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 | ## Task 2.
Let $a \geqslant 2018$ be a real number. In each of 2018 jars, there is a finite number of balls, with the mass of each ball being of the form $a^{k}$, where $k \in \mathbb{Z}$. The total mass of the balls in each jar is the same. What is the minimum number of balls of the same mass among the balls in the j... | 2018 | 101 | 4 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n}^{2}=a_{n+1} a_{n}-1\left(n \in \mathbf{Z}_{+}\right) \text {, and } a_{1}=\sqrt{2} \text {. }
$$
Then the natural number closest to $\sqrt{a_{2014}}$ is $\qquad$ | 8 | 96 | 1 |
math | ## 176. Math Puzzle $1 / 80$
If in the kindergarten of a small town 63 children are present, it is considered to be 84 percent occupied.
What is its capacity, i.e., the number of children at 100 percent occupancy? | 75 | 61 | 2 |
math | 1. Let real numbers $x, y$ satisfy the equation $9 x^{2}+4 y^{2}-3 x+2 y=0$. Then the maximum value of $z=3 x+2 y$ is $\qquad$ . | 1 | 53 | 1 |
math | ## PROBLEM 2
Knowing that $3^{6 n+12}+9^{3 n+6}+27^{2 n+4}=3^{4(n+3)+255}, n \in \mathbb{N}$.
Find the remainder of the division of A by 5, where $A=2^{n}+3^{n}+4^{n}+7^{n}$. | 2 | 91 | 1 |
math | G2.4 On the coordinate plane, a circle with centre $T(3,3)$ passes through the origin $O(0,0)$. If $A$ is a point on the circle such that $\angle A O T=45^{\circ}$ and the area of $\triangle A O T$ is $Q$ square units, find the value of $Q$. | 9 | 80 | 1 |
math | ## Aufgabe 11/81
Gesucht ist die kleinste Potenz $11^{n}(n \in N, n>0)$ in dekadischer Schreibweise, die auf die Ziffernfolge 001 endet.
| 11^{50} | 57 | 6 |
math | Solve the following equation:
$$
x^{4}+2 x^{3}-x^{2}-2=0
$$ | x_{1}=1,\quadx_{2}\approx-2.56 | 27 | 17 |
math | 31. a) Find all integers that start with the digit 6 and decrease by 25 times when this digit is erased.
b) Prove that there do not exist integers that decrease by 35 times when the first digit is erased. | 6250\ldots0(n=0,1,2,\ldots) | 52 | 19 |
math | 10,11
The base of an inclined parallelepiped is a rectangle $A B C D$; $A A 1, B B 1, C C 1$ and $D D 1$ are the lateral edges. A sphere with center at point $O$ touches the edges $B C, A 1 B 1$ and $D D 1$ at points $B, A 1$ and $D 1$, respectively. Find $\angle A 1 O B$, if $A D=4$, and the height of the parallelepi... | 2\arcsin\frac{1}{\sqrt{5}} | 127 | 15 |
math | 3. determine all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ for which the following applies
$$
\frac{1}{x} f(-x)+f\left(\frac{1}{x}\right)=x \quad \forall x \in \mathbb{R} \backslash\{0\}
$$
## Solution | f(x)=\frac{1}{2}(x^{2}+\frac{1}{x}) | 87 | 21 |
math | 8.394. $\left\{\begin{array}{l}\sin x+\cos y=0, \\ \sin ^{2} x+\cos ^{2} y=\frac{1}{2} .\end{array}\right.$ | x_{1}=(-1)^{k+1}\frac{\pi}{6}+\pik,y_{1}=\\frac{\pi}{3}+2\pin,\quadx_{2}=(-1)^{k}\frac{\pi}{6}+\pik,\quady_{2}=\\frac{2}{3}\pi+2\pin,\text{where}k\text{} | 54 | 84 |
math | 1. Given that $a$, $b$, $c$, and $d$ are prime numbers, and $a b c d$ is the sum of 77 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$ | 32 | 55 | 2 |
math | 8. Let $A B C$ be a triangle with sides $A B=6, B C=10$, and $C A=8$. Let $M$ and $N$ be the midpoints of $B A$ and $B C$, respectively. Choose the point $Y$ on ray $C M$ so that the circumcircle of triangle $A M Y$ is tangent to $A N$. Find the area of triangle $N A Y$. | \frac{600}{73} | 97 | 10 |
