task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | G6.1 An $n$-sided convex polygon has 20 diagonals. Find $n$.
G6.2 Two dice are thrown. The probability of getting a total of $n$ is $\frac{k}{36}$. Find $k$.
G6.3 A man drives at $25 \mathrm{~km} / \mathrm{h}$ for 3 hours and then at $50 \mathrm{~km} / \mathrm{h}$ for 2 hours.
His average speed for the whole journey is... | 8,5,35,1 | 170 | 8 |
math | 11. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is
保留源文本的换行和格式，翻译结果如下：
11. (10 points) The largest odd number that cannot be written as the sum of three distinct composite numbers is | 17 | 64 | 2 |
math | 2 - 83 The sum of several positive integers is 1976. Find the maximum value of the product of these positive integers.
Several positive integers add up to 1976. Find the maximum value of the product of these positive integers. | 2 \times 3^{658} | 54 | 10 |
math | 8. (10 points) In $\triangle A B C$, $B D=D E=E C$, $C F: A C=1: 3$. If the area of $\triangle A D H$ is 24 square centimeters more than the area of $\triangle H E F$, find the area of triangle $A B C$ in square centimeters? | 108 | 77 | 3 |
math | 8.286. $\operatorname{tg} 5 z-\operatorname{tg} 3 z-2 \operatorname{tg} 2 z=0$.
8.286. $\tan 5z - \tan 3z - 2 \tan 2z = 0$. | z_{1}=\pik;z_{2}=\frac{\pi}{16}(2n+1),kn\inZ | 68 | 28 |
math | 8. It is known that two engineering teams, Team A and Team B, have several people each. If 90 people are transferred from Team A to Team B, then the total number of people in Team B will be twice that of Team A; if some people are transferred from Team B to Team A, then the total number of people in Team A will be 6 ti... | 153 | 95 | 3 |
math | 10.1. Having found some polynomial of the sixth degree $x^{6}+a_{1} x^{5}+\ldots+a_{5} x+a_{6}$ with integer coefficients, one of the roots of which is the number $\sqrt{2}+\sqrt[3]{5}$, write in the answer the sum of its coefficients $a_{1}+a_{2}+\ldots+a_{6}$. | -47 | 92 | 3 |
math | 10. Given the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy: $a_{1}=-1, b_{1}=2, a_{n+1}=-b_{n}, b_{n+1}=2 a_{n}-3 b_{n}\left(n \in \mathbf{N}^{*}\right)$, then $b_{2015}+b_{2016}=$ $\qquad$ . | -3\cdot2^{2015} | 110 | 11 |
math | # 4.1. Condition:
In front of the elevator stand people weighing 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to get everyone up? | 4 | 84 | 1 |
math | Anetka's uncle has his birthday on the same day of the year as Anetka's aunt. The uncle is older than the aunt, but not by more than ten years, and both are adults. At the last celebration of their birthdays, Anetka realized that if she multiplied their celebrated ages and then multiplied the resulting product by the n... | 1or4 | 110 | 3 |
math | Example 8. The random variable $X$ is uniformly distributed on the interval $[-3,2]$. Find the distribution function $F(x)$ of this random variable. | F(x)={\begin{pmatrix}0&\text{for}&x\leq-3\\\frac{x+3}{5}&\text{for}&-3<x<2\\1&\text{for}&x\geq2\end{pmatrix}.} | 36 | 61 |
math | 2. For a regular quadrilateral pyramid, the ratio of the area of a diagonal section to the area of a side face is $\sqrt{6}: 2$, then the angle between the side face and the base is . $\qquad$ | \frac{\pi}{3} | 50 | 7 |
math | 15. Given $f(x)=x^{2}+c$, and $f(f(x))=f\left(x^{2}+1\right)$.
