frac{1}{p}(C_{2p}^{p}-2)+2 | 86 | 18 |
math | 6. Let the circumcenter of the tetrahedron $S-ABC$ be $O$, the midpoints of $SB$ and $AC$ be $N$ and $M$ respectively, and the midpoint of segment $MN$ be $P$. Given that $SA^2 + SB^2 + SC^2 = AB^2 + BC^2 + AC^2$, if $SP = 3\sqrt{7}$ and $OP = \sqrt{21}$, then the radius of the sphere $O$ is $\qquad$ | 2\sqrt{21} | 117 | 7 |
math | 11. (15 points) Given $\odot O: x^{2}+y^{2}=4$ and the curve $C: y=3|x-t|, A(m, n) 、 B(s, p)(m 、 n 、 s 、 p \in$ $\left.\mathbf{Z}_{+}\right)$ are two points on the curve $C$, such that the ratio of the distance from any point on $\odot O$ to point $A$ and to point $B$ is a constant $k(k>1)$. Find the value of $t$. | \frac{4}{3} | 128 | 7 |
math | Example 2 Given the sequence $\left\{x_{n}\right\}$ satisfies $x_{0}=0, x_{n+1}=$ $x_{n}+a+\sqrt{b^{2}+4 a x_{n}}, n=0,1,2, \cdots$, where $a$ and $b$ are given positive real numbers. Find the general term of this sequence. | x_{n}=a n^{2}+b n | 87 | 12 |
math | Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 191 | 55 | 3 |
math | 14. (i) Find the least non-negative residue of $2^{400}$ modulo 10;
(ii) Find the last two digits in the decimal representation of $2^{1000}$;
(iii) Find the last two digits in the decimal representation of $9^{9^{9}}$ and $9^{9^{9^{9}}}$;
(iv) Find the least non-negative residue of $\left(13481^{56}-77\right)^{28}$ wh... | 6, 76, 89, 70, 6 | 153 | 16 |
math | A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i] | 91 | 113 | 2 |
math | $\underline{\text { Folklore }}$
Students from different cities came to the tournament. One of the organizers noticed that they could form 19 teams of 6 people each, and at the same time, less than a quarter of the teams would have a reserve player. Another suggested forming 22 teams of 5 or 6 people each, and then mo... | 118 | 95 | 3 |
math | Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$. | 30^\circ | 90 | 4 |
math | Example 2. Let $x$ be a positive real number, then the minimum value of the function $y=x^{2}-x+\frac{1}{x}$ is $\qquad$ - (1995, National Junior High School Competition) | 1 | 53 | 1 |
math | 3. Determine the subsets $A$ of $\mathbb{Z}_{6}$ with the property that $A$ has three elements and there exists a function $f: A \rightarrow A, f(x)=x+a, a \in \mathbb{Z}_{6} \backslash\{\hat{0}\}$.