math | 5 Given a circle $(x-14)^{2}+(y-12)^{2}=36^{2}$, a point $C(4,2)$ inside the circle, and two moving points $A, B$ on the circumference, such that $\angle A C B=90^{\circ}$. The equation of the locus of the midpoint of the hypotenuse $A B$ is $\qquad$ . | (x-9)^{2}+(y-7)^{2}=13\times46 | 92 | 21 |
math | Example 5 Lift Your Veil
A 101-digit natural number $A=\underbrace{88 \cdots 8}_{\text {S0 digits }} \square \underbrace{99 \cdots 9}_{\text {S0 digits }}$ is divisible by 7. What is the digit covered by $\square$? | 5 | 75 | 1 |
math | 21. Five participants of the competition became its prize winners, scoring 20, 19, and 18 points and taking first, second, and third places, respectively. How many participants won each prize place if together they scored 94 points? | 1,2,2 | 55 | 5 |
math | Example 4.11 Select $k$ elements from an $n$-element permutation to form a combination, such that any two elements $a$ and $b$ in the combination satisfy the condition: in the original permutation, there are at least $r$ elements between $a$ and $b$ (with $n+r \geqslant k r+k$). Let $f_{r}(n, k)$ denote the number of d... | f_{r}(n,k)=\binom{n-kr+r}{k} | 111 | 17 |
math | [ Unusual Constructions Common Fractions
How, without any measuring tools, can you measure 50 cm from a string that is $2 / 3$ of a meter long?
# | \frac{1}{2} | 40 | 7 |
math | 9. Given the parabola $C: x^{2}=2 p y(p>0)$, draw two tangent lines $R A$ and $R B$ from the point $R(1,-1)$ to the parabola $C$, with the points of tangency being $A$ and $B$. Find the minimum value of the area of $\triangle R A B$ as $p$ varies. | 3\sqrt{3} | 87 | 6 |
math | (2) A line segment of length $3 \mathrm{~cm}$ is randomly divided into three segments. The probability that these three segments can form a triangle is $\qquad$. | \frac{1}{4} | 38 | 7 |
math | 3. Given the equation $x^{2}+(b+2) x y+b y^{2}=0$ $(b \in \mathbf{R})$ represents two lines. Then the range of the angle between them is $\qquad$ . | \left[\arctan \frac{2 \sqrt{5}}{5}, \frac{\pi}{2}\right] | 53 | 27 |
math | 11. Find the value of
$$
\frac{2006^{2}-1994^{2}}{1600} .
$$ | 30 | 35 | 2 |
math | 4 In a regular $\triangle A B C$, $D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. The eccentricity of the hyperbola with foci at $B$ and $C$ and passing through points $D$ and $E$ is $\qquad$. | \sqrt{3}+1 | 68 | 7 |
math | Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity.
[i]Proposed by Mr.Etesami[/i] | n = 2^k - 2 | 72 | 10 |
math | 5. In a regular tetrahedron $ABCD$, $E$ and $F$ are on edges $AB$ and $AC$ respectively, satisfying $BE=3$, $EF=4$, and $EF$ is parallel to plane $BCD$. Then the area of $\triangle DEF$ is $\qquad$. | 2\sqrt{33} | 68 | 7 |
math | A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$... | 173 | 94 | 3 |
math | Question 6 Let $A \subset S$ and $A \neq \varnothing$, we call
$$
D_{\mathbf{A}}=\frac{1}{|A|} \sum_{\mathbf{a} \in \mathrm{A}} a_{\mathbf{.}}
$$
the average value of $A$. Find the arithmetic mean of all $D_{\mathrm{A}}$. | \frac{n+1}{2} | 90 | 8 |
math | 4. Find the integer solutions $(x, y, z)$ of the system of equations
$$
\left\{\begin{array}{l}
x+y+z=0, \\
x^{3}+y^{3}+z^{3}=-18
\end{array}\right.