(1) Let $g(x)=f(f(x))$, find the analytical expression of the function $g(x)$;
(2) Let $\varphi(x)=g(x)-\lambda f(x)$, try to find the value of the real number $\lambda$ such that $\varphi(x)$ is a decreasing function on $(-\infty,-1]$ and an increasing fun... | 4 | 122 | 1 |
math | 19.1.11 * Find all positive integers $n$ such that $(n-36)(n-144)-4964$ is a perfect square. | 2061,1077,489,297 | 39 | 17 |
math | Three, (50 points) Try to find the number of all sequences that satisfy the following conditions: (1) each term is an integer no less than 2;
(2) the sum of all terms equals a fixed value $m$.
| \frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{-1}-(\frac{1-\sqrt{5}}{2})^{-1}] | 51 | 42 |
math | $4 \cdot 31$ Solve the equation $\left[x^{2}-2 x\right]=[x]^{2}-2[x]$ for real numbers.
Find the real solutions to the equation $\left[x^{2}-2 x\right]=[x]^{2}-2[x]$. | x\in[n+1,1+\sqrt{1+n^{2}})foranyn\in\mathbb{N}\cup{0}orxisnegativeintegeror0 | 61 | 38 |
math | 11. In an isosceles right $\triangle ABC$, it is known that $\angle ABC=90^{\circ}$, and the coordinates of points $A, B$ are $A(1,0), B(3,1)$, then the coordinates of vertex $C$ are $\qquad$ | (2,3)or(4,-1) | 67 | 11 |
math | Exercise 6. Determine all sequences $\left(a_{n}\right)_{n \geqslant 1}$ of real numbers such that $a_{i}=a_{i+2020}$ for all integers $i \geqslant 1$, and such that
$$
a_{j}+2 a_{j+2} \geqslant a_{j+1}^{2}+a_{j+1}+1
$$
for all integers $\mathrm{j} \geqslant 1$. | 1 | 116 | 1 |
math | Solve the following equations:
a) $2+\frac{5}{4 x}-\frac{15}{4 x(8 x+3)}=\frac{2(7 x+1)}{7 x-3}$,
b) $\frac{2}{x}+\frac{1}{x^{2}}-\frac{7+10 x}{x^{2}\left(x^{2}+7\right)}=\frac{2}{x+\frac{3}{x+\frac{4}{x}}}$. | 4 | 110 | 1 |
math | Example 3 Given two quadratic functions $y_{1}$ and $y_{2}$, when $x$ $=\alpha(\alpha>0)$, $y_{1}$ reaches its maximum value of 5, and $y_{2}=25$; also, the minimum value of $y_{2}$ is $-2, y_{1}+y_{2}=x^{2}+16 x+$ 13. Find the value of $\alpha$ and the analytical expressions of the quadratic functions $y_{1}$ and $y_{... | y_{1}=-2 x^{2}+4 x+3, y_{2}=3 x^{2}+12 x+10 | 120 | 32 |
math | ## Task B-2.5.
A bus left from place $A$ to place $B$. 50 minutes later, a car left from place $A$ and arrived at place $B$ 10 minutes before the bus. If they had left simultaneously, one from place $A$ and the other from place $B$ (one heading towards the other), they would have met after one hour and 12 minutes. If ... | 3 | 122 | 1 |
math | 2. Solve the inequality $\log _{x}(6 x-5)>2$.
# | x\in(5/6;1)\cup(1;5) | 20 | 16 |
math | 5. Find all 4-digit numbers that are 7182 less than the number written with the same digits in reverse order.
