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 | 4. In the sequence $\left\{a_{n}\right\}$, $a_{k}+a_{k+1}=2 k+1\left(k \in \mathbf{N}^{*}\right)$, then $a_{1}+a_{100}$ equals | 101 | 64 | 3 |
math | 6. If a sequence of numbers $a_{1}, a_{2}, \cdots$ satisfies that for any positive integer $n$,
$$
a_{1}+a_{2}+\cdots+a_{n}=n^{3}+3 n^{2},
$$
then $\frac{1}{a_{2}+2}+\frac{1}{a_{3}+2}+\cdots+\frac{1}{a_{2015}+2}=$ $\qquad$ | \frac{1007}{6048} | 108 | 13 |
math | In the isosceles triangle $A B C$, $A C B \angle = A B C \angle = 40^{\circ}$. On the ray $A C$, we measure and mark a segment $A D$ equal to $B C$. What is the measure of $\angle B D C$? | 30 | 68 | 2 |
math | 5. The maximum value of the function $f(x)=(6-x)^{3}(x-1)^{\frac{2}{3}}(1 \leqslant x \leqslant 6)$ is
保留了源文本的换行和格式。 | \frac{3^{6} \times 5^{3}}{11^{4}} \sqrt[3]{1100} | 57 | 30 |
math | Example 1.40 A centipede with 40 legs and a dragon with 3 heads are in the same cage, with a total of 26 heads and 298 legs. If the centipede has 1 head, then the dragon has $\qquad$ legs. (1988, Shanghai Junior High School Competition)
If the centipede has $x$ individuals, and the dragon has $y$ individuals, and each ... | 14 | 266 | 2 |
math | 134. Someone said to his friend: "Give me 100 rupees, and I will be twice as rich as you," to which the latter replied: "If you give me just 10 rupees, I will be six times richer than you." The question is: how much did each have? | 40;170 | 67 | 6 |
math | $4 \cdot 61$ For what real values of $a$, $b$, and $c$, does the equation
$$
\begin{aligned}
& |a x+b y+c z|+|b x+c y+a z|+|c x+a y+b z| \\
= & |x|+|y|+|z|
\end{aligned}
$$
hold for all real numbers $x, y, z$? | ,b,(0,0,1),(0,0,-1),(0,1,0),(0,-1,0),(1,0,0),(-1,0,0) | 95 | 39 |
math | Example 6. Try to find the equations of all ellipses with foci at $(\pm 3,0)$, and write down the conditions under which the equation can represent a hyperbola with the same foci as the ellipse. | a>3 | 51 | 3 |
math | Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$ | (1, 1, 1) | 82 | 10 |
math | 3. The sum of the digits of the number $X$ is $Y$, and the sum of the digits of the number $Y$ is $Z$. If $X+Y+Z=60$, determine the number $X$.
| 44,50,47 | 51 | 8 |
math | 1. Calculate: $\frac{2 \frac{5}{8}-\frac{2}{3} \times 2 \frac{5}{14}}{\left(3 \frac{1}{12}+4.375\right) \div 19 \frac{8}{9}}$ | 2\frac{17}{21} | 68 | 10 |
math | # Problem 8. (4 points)
Solve in integers: $6 x^{2}+5 x y+y^{2}=6 x+2 y+7$
Indicate the answer for which the value of $|x|+|y|$ is the greatest. Write the answer in the form ( $x ; y$ ). | (-8;25) | 70 | 6 |
math | Example 8 Find the real solutions of the system of equations $\left\{\begin{array}{l}x^{3}+x^{3} y^{3}+y^{3}=17, \\ x+x y+y=5\end{array}\right.$. | x_{1}=1, y_{1}=2 ; x_{2}=2, y_{2}=1 | 57 | 23 |
math | B1. What is the smallest positive integer consisting of the digits 2, 4, and 8, where each of these digits appears at least twice and the number is not divisible by 4? | 244882 | 42 | 6 |
math | 167. To transport 60 tons of cargo from one place to another, a certain number of trucks were required. Due to the poor condition of the road, each truck had to carry 0.5 tons less than originally planned, which is why 4 additional trucks were required. How many trucks were initially required? | 20 | 67 | 2 |
math | 88. Among the balls that differ only in color, there are 6 white, 4 black, and 8 red. In how many ways can two boys divide these balls (not necessarily equally) between themselves so that each of them gets at least two balls of each color? | 15 | 58 | 2 |
math | 3. For the quadratic trinomials $f_{1}(x)=a x^{2}+b x+c_{1}, f_{2}(x)=a x^{2}+b x+c_{2}$, $\ldots, f_{2020}(x)=a x^{2}+b x+c_{2020}$, it is known that each of them has two roots.