$$ | (1,2,-3), (2,1,-3), (1,-3,2), (2,-3,1), (-3,1,2), (-3,2,1) | 64 | 42 |
math | 319. $\sqrt{5-x}+2=7$.
319. $\sqrt{5-x}+2=7$.
The equation is already in English, so no translation was needed for the mathematical expression. However, if you meant to have the problem solved, here is the solution:
To solve the equation $\sqrt{5-x}+2=7$, we first isolate the square root term:
1. Subtract 2 from b... | -20 | 205 | 3 |
math | 6th Irish 1993 Problem A1 The real numbers x, y satisfy x 3 - 3x 2 + 5x - 17 = 0, y 3 - 3y 2 + 5y + 11 = 0. Find x + y. | x+y=2 | 65 | 4 |
math | With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | 91 | 56 | 2 |
math | 3. Determine a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, gives six-digit numbers written with the same digits as the original number. | 142857 | 42 | 6 |
math | The function $f:\mathbb{N} \rightarrow \mathbb{N}$ verifies:
$1) f(n+2)-2022 \cdot f(n+1)+2021 \cdot f(n)=0, \forall n \in \mathbb{N};$
$2) f(20^{22})=f(22^{20});$
$3) f(2021)=2022$.
Find all possible values of $f(2022)$. | 2022 | 113 | 4 |
math | Solve the equation $\left(x^{2}+y\right)\left(x+y^{2}\right)=(x-y)^{3}$ in the set of integers. | (x,y)=(0,0),(8,-10),(9,-6),(9,-21),(-1,-1),(x,0) | 35 | 30 |
math | G3.3 Let $x$ and $y$ be positive real numbers with $x<y$. If $\sqrt{x}+\sqrt{y}=1$ and $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{10}{3}$ and $x<y$, find the value of $y-x$. | \frac{1}{2} | 72 | 7 |
math | 4. In the company, several employees have a total monthly salary of 10000 dollars. A kind manager proposes to double the salary for everyone earning up to 500 dollars, and increase the salary by 500 dollars for the rest, so the total salary will become 17000 dollars. A mean manager proposes to reduce the salary to 500 ... | 7000 | 111 | 4 |
math | 2. Let $A B C D$ be an isosceles trapezoid such that $A D=B C, A B=3$, and $C D=8$. Let $E$ be a point in the plane such that $B C=E C$ and $A E \perp E C$. Compute $A E$. | 2\sqrt{6} | 72 | 6 |
math | 7. Given that the 6027-digit number $\frac{a b c a b c \cdots a b c}{2000 \uparrow a b c}$ is a multiple of 91. Find the sum of the minimum and maximum values of the three-digit number $\overline{a b c}$. | 1092 | 70 | 4 |
math | Example 4 (Adapted from the 26th International Mathematical Olympiad 1994) Let $f:(-1,+\infty) \rightarrow (-1,+\infty)$ be a continuous and monotonic function, with $f(0)=0$, and satisfying
$$
f[x+f(y)+x f(y)] \geqslant y+f(x)+y f(x) \text {, for } \forall x, y \in(-1,+\infty),
$$
Find $f(x)$. | f(x)=-\frac{x}{1+x};f(x)=x | 114 | 15 |
math | 16th Irish 2003 Problem A1 Find all integral solutions to (m 2 + n)(m + n 2 ) = (m + n) 3 . | (,n)=(0,n),(,0),(-5,2),(-1,1),(1,-1),(2,-5),(4,11),(5,7),(7,5),(11,4) | 39 | 46 |
math | An integer $n>0$ is written in decimal system as $\overline{a_ma_{m-1}\ldots a_1}$. Find all $n$ such that
\[n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)\] | 18 | 66 | 2 |
math | 17. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the greatest possible value of the sum $a+b$.