ANSWER: 1909 | 1909 | 36 | 4 |
math | 18. In $\triangle A B C$, the three sides $a$, $b$, $c$ satisfy $2 b=a+c$. Find the value of $5 \cos A-4 \cos A \cos C+5 \cos C$. | 4 | 52 | 1 |
math | 8.2. For different numbers $a$ and $b$, it is known that $\frac{a}{b}+a=\frac{b}{a}+b$. Find $\frac{1}{a}+\frac{1}{b}$. | -1 | 53 | 2 |
math | 7.168. $\log _{4 x+1} 7+\log _{9 x} 7=0$. | \frac{1}{12} | 29 | 8 |
math | Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he... | 10 | 118 | 2 |
math | 6. Let the set $I=\{1,2, \cdots, n\}(n \geqslant 3)$. If two non-empty proper subsets $A$ and $B$ of $I$ satisfy $A \cap B=\varnothing, A \cup$ $B=I$, then $A$ and $B$ are called a partition of $I$. If for any partition $A, B$ of the set $I$, there exist two numbers in $A$ or $B$ such that their sum is a perfect square... | 15 | 129 | 2 |
math | Example 9 In the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, $a_{1}=b_{1}=1$, and $\left\{\begin{array}{l}a_{n+1}=b_{n}+3 \cdots \text { (1) } \\ b_{n+1}=2 a_{n}-1 \cdots \text { (2) }\end{array}\left(n \in \mathbf{N}^{*}\right)\right.$, find $a_{n}$ and $b_{n}$. | a_{n}=\frac{1-(-1)^{n}}{2}(3\cdot2^{\frac{n-1}{2}})+\frac{1+(-1)^{n}}{2}(3\cdot2^{\frac{n}{2}}-2),\b_{n}=\frac{1-(-1)^{n}}{2}(3\cdot2^ | 132 | 84 |
math | Four, (20 points) Given that $x$ and $y$ are real numbers, and satisfy
$$
\begin{array}{l}
x y + x + y = 17, \\
x^{2} y + x y^{2} = 66 .
\end{array}
$$
Find the value of $x^{4} + x^{3} y + x^{2} y^{2} + x y^{3} + y^{4}$. | 12499 | 103 | 5 |
math | ## 18. Essay in Latin
An essay in Latin is scored on a scale from 0 to $20^{1}$. Michel's score is above the average, while Claude's score is below the average. What score did each of them receive, if it is known that when one third of the smaller of these two scores is subtracted from each of them, one of the resulti... | =14p=6 | 91 | 6 |
math | 8 The number of non-empty subsets of the set $\{1,2,3, \cdots, 2009\}$ whose elements sum to an odd number is $\qquad$ . | 2^{2008} | 42 | 7 |
math | 10. (20 points) Let the left vertex of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ be $A$, and the right focus be $F(c, 0)$, and $2b$, $a$, $c$ form a geometric sequence. A line passing through point $F$ intersects the ellipse at points $M$ and $N$, and the lines $AM$ and $AN$ intersect the right dir... | 90 | 134 | 2 |
math | 4. Given $n(n \geqslant 3)$ lines where exactly $m(m \geqslant 2)$ lines are parallel, and no three lines intersect at the same point. The maximum number of regions these $n$ lines can divide the plane into is $\qquad$ | \frac{1}{2}\left(n^{2}+n-m^{2}+m\right)+1 | 62 | 24 |
math | 4. Without calculating the product: 1.2.3.4.5.6.7.8.9.10.11$\cdot$12.13, find its last two digits. | 0 | 46 | 1 |
math | 2. Compute
$$
\sum_{n_{60}=0}^{2} \sum_{n_{59}=0}^{n_{60}} \cdots \sum_{n_{2}=0}^{n_{3}} \sum_{n_{1}=0}^{n_{2}} \sum_{n_{0}=0}^{n_{1}} 1 .
$$ | 1953 | 84 | 4 |
math | Example 4. Find $\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-3}\right)^{x}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the text is already in English, there's no need for translation. If you meant to tr... | e^5 | 149 | 3 |
math | 14. (5 points) Wang Yu is playing a balloon shooting game, which has two levels, and the number of balloons in each level is the same. If in the first level, the number of balloons Wang Yu hits is 2 more than 4 times the number of balloons he misses; in the second level, the number of balloons he hits is 8 more than in... | 147 | 111 | 3 |
math | Let $A B C$ be the angles of a triangle and $R$ the radius of the circle passing through the vertices. Connect the feet of the altitudes of the triangle and calculate the sides, angles, and the radius of the circumscribed circle of the resulting triangle (the pedal triangle). | \frac{R}{2} | 61 | 7 |
math | Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$ | f(x) = \pm x + C | 60 | 10 |
math | Three. (50 points) Given a finite set of planar vectors $M$, for any three elements chosen from $M$, there always exist two elements $\boldsymbol{a}, \boldsymbol{b}$ such that $\boldsymbol{a}+\boldsymbol{b} \in M$. Try to find the maximum number of elements in $M$.
| 7 | 75 | 1 |
math | ## Task 3 - 170623
In the turning shop of a company, individual parts are turned from lead rods. Each lead rod produces one individual part.