Prof. Bene Marius, Alexandria | A={\hat{0},\hat{2},\hat{4}},A={\hat{1},\hat{3},\hat{5}} | 75 | 34 |
math | Find the solutions of the equation
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{n} x+3}}}}}=1+\sqrt{x}
$$ | 2^{-2n} | 47 | 5 |
math | 3. In $\triangle A B C$, $A B$ is the longest side, $\sin A \sin B=$ $\frac{2-\sqrt{3}}{4}$. Then the maximum value of $\cos A \cos B$ is $\qquad$ . | \frac{2+\sqrt{3}}{4} | 56 | 12 |
math | 7.2. Petya wrote down all natural numbers from 1 to $n$ in a row on the board and counted the total number of digits written. It turned out to be 777. What is $n$? | 295 | 50 | 3 |
math | Find $f(x)$ such that $f(x)^{2} f\left(\frac{1-x}{1+x}\right)=64 x$ for $x$ not $0, \pm 1$. | f(x)=4\left(\frac{x^{2}(1+x)}{1-x}\right)^{1/3} | 45 | 26 |
math | 1. If the system of inequalities $\left\{\begin{array}{l}x-1000 \geqslant 1018, \\ x+1 \leqslant a\end{array}\right.$ has only a finite number of real solutions, then the value of $a$ is $\qquad$. | 2019 | 72 | 4 |
math | 5. Simplify: $\frac{\sqrt{3}+\sqrt{5}}{3-\sqrt{6}-\sqrt{10}+\sqrt{15}}$. | \sqrt{3}+\sqrt{2} | 37 | 10 |
math | 2.051. $\frac{\left(a^{2}-b^{2}\right)\left(a^{2}+\sqrt[3]{b^{2}}+a \sqrt[3]{b}\right)}{a \sqrt[3]{b}+a \sqrt{a}-b \sqrt[3]{b}-\sqrt{a b^{2}}}: \frac{a^{3}-b}{a \sqrt[3]{b}-\sqrt[6]{a^{3} b^{2}}-\sqrt[3]{b^{2}}+a \sqrt{a}} ;$
$$
a=4.91 ; b=0.09
$$ | 5 | 143 | 1 |
math | 2.289. Represent the polynomial $x^{8}-16$ as a product of polynomials of the second degree. | (x^{2}-2)(x^{2}+2)(x^{2}-2x+2)(x^{2}+2x+2) | 28 | 32 |
math | Five. (13 points) Let the quadratic function
$$
f(x)=x^{2}+b x+c(b, c \in \mathbf{R})
$$
intersect the $x$-axis. If for all $x \in \mathbf{R}$, we have
$$
f\left(x+\frac{1}{x}\right) \geqslant 0 \text {, and } f\left(\frac{2 x^{2}+3}{x^{2}+1}\right) \leqslant 1 .
$$
Find the values of $b$ and $c$. | b=-4, c=4 | 135 | 7 |
math | Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\]
the following inequality also holds
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{... | \left(\frac{3}{4}\right)^n | 137 | 13 |
math | ## Task 2
Subtract from 3182100 the product of the numbers 56823 and 56. By how much is the difference greater than 10? | 2 | 44 | 1 |
math | 5. Given $x, y \in \mathbf{R}$. Then
$$
\cos (x+y)+2 \cos x+2 \cos y
$$
the minimum value is $\qquad$ $\therefore$. | -3 | 50 | 2 |
math | ## Problem Statement
Write the canonical equations of the line.
$$
\begin{aligned}
& 4 x+y-3 z+2=0 \\
& 2 x-y+z-8=0
\end{aligned}
$$ | \frac{x-1}{-2}=\frac{y+6}{-10}=\frac{z}{-6} | 49 | 28 |
math | Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
[i]Proposed by Evan Chen[/i] | 1186 | 86 | 4 |
math | Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$ | -(x - 1)^2 e^x + C | 107 | 12 |
math | Solve the following system of equations:
$$
\begin{gathered}
x^{2}-y z=-23 \\
y^{2}-z x=-4 \\
z^{2}-x y=34
\end{gathered}
$$ | \5,\6,\8 | 52 | 6 |
math | 6.083. $\left\{\begin{array}{l}12(x+y)^{2}+x=2.5-y, \\ 6(x-y)^{2}+x=0.125+y .\end{array}\right.$ | (\frac{1}{4};\frac{1}{6}),(\frac{1}{12};\frac{1}{3}),(-\frac{5}{24};-\frac{7}{24}),(-\frac{3}{8};-\frac{1}{8}) | 58 | 61 |
math | $\sin (\alpha+\beta) \sin (\alpha-\beta)-(\sin \alpha+\sin \beta)(\sin \alpha-\sin \beta)=?$ | 0 | 33 | 1 |
math | 13. (7 points) 8 years ago, the sum of the father and son's ages was 80 years old, this year the father's age is 3 times the son's age, how old is the son this year? | 24 | 51 | 2 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, we have
$$
f(f(x)+y)=2 x+f(f(y)-x)
$$ | f(x)=x+ | 57 | 5 |
math | 11. The 8 vertices of a cube can form
$\qquad$ non-equilateral triangles. | 48 | 22 | 2 |
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