| param1 | param2 | |
| :---: | :---: | :---: |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-3 x^{2}+5 x+11=0$ | |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+3 x^{2}+... | 4 | 242 | 1 |
math | 3. Given a cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the distance between the skew lines $A C_{1}$ and $B_{1} C$ is $\qquad$ | \frac{\sqrt{6}}{6} | 59 | 10 |
math | 7. Let $H$ be the orthocenter of $\triangle A B C$, and
$$
3 \overrightarrow{H A}+4 \overrightarrow{H B}+5 \overrightarrow{H C}=\mathbf{0} \text {. }
$$
Then $\cos \angle A H B=$ $\qquad$ | -\frac{\sqrt{6}}{6} | 73 | 10 |
math | $\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? | 1440 | 69 | 4 |
math | 2B. Determine all solutions ( $x, y$ ) of the equation $\frac{x+6}{y}+\frac{13}{x y}=\frac{4-y}{x}$, in the set of real numbers. | (-3,2) | 49 | 5 |
math | Problem 10.1. Consider the inequality $\left|x^{2}-5 x+6\right| \leq x+a$, where $a$ is a real parameter.
a) Solve the inequality for $a=0$.
b) Find the values of $a$ for which the inequality has exactly three integer solutions.
Stoyan Atanassov | \in[-2,1) | 76 | 7 |
math | Among the fractions with positive denominators less than 1001, which one has the smallest difference from $\frac{123}{1001}$? | \frac{94}{765} | 35 | 10 |
math | 3. (4 points) Solve the equation of the form $f(f(x))=x$, given that $f(x)=x^{2}+2 x-5$
# | \frac{1}{2}(-1\\sqrt{21}),\frac{1}{2}(-3\\sqrt{17}) | 37 | 30 |
math | Example 8 Let $F=\max _{1 \leqslant x \leqslant 3}\left|x^{3}-a x^{2}-b x-c\right|$. Find the minimum value of $F$ when $a, b, c$ take all real numbers. (2001, IMO China National Training Team Selection Exam) | \frac{1}{4} | 77 | 7 |
math | 8. Let point $P$ be on the ellipse $\frac{x^{2}}{5}+y^{2}=1$, $F_{1}, F_{2}$ are the two foci of the ellipse, if the area of $\triangle F_{1} P F_{2}$ is $\frac{\sqrt{3}}{3}$, then $\angle F_{1} P F_{2}=$ $\qquad$ | 60 | 89 | 2 |
math | Example 6 The number of non-negative integer solutions to the equation $2 x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+x_{9}+x_{10}=3$ is?
(1985 National High School League Question) | 174 | 80 | 3 |
math | 9. Place 3 identical white balls, 4 identical red balls, and 5 identical yellow balls into three different boxes, allowing some boxes to contain balls of different colors. The total number of different ways to do this is (answer in numbers).
允许有的盒子中球的颜色不全的不同放法共有种 (要求用数字做答).