The swarf obtained from the production of every 6 individual parts can be melted down to make one lead rod. (Any smaller amount of swarf is insufficient for this purpose.)
What... | 43 | 97 | 2 |
math | 17. In triangle $A B C, \angle B A C$ is $120^{\circ}$. The length of $A B$ is 123 . The point $M$ is the midpoint of side $B C$. The line segments $A B$ and $A M$ are perpendicular. What is the length of side $A C$ ? | 246 | 79 | 3 |
math | Find the pairs $(a, b)$ of non-zero natural numbers, with $b \neq 1$, such that the numbers $\frac{a^{3} b-1}{a+1}$ and $\frac{b^{3} a+1}{b-1}$ are non-zero natural numbers.
## 2 Solution | (1,3),(2,2),(3,3) | 68 | 13 |
math | 1. The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ is | 2\leqslantx<4 | 47 | 9 |
math | 6. (2003 Bulgarian Mathematical Competition) The sequence $\left\{y_{n}\right\}$ is defined as follows:
$$
y_{1}=y_{2}=1, y_{n+2}=(4 k-5) y_{n+1}-y_{n}+4-2 k, n=1,2, \cdots
$$
Find all integers $k$ such that every term in the sequence $\left\{y_{n}\right\}$ is a perfect square. | k=1ork=3 | 110 | 6 |
math | In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$. | 30^\circ | 51 | 4 |
math | 2. If the difference between the maximum and minimum elements of the real number set $\{1,2,3, x\}$ equals the sum of all elements in the set, then the value of $x$ is $\qquad$ ـ. | -\frac{3}{2} | 52 | 7 |
math | 5. The maximum value of the algebraic expression $a \sqrt{2-b^{2}}+b \sqrt{2-a^{2}}$ is $\qquad$ . | 2 | 37 | 1 |
math | 74. The cathetus of a right-angled triangle is a perfect cube, the other cathetus represents the difference between this cube and its side (i.e., the first power), and the hypotenuse is the sum of the cube and its side. Find the sides.
## Problems of Iamblichus. | 10,6,8 | 65 | 6 |
math | ## Task 5 - 210835
Someone withdraws a certain amount of money from their savings account. They receive this amount paid out in a total of 29 banknotes, exclusively in 10-mark notes, 20-mark notes, and 50-mark notes. The number of 10-mark notes is 1 less than the number of 20-mark notes. The number of 50-mark notes is... | 1000\mathrm{M} | 119 | 9 |
math | The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there? | 432 | 50 | 3 |
math | \section*{Problem 2 - 101012}
If \(n\) is a positive integer, then \(s_{n}\) denotes the sum of all positive integers from 1 to \(n\).
a) For which positive integer \(n\) do we get \(s_{n}=2415\)?
b) For which positive integer \(m\) is \(s_{m}\) exactly 69 times as large as \(m\)?
\section*{a) It holds that \(s_{n}... | 137 | 124 | 3 |
math | Example 3 Find the positive integer solutions of the equation $\frac{x y}{z}+\frac{x z}{y}+\frac{y z}{x}=3$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (1,1,1) | 61 | 7 |
math | ## Problem Statement
Find the derivative.