Denote by $x_{i}$ one of the roots of $f_{i}(x)$, where $i=1,2, \ldots, 2020$. Find the value
$$
f_{2}\left(x_{1}\right)+f_{3}\left(x_{2}\r... | 0 | 189 | 1 |
math | 9th Chinese 1994 Problem A3 X is the interval [1, ∞). Find all functions f: X → X which satisfy f(x) ≤ 2x + 2 and x f(x + 1) = f(x) 2 - 1 for all x. | f(x)=x+1 | 62 | 6 |
math | 7. Place eight "Rooks" on an $8 \times 8$ chessboard. If no two rooks can "capture" each other, then the eight rooks are in a "safe state". There are $\qquad$ different safe states. | 40320 | 54 | 5 |
math | 86. Using the equality $x^{3}-1=(x-1)\left(x^{2}+x+1\right)$, solve the equation
$$
x^{3}-1=0
$$
Applying the obtained formulas, find the three values of $\sqrt[3]{1}$ in 103 -arithmetic. | x_{1}=1,\quadx_{2}=56,\quadx_{3}=46 | 73 | 21 |
math | In how many ways can 105 be expressed as the sum of at least two consecutive positive integers? | 7 | 22 | 1 |
math | 7.1. (13 points) Find $\frac{S_{1}}{S_{2}}$, where
$$
S_{1}=\frac{1}{2^{18}}+\frac{1}{2^{17}}-\frac{1}{2^{16}}+\ldots+\frac{1}{2^{3}}+\frac{1}{2^{2}}-\frac{1}{2}, \quad S_{2}=\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+\ldots+\frac{1}{2^{16}}+\frac{1}{2^{17}}-\frac{1}{2^{18}}
$$
(in both sums, the signs of the terms ... | -0.2 | 182 | 4 |
math | 395. Find the values of $\cos x, \operatorname{tg} x, \operatorname{ctg} x$, if it is known that $\sin x=-3 / 5, 0<x<3 \pi / 2$. | \cosx=-\frac{4}{5},\operatorname{tg}x=\frac{3}{4},\operatorname{ctg}x=\frac{4}{3} | 54 | 40 |
math | 3. Let the sides of the cyclic quadrilateral $ABCD$ be $AB=3, BC=4, CD=5, DA=6$, then the area of quadrilateral $ABCD$ is $\qquad$ . | 6\sqrt{10} | 48 | 7 |
math | $6 \cdot 9$ Suppose there are $n$ teams participating in a volleyball tournament, and the tournament is conducted in a knockout (elimination) format. In the tournament, every two teams that meet play one match, and each match must have a winner and a loser, with the losing team being eliminated from the tournament. How... | n-1 | 84 | 3 |
math | 15 There are 10 players $A_{1}, A_{2}, \cdots, A_{10}$, their points are $9, 8, 7, 6, 5, 4, 3, 2, 1, 0$, and their ranks are 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th. Now a round-robin tournament is held, that is, each pair of players will play exactly one match, and each match must have a winner. If the playe... | 12 | 255 | 2 |
math | Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$. | y^3 + 15y^2 - 198y + 1 | 38 | 19 |
math | 23. Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1,1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. (For example, the following are two distinct ways of using two tiles of size $1 \times 1$, two tiles of size $1 \times 2$ and one tile of size $1... | 169 | 151 | 3 |
math | 5.103 Suppose there are 100 mutually hostile countries on Mars. To maintain peace, it is decided to form several alliances, with each alliance including at most 50 countries, and any two countries must belong to at least one alliance. Try to answer the following questions:
(1) What is the minimum number of alliances ne... | 6 | 106 | 1 |
math | Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots. | 4027 | 76 | 4 |
math | We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24$?