Allowing some boxes to contain balls of diff... | 3150 | 97 | 4 |
math | ## Subject IV. (20 points)
Emil is waiting in line at a ticket booth, along with other people, standing in a row. Andrei, who is right in front of Emil, says: "Behind me, there are 5 times as many people as in front of me." Mihai, who is right behind Emil, says: "Behind me, there are 3 times as many people as in front... | 25 | 157 | 2 |
math | 5. Given two propositions, proposition $p$ : the function $f(x)=\log _{a} x(x>0)$ is monotonically increasing; proposition $q$ : the function $g(x)=x^{2}+a x+1>0$ $(x \in \mathbf{R})$. If $p \vee q$ is a true proposition, and $p \wedge q$ is a false proposition, then the range of real number $a$ is $\qquad$ . | (-2,1]\cup[2,+\infty) | 109 | 13 |
math | 1. A circle with radius 12 and center at point $O$ and a circle with radius 3 touch internally at point $H$. The line $X H$ is their common tangent, and the line $O X$ is tangent to the smaller circle. Find the square of the length of the segment $O X$. | 162 | 68 | 3 |
math | Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $... | 5000 | 104 | 4 |
math | 7. In $\triangle A B C$, the maximum value of $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ is | \frac{1}{8} | 41 | 7 |
math | 9. Given a complex number $z$ satisfying $|z|=1$, find the maximum value of $u=\left|z^{3}-3 z+2\right|$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 3\sqrt{3} | 63 | 6 |
math |
Problem 1. The function $f(x)=\sqrt{1-x}(x \leq 1)$ is given. Let $F(x)=f(f(x))$.
a) Solve the equations $f(x)=x$ and $F(x)=x$.
b) Solve the inequality $F(x)>x$.
c) If $a_{0} \in(0,1)$, prove that the sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$, determined by $a_{n}=f\left(a_{n-1}\right)$ for $n=$ $1,2, \ldots$,... | \alpha=\frac{-1+\sqrt{5}}{2} | 149 | 14 |
math | 10. (20 points) Find all values of the parameter $a$ for which the equation
$$
5|x-4 a|+\left|x-a^{2}\right|+4 x-4 a=0
$$
has no solution. | (-\infty,-8)\cup(0,+\infty) | 54 | 15 |
math | 1.29 Calculate the sum of the cubes of two numbers if their sum and product are 11 and 21, respectively. | 638 | 29 | 3 |
math | 1. 15 Choose any 1962-digit number that is divisible by 9, and let the sum of its digits be $a$, the sum of the digits of $a$ be $b$, and the sum of the digits of $b$ be $c$. What is $c$? | 9 | 65 | 1 |
math | Example 4 Given $x, y \in(-2,2)$, and $x y=-1$. Find the minimum value of the function $u=\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}$. (2003, National High School Mathematics Competition) | \frac{12}{5} | 65 | 8 |
math | [ Rectangular parallelepipeds ] [ Angles between lines and planes $]$
The sides of the base of a rectangular parallelepiped are equal to $a$ and $b$. The diagonal of the parallelepiped is inclined to the base plane at an angle of $60^{\circ}$. Find the lateral surface area of the parallelepiped. | 2(+b)\sqrt{3(^{2}+b^{2})} | 75 | 17 |
math | Example 6 Let $a, b, c$ be distinct real numbers, and satisfy the relation
$$
b^{2}+c^{2}=2 a^{2}+16 a+14,
$$
and $bc=a^{2}-4a-5$.
Find the range of values for $a$.