$y=\frac{4+x^{4}}{x^{3}} \cdot \operatorname{arctg} \frac{x^{2}}{2}+\frac{4}{x}$ | \frac{x^{4}-12}{x^{4}}\cdot\operatorname{arctg}\frac{x^{2}}{2} | 49 | 31 |
math |
4. Find the largest positive integer $n$ such that
$$
\lfloor\sqrt{1}\rfloor+\lfloor\sqrt{2}\rfloor+\lfloor\sqrt{3}\rfloor+\cdots+\lfloor\sqrt{n}\rfloor
$$
is a prime $(\lfloor x\rfloor$ denotes the largest integer not exceeding $x)$.
(Patrik Bak)
| 47 | 88 | 2 |
math | 13.315. The initial cost price of a unit of product was 50 rubles. During the first year of production, it increased by a certain percentage, and during the second year, it decreased (relative to the increased cost price) by the same percentage, as a result of which it became 48 rubles. Determine the percentages of the... | 20 | 89 | 2 |
math | Dudeney, Amusements in Mathematics Problem 17 Four brothers - named John, William, Charles and Thomas - had each a money-box. They boxes were all given to them on the same day, and they had at once put what money they had into them; only, as the boxes were not very large, they first changed the money into as few coins ... | C=5,J=8,W=12,T=20 | 194 | 14 |
math | 1. (10 points) Calculate: $19 \times 0.125+281 \times \frac{1}{8}-12.5=$ $\qquad$ | 25 | 42 | 2 |
math | 20 Let $a, b, c$ be integers satisfying the inequality $1<a<b<c$, and $(a b-1)(b c-1)(c a-1)$ is divisible by $a b c$. Find the values of $a, b, c$. | =2,b=3,=5 | 57 | 8 |
math | Example 2 Given the ellipse $C: \frac{x^{2}}{24}+\frac{y^{2}}{16}=1$, the line $l: \frac{x}{12}+\frac{y}{8}=1$, and a point $P$ on $l$, the ray $O P$ intersects the ellipse $C$ at $R$. Also, point $Q$ is on $O P$ such that $|O Q| \cdot|O P|=|O R|^{2}$. Find the locus of point $Q$ as $P$ moves along $l$, and identify wh... | \frac{(x-1)^{2}}{\frac{5}{2}}+\frac{(y-1)^{2}}{\frac{5}{3}}=1 | 139 | 36 |
math | 1. (10 points) Calculate: $19 \times 75 + 23 \times 25=$ | 2000 | 27 | 4 |
math | 1. Given the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for any $x, y \in$ $\mathbf{R}$, we have
$$
f(2 x+f(y))=x+y+f(x) .
$$
Find $f(x)$. | f(x)=x(x\in{R}) | 66 | 10 |
math | 4.97 Real number $p \geqslant \frac{1}{4}$. Find all positive real solutions $x$ of the following equation:
$$\log _{\sqrt{2}}^{2} x+2 \log _{\sqrt{2}} x+2 \log _{\sqrt{2}}\left(x^{2}+p\right)+p+\frac{15}{4}=0$$ | x=\frac{1}{2} | 90 | 8 |
math | Example 8 Given real numbers $x, y, z$ satisfy
$$
\begin{array}{l}
x+y+z=5, \\
x y+y z+z x=3 .
\end{array}
$$
Then the maximum value of $z$ is | \frac{13}{3} | 56 | 8 |
math | 457*. 80 students are arranged in a rectangle $8 \times 10$. All of them are of different heights. In each transverse row, the tallest student was chosen. The shortest of them turned out to be Andreev. In each longitudinal row, the shortest student was chosen. The tallest of them turned out to be Borisov. Who is taller... | Andreev | 86 | 3 |
math | 7.048. $0.5\left(\lg \left(x^{2}-55 x+90\right)-\lg (x-36)\right)=\lg \sqrt{2}$. | 54 | 47 | 2 |
math | 9.3. Vasya must write one digit on each face of several dice so that any ordered combination of three digits from 000 to 999 inclusive can be obtained by selecting some three different dice and placing them with the appropriate sides up in the correct order. At the same time, the digits 6 and 9 do not transform into ea... | 5 | 97 | 1 |
math | 6.003. $\frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$.