[i]2020 CCA Math Bonanza Lightning Round #1.1[/i] | 1 | 77 | 1 |
math | 13. Given the function $f(x)=4 \cos x \cdot \sin \left(x+\frac{7 \pi}{6}\right)+a$ has a maximum value of 2.
(1) Find the value of $a$ and the smallest positive period of $f(x)$; (2) Find the intervals where $f(x)$ is monotonically decreasing. | =1,T=\pi,[-\frac{\pi}{3}+k\pi,\frac{\pi}{6}+k\pi],k\in{Z} | 80 | 36 |
math | 1. (2 points) Boy Vasya tried to recall the distributive law of multiplication and wrote the formula: $a+(b \times c)=(a+b) \times(a+c)$. Then he substituted three non-zero numbers into this formula and found that it resulted in a true equality. Find the sum of these numbers. | 1 | 68 | 1 |
math | 11.4. In a semicircle with a radius of 18 cm, a semicircle with a radius of 9 cm is constructed on one of the halves of the diameter, and a circle is inscribed, touching the larger semicircle from the inside, the smaller semicircle from the outside, and the second half of the diameter. Find the radius of this circle.
... | 8 | 83 | 1 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=2 p+3 q$
$b=p-2 q$
$|p|=2$
$|q|=1$
$(\widehat{p, q})=\frac{\pi}{3}$ | 7\sqrt{3} | 65 | 6 |
math | 163. Reduced Share. A father gave his children 6 dollars for entertainment, which was to be divided equally. But two young cousins joined the company. The money was divided equally among all the children, so that each child received 25 cents less than originally intended. How many children were there in total? | 8 | 65 | 1 |
math | Can a regular triangle be divided into
a) 2007,
b) 2008 smaller regular triangles? | 2007 | 27 | 4 |
math | 1. Do there exist natural numbers $a, b$, and $c$ such that each of the equations
$$
\begin{array}{ll}
a x^{2}+b x+c=0, & a x^{2}+b x-c=0 \\
a x^{2}-b x+c=0, & a x^{2}-b x-c=0
\end{array}
$$
has both roots as integers? | =1,b=5,=6 | 93 | 8 |
math | Example 11 Let natural numbers $a, b, c, d$ satisfy $\frac{a}{b}+\frac{c}{d}<1$, and $a+c=20$, find the maximum value of $\frac{a}{b}+\frac{c}{d}$. | \frac{1385}{1386} | 61 | 13 |
math | 8.96 Find all sequences of natural numbers $a_{1}, a_{2}, a_{3}, \cdots$ that satisfy the following two conditions:
(1) $a_{n} \leqslant n \sqrt{n}, n=1,2,3, \cdots$
(2) For any different $m, n$,
$$m-n \mid a_{m}-a_{n}$$ | a_{n}=1 \text{, for any } n \geqslant 1 \text{; } a_{n}=n, \text{ for any } n \geqslant 1 | 90 | 45 |
math | Solve the following equation:
$$
\sqrt{\frac{x-1991}{10}}+\sqrt{\frac{x-1990}{11}}=\sqrt{\frac{x-10}{1991}}+\sqrt{\frac{x-11}{1990}} .
$$ | 2001 | 66 | 4 |
math | 18. Master Li made 8 identical rabbit lanterns in 3 days, making at least 1 lantern each day. Master Li has ( ) different ways to do this. | 21 | 37 | 2 |
math | ## Task 3 - 030933
Which points $P(x ; 0)$ are twice as far from the point $P_{1}(a ; 0)$ as from $P_{2}(b ; 0)$?