(2006, Hunan Province Junior High School Mathematics Competition) | a > -1 \text{ and } a \neq \frac{1 \pm \sqrt{21}}{4}, a \neq -\frac{5}{6} | 85 | 40 |
math | What integer is equal to $1^{1}+2^{2}+3^{3}$ ? | 32 | 21 | 2 |
math | 2. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
\left(x^{2}+1\right)\left(y^{2}+1\right)+24 x y & =0 \\
\frac{12 x}{x^{2}+1}+\frac{12 y}{y^{2}+1}+1 & =0
\end{aligned}
$$ | (3\\sqrt{8},-2\\sqrt{3})(-2\\sqrt{3},3\\sqrt{8}) | 91 | 27 |
math | 8. Let $\triangle A B C$ have internal angles $A, B, C$ with opposite sides $a, b, c$ respectively, and satisfy $a \cos B-b \cos A=\frac{4}{5} c$, then $\frac{\tan A}{\tan B}=$
$\qquad$ . | 9 | 69 | 1 |
math | Determine all solutions $(m, n) \in \mathbf{N}^{2}$ of the equation:
$$
7^{m}-3 \times 2^{n}=1
$$
## SOLUTIONS | (1,1)(2,4) | 45 | 9 |
math | I2.1 If the solution of the system of equations $\left\{\begin{array}{r}x+y=P \\ 3 x+5 y=13\end{array}\right.$ are positive integers, find the value of $P$. | 3 | 53 | 1 |
math | For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $... | 10 | 146 | 2 |
math | Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$. | \frac{3\sqrt{3}}{2} | 63 | 12 |
math | 9.2. The number $a$ is a root of the quadratic equation $x^{2}-x-100=0$. Find the value of $a^{4}-201 a$ | 10100 | 43 | 5 |
math | 2. Solve the system of equations
$$
\left\{\begin{array}{l}
\left(1+4^{2 x-y}\right) 5^{1-2 x+y}=1+2^{2 x-y+1}, \\
y^{3}+4 x+1+\ln \left(y^{2}+2 x\right)=0 .
\end{array}\right.
$$
(1999, Vietnam Mathematical Olympiad) | x=0, y=-1 | 98 | 7 |
math |
NT1 Solve in positive integers the equation $1005^{x}+2011^{y}=1006^{z}$.
| (2,1,2) | 33 | 7 |
math | 3+
The center of the circumscribed circle of a triangle is symmetric to its incenter with respect to one of the sides. Find the angles of the triangle.
# | 36,36,108 | 35 | 9 |
math | 14. (6 points) Given that $\alpha$ is an acute angle, $\beta$ is an obtuse angle, 4 students calculated $0.25(\alpha+\beta)$ and obtained the results $15.2^{\circ} 、 45.3^{\circ} 、 78.6^{\circ} 、 112^{\circ}$, respectively. The possible correct result is $\qquad$ . | 45.3 | 98 | 4 |
math | 21. 2.9 * Color the vertices of the $n$-sided pyramid $S-A_{1} A_{2} \cdots A_{n}$, with each vertex being a different color from its adjacent vertices on the same edge. Given $n+1$ colors to use, how many different coloring methods are there? (When $n=4$, this is a problem from the 1995 National High School Competitio... | (n+1)[(n-1)^{n}+(-1)^{n}(n-1)] | 97 | 23 |
math | 2. Given: $b_{1}, b_{2}, b_{3}, b_{4}$ are positive integers, the polynomial $g(z)=(1-z)^{b_{1}}\left(1-z^{2}\right)^{b_{2}}\left(1-z^{3}\right)^{b_{3}}\left(1-z^{4}\right)^{b_{4}}$ when expanded and terms higher than 4th degree are omitted, becomes $1-2 z$. Also, $\alpha$ is the largest root of the polynomial $f(x)=x^... | 5 | 180 | 1 |
math | Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of:
$ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$ | 1 | 142 | 1 |
math | # Task 2. (10 points)
Find the value of the parameter $p$ for which the equation $p x^{2}=|x-1|$ has exactly three solutions.