6.003. $\frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$. | x_{1}=-5,x_{2}=1,x_{3,4}=-1\\sqrt{6} | 79 | 24 |
math | 10. If a four-digit number satisfies the following conditions:
（1）The thousands digit is the same as the hundreds digit, and the tens digit is the same as the units digit;
（2）The difference between this four-digit number and 99 is a perfect square. Find this four-digit number. | 1188,4455,9900 | 64 | 14 |
math | An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm? | 10 | 69 | 2 |
math | 1st Putnam 1938 Problem B6 What is the shortest distance between the plane Ax + By + Cz + 1 = 0 and the ellipsoid x 2 /a 2 + y 2 /b 2 + z 2 /c 2 = 1. You may find it convenient to use the notation h = (A 2 + B 2 + C 2 ) -1/2 , m = (a 2 A 2 + b 2 B 2 + c 2 C 2 ) 1/2 . What is the algebraic condition for the plane not to... | (1-)if<1,otherwise0 | 137 | 9 |
math | (1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$.
(2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^... | f(x) = 2x - \frac{1}{2} | 223 | 16 |
math | Solve the following equation:
$$
\log _{x} 10+2 \log _{10 x} 10+3 \log _{100 x} 10=0
$$ | x_{1}=10^{\frac{-5+\sqrt{13}}{6}},\quadx_{2}=10^{\frac{-5-\sqrt{13}}{6}} | 48 | 42 |
math | An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. | 144 | 64 | 3 |
math | Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have. | 7 | 69 | 1 |
math | Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$. | 126 | 102 | 3 |
math | The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in ... | 90^\circ - \frac{\angle BAC}{2} | 127 | 14 |
math | 18. Let $f(x), g(x)$ be odd and even functions defined on $R$, respectively, and $f(x)+g(x)=2^{x}$. If for $x \in\left[\frac{1}{2}, 2\right]$, the inequality $a f(x)-f(3 x) \leqslant 2 g(2 x)$ always holds, find the range of the real number $a$. | (-\infty,10] | 94 | 8 |
math |
C4. We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \%$ have sent a message to $A$. How many possible two-digit values of $n$ are there?
| 26 | 78 | 2 |
math | 13.325. 9000 parts can be manufactured on several new machines of the same design and one machine of the old design, which works twice as slowly as each of the new machines. The old machine can also be replaced by a new machine of the same design as the others. In this second option, each machine would produce 200 fewe... | 5 | 95 | 1 |
math | [ Isosceles, Inscribed, and Circumscribed Trapezoids ]
[Properties and characteristics of isosceles triangles. ]
Let $M$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$, in which sides $AB$, $AD$, and $BC$ are equal to each other.
Find the angle $CMD$, given that $DM = MC$, and $\angle... | 120 | 95 | 3 |
math | Michelle has a word with $2^n$ letters, where a word can consist of letters from any alphabet. Michelle performs a swicheroo on the word as follows: for each $k = 0, 1, \ldots, n-1$, she switches the first $2^k$ letters of the word with the next $2^k$ letters of the word. For example, for $n = 3$, Michelle changes \[
A... | 2^n | 171 | 2 |
math | 7.146. $\left\{\begin{array}{l}\log _{\sqrt{x}}(x y)=8 \\ \log _{3} \log _{1 / 9} \frac{x}{y}=0 .\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
\log _{\sqrt{x}}(x y)=8 \\
\log _{3} \log _{1 / 9} \frac{x}{y}=0 .
\end{array}\right.