Determine the abscissas of these points! $(b>a)$ | +\frac{2}{3}(b-)2b- | 67 | 12 |
math | 8. Solve the system $\left\{\begin{array}{l}\frac{1}{2} \log _{2} x-\log _{2} y=0 \\ x^{2}-2 y^{2}=8 .\end{array}\right.$ | 4;2 | 56 | 3 |
math | 39. Find a two-digit number that has the following properties:
a) if the desired number is multiplied by 2 and 1 is added to the product, the result is a perfect square;
b) if the desired number is multiplied by 3 and 1 is added to the product, the result is a perfect square. | 40 | 68 | 2 |
math | Note that if we flip the sheet on which the digits are written, the digits $0,1,8$ will not change, 6 and 9 will swap places, and the others will lose their meaning. How many nine-digit numbers exist that do not change when the sheet is flipped? | 1500 | 60 | 4 |
math | A regular hexagon and an equilateral triangle have the same perimeter. What is the ratio of their areas? | \frac{3}{2} | 22 | 7 |
math | 10. (12 points) In the Sheep Sheep Sports Meet, Happy Sheep, Boiling Sheep, Lazy Sheep, Warm Sheep, and Big Bad Wolf participated in a 400-meter race. After the race, the five of them discussed the results.
First place said: “Happy Sheep ran faster than Lazy Sheep.”
Second place said: “I ran faster than Warm Sheep.”
Th... | 2 | 143 | 1 |
math | While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
| 35 | 57 | 2 |
math | There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time? | n \neq 1, 3 | 74 | 10 |
math | Let $A B C D$ be a parallelogram. Let $E$ be a point such that $A E=B D$ and $(A E)$ is parallel to $(B D)$. Let $F$ be a point such that $E F=A C$ and $(E F)$ is parallel to $(A C)$. Express the area of the quadrilateral $A E F C$ in terms of the area of $A B C D$. | 2\timesAreaofABCD | 94 | 7 |
math | 7. (4 points) In an opaque cloth bag, there are 10 chopsticks each of black, white, and yellow colors. The minimum number of chopsticks to be taken out to ensure that there is a pair of chopsticks of the same color is $\qquad$. | 4 | 58 | 1 |
math | Example 2. Let $A, B, C$ be the three interior angles of a triangle. Find the maximum value of $\sin A+\sin B+\sin C$.
untranslated text remains the same as requested, only the problem statement is translated. | \frac{3 \sqrt{3}}{2} | 52 | 12 |
math | 8、Car A and Car B start from A and B respectively at the same time, heading towards each other, and meet after 4 hours. Car A then takes another 3 hours to reach B. If Car A travels 20 kilometers more per hour than Car B, the distance between A and B is ( ) kilometers. | 560 | 68 | 3 |
math | Example 2 A class participated in a math competition, with a total of $a$, $b$, and $c$ three questions. Each question either scores full marks or 0 points, where question $a$ is worth 20 points, and questions $b$ and $c$ are worth 25 points each. After the competition, every student answered at least one question corr... | 42 | 220 | 2 |
math | 10.1. A motorist and a cyclist set off from points A and B, not at the same time, towards each other. Meeting at point C, they immediately turned around and continued with the same speeds. Upon reaching their respective points A and B, they turned around again and met for the second time at point D. Here they turned ar... | C | 96 | 1 |
math | 4.10. From point $A$, two rays are drawn intersecting a given circle: one - at points $B$ and $C$, the other - at points $D$ and $E$. It is known that $A B=7, B C=7, A D=10$. Determine $D E$. | 0.2 | 69 | 3 |
math | 377. Divide the given number into two parts so that their product is the greatest possible (solve in an elementary way). | \frac{N}{2} | 26 | 7 |
math | 5. Determine the real numbers $a$ and $b, a \leq b$ for which the following conditions are satisfied:
a) for any real numbers $x$ and $y$ such that $0 \leq y \leq x$ and $x+5 y=105$ it holds that $a \leq x+y \leq b$,
b) the difference $b-a$ is minimal. | 70 | 91 | 2 |
math | 585. Find the one-sided limits of the function
$$
f(x)=\operatorname{arctg} \frac{1}{x-1}(x \neq 1) \text { as } x \rightarrow 1
$$ | f(1+0)=\frac{\pi}{2},\quadf(1-0)=-\frac{\pi}{2} | 54 | 29 |
math | 2. It is known that $\frac{\cos x-\sin x}{\cos y}=\frac{\sqrt{2}}{3} \operatorname{tg} \frac{x+y}{2}$ and $\frac{\sin x+\cos x}{\sin y}=3 \sqrt{2} \operatorname{ctg} \frac{x+y}{2}$. Find all possible values of the expression $\operatorname{tg}(x+y)$, given that there are at least three. | 1,\frac{3}{4},-\frac{3}{4} | 102 | 15 |
math | (4) If real numbers $a$, $b$, $c$ satisfy: $a+b+c=a^{2}+b^{2}+c^{2}$, then the maximum value of $a+b+c$ is $\qquad$ . | 3 | 52 | 1 |
math | ## Task 3 - 280513
Four equally sized boxes with the same contents have a combined mass of $132 \mathrm{~kg}$.
What is the mass of the contents of one box, if the total mass of all four empty boxes is 12 $\mathrm{kg}$? | 30\mathrm{~} | 67 | 7 |
math | 14. The set $P$ consisting of certain integers has the following properties:
(1) $P$ contains both positive and negative numbers;
(2) $P$ contains both odd and even numbers;
(3) $1 \notin P$;
(4) If $x, y \in P$, then $x+y \in P$.
Determine the relationship between 0, 2 and the set $P$. | 0\inP,2\notinP | 90 | 10 |
math | 7、A and B read a 120-page book, starting on October 1st. A reads 8 pages every day; B reads 13 pages every day, but he takes a break every two days. By the end of the long holiday on October 7th, A and B $\qquad$ compared $\qquad$ who read more, and by how many pages. | 9 | 82 | 1 |
math | Given the following system of equations:
$$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$
Find the value of L that makes $LMT +SPR I NG = 2023$ true.
| \frac{341}{40} | 101 | 10 |
math | |
At a familiar factory, they cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 disks wi... | 4 | 101 | 1 |
math | 8,9
Find the sum of the exterior angles of a convex $n$-gon, taken one at each vertex. | 360 | 26 | 3 |
math | 14. Given the function $f(x)=\sin ^{4} x$.
(1) Let $g(x)=f(x)+f\left(\frac{\pi}{2}-x\right)$, find the maximum and minimum values of $g(x)$ in the interval $\left[\frac{\pi}{6}, \frac{3 \pi}{8}\right]$;
(2) Find the value of $\sum_{k=1}^{89} f\left(\frac{k \pi}{180}\right)$. | \frac{133}{4} | 114 | 9 |
math | Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime. | 591 \times 1990^{1989} | 52 | 18 |
math | 6. Problem: How many pairs of positive integers $(a, b)$ with $\leq b$ satisfy $\frac{1}{a}+\frac{1}{b}=\frac{1}{6}$ ? | 5 | 44 | 1 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\text { GCD }(a ; b ; c)=33, \\
\text { LCM }(a ; b ; c)=3^{19} \cdot 11^{15} .
\end{array}\right.
$$ | 9072 | 91 | 4 |
math | Let $z$ be a complex number such that $|z| = 1$ and $|