# | \frac{1}{4} | 40 | 7 |
math | 6.217. $\left\{\begin{array}{l}x^{4}+x^{2} y^{2}+y^{4}=91 \\ x^{2}+x y+y^{2}=13\end{array}\right.$ | (1;3),(3;1),(-1;-3),(-3;-1) | 57 | 19 |
math | 4. It is known that construction teams A and B each have several workers. If team A lends 90 workers to team B, then the total number of workers in team B will be twice that of team A; if team B lends a certain number of workers to team A, then the total number of workers in team A will be 6 times that of team B. Team ... | 153 | 89 | 3 |
math | Let $\triangle ABC$ be a triangle with $AB=85$, $BC=125$, $CA=140$, and incircle $\omega$. Let $D$, $E$, $F$ be the points of tangency of $\omega$ with $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively, and furthermore denote by $X$, $Y$, and $Z$ the incenters of $\triangle AEF$, $\triangle BFD$, and $\t... | 30 | 138 | 2 |
math | A train line is divided into 10 segments by stations $A, B, C, D, E, F, G, H, I, J$ and $K$. The distance from $A$ to $K$ is equal to $56 \mathrm{~km}$. The route of two consecutive segments is always less than or equal to $12 \mathrm{~km}$ and the route of three consecutive segments is always greater than or equal to ... | )5,b)22,)29 | 144 | 9 |
math | 4. The smallest positive integer $a$ that makes the inequality
$$
\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}<a-2007 \frac{1}{3}
$$
hold for all positive integers $n$ is $\qquad$ | 2009 | 76 | 4 |
math | 7. For the tetrahedron $ABCD$, $AB \perp BC$, $CD \perp BC$, $BC=2$, and the angle between the skew lines $AB$ and $CD$ is $60^{\circ}$. If the radius of the circumscribed sphere of the tetrahedron $ABCD$ is $\sqrt{5}$, then the maximum volume of the tetrahedron $ABCD$ is $\qquad$ . | 2\sqrt{3} | 100 | 6 |
math | 4. There are 5 fractions: $\frac{2}{3}, \frac{5}{8}, \frac{15}{23}, \frac{10}{17}, \frac{12}{19}$, if arranged in order of size, which number is in the middle? | \frac{12}{19} | 64 | 9 |
math | Example 1.31. Find the coordinates of the point $M$ that bisects the segment of the line
$$
\frac{x-2}{3}=\frac{y+1}{5}=\frac{z-3}{-1}
$$
enclosed between the planes $x o z$ and $x o y$. | M(6.8;7;1.4) | 73 | 12 |
math | 6. Find $\cos \frac{\pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{5 \pi}{7}=$ | \frac{1}{2} | 35 | 7 |
math | 5. The number of common terms (terms with the same value) in the arithmetic sequences $2,5,8, \cdots, 2015$ and $4,9,14, \cdots, 2014$ is $\qquad$ | 134 | 59 | 3 |
math | A tournament will take place with 100 competitors, all with different skill levels. The most skilled competitor always wins against the least skilled competitor. Each participant plays exactly twice, with two randomly drawn opponents (once against each). A competitor who wins two matches receives a medal. Determine the... | 1 | 70 | 1 |
math | Folkcor $^{2}$
Find the maximum value of the expression $x+y$, if $(2 \sin x-1)(2 \cos y-\sqrt{3})=0, x \in[0,3 \pi / 2], y \in$ $[\pi, 2 \pi]$. | \frac{10\pi}{3} | 67 | 10 |
math | 4. Find the minimum value of the function $f(x)=\sqrt{4 x^{2}-12 x+8}+\sqrt{4+3 x-x^{2}}$. | \sqrt{6} | 39 | 5 |
math | 3A. Determine all arithmetic progressions with difference $d=2$, such that the ratio $\frac{S_{3 n}}{S_{n}}$ does not depend on $n$. | 1,3,5,\ldots | 40 | 8 |
math | 10.2 It is known that $10 \%$ of people own no less than $90 \%$ of all the money in the world. For what minimum percentage of all people can it be guaranteed that these people own $95 \%$ of all the money? | 55 | 57 | 2 |
math | Example 3 Let $M=\{1,2, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: when $x \in A$, $15 x \notin A$. Then the maximum number of elements in $A$ is $\qquad$
(1995, National High School Mathematics Competition) | 1870 | 79 | 4 |
math | [ [Geometric progression]
It is known that the sum of the first n terms of a geometric progression consisting of positive numbers is $\mathrm{S}$, and the sum of the reciprocals of the first n terms of this progression is R. Find the product of the first n terms of this progression.
# | (S/R)^{n/2} | 64 | 8 |
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