\] | (3;27) | 123 | 6 |
math | 2 . If $x_{n+1}=\frac{3 x_{n}+4}{2 x_{n}+1}, x_{1}=1$, find $x_{n}$.
| x_{n}=\frac{4(-5)^{n}-1}{2(-5)^{n}+1} | 42 | 26 |
math | Subject 3. In the equilateral triangle $\mathrm{ABC}$ with $\mathrm{AB}=6 \mathrm{~cm}$, perpendiculars $\mathrm{A}^{\prime} \mathrm{A} \perp(\mathrm{ABC}), \mathrm{B}^{\prime} \mathrm{B} \perp(\mathrm{ABC})$ are raised from the same side of the plane (ABC), such that $\mathrm{AA}^{\prime}=\mathrm{BB}^{\prime}=6 \sqrt{... | \frac{\sqrt{39}}{8} | 150 | 11 |
math | 11. In rectangle $A B C D$, it is known that $A B=2, A D<\sqrt{2}$, and an ellipse $K$ is constructed with side $A B$ as the major axis such that the length of the minor axis of ellipse $K$ is $\sqrt{2}|A D|$. Take a point $P$ on ellipse $K$ different from the endpoints, and connect $P C$ and $P D$, intersecting $A B$ ... | 4 | 131 | 1 |
math | Circles $ A$, $ B$ and $ C$ are externally tangent to each other and internally tangent to circle $ D$. Circles $ B$ and $ C$ are congruent. Circle $ A$ has radius $ 1$ and passes through the center of $ D$. What is the radius of circle $ B$?
[asy]
unitsize(15mm);
pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
draw(Ci... | \frac{8}{9} | 261 | 7 |
math | There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$.
([i]Note. An integer point is a point with integer coordinates.[/i])
[i]Proposed by CSJL.[/i] | 3954 | 75 | 4 |
math | 6. Choose three different numbers from $1,2, \cdots, 20$, then the probability that these three numbers form an arithmetic sequence is $\qquad$ | \frac{3}{38} | 36 | 8 |
math | A4. Twenty students go abseiling during a school trip. In each round, one student gets a turn to abseil, so after twenty rounds, everyone has safely descended. To determine who goes first in round 1, cards with numbers 1 to 20 are distributed to the students. The one who gets the 1 starts. In round 2, cards with number... | 66 | 176 | 2 |
math | A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}},
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The an... | 2^{101}-1 | 111 | 7 |
math | Let $p$ be an odd prime number. Let $g$ be a primitive root of unity modulo $p$. Find all the values of $p$ such that the sets $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$ and $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$ are equal modulo $p$. | p = 3 | 101 | 5 |
math | If $x+y=1$ and $x^{2}+y^{2}=2$, calculate $x^{3}+y^{3}$. | \frac{5}{2} | 32 | 7 |
math | 66. Solve the system of equations using the matrix method
$$
\left\{\begin{array}{cc}
x_{1}+2 x_{2} & =10 \\
3 x_{1}+2 x_{2}+x_{3} & =23 \\
x_{2}+2 x_{3} & =13
\end{array}\right.
$$ | x_{1}=4,x_{2}=3,x_{3}=5 | 84 | 15 |
math | Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$. | \frac{1}{2} | 69 | 7 |
math | Let's determine the
$$
\lim _{n \rightarrow+\infty} \frac{\sqrt{n^{2}-1^{2}}+\sqrt{n^{2}-2^{2}}+\ldots+\sqrt{n^{2}-(n-1)^{2}}}{n^{2}}
$$
limit. | \frac{\pi}{4} | 66 | 7 |
math | Question 109, If the complex number $z$ satisfies $|z-\sqrt{3}|+|z+\sqrt{3}|=4$, then the maximum value of $|z+i|$ is
保留源文本的换行和格式，直接输出翻译结果。
(Here, the last sentence is a note to the translator, not part of the translation. The actual translation is above.) | \frac{4\sqrt{3}}{3} | 85 | 12 |
math | 5. In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, points $X$ and $Y$ are the centers of the squares $A A_{1} B_{1} B$ and $B B_{1} C_{1} C$, respectively, and point $Z$ is on the diagonal $B D$ such that $D Z=3 Z B$. Then the area of the section cut by the plane $X Y Z$ from the circumscribed sphere of... | \frac{5\pi}{8} | 125 | 9 |
math | 1. If real numbers $x, y$ satisfy $x^{2}+y^{2}+x y=1$, then the maximum value of $x+y$ is $\qquad$ . | \frac{2\sqrt{3}}{3} | 42 | 12 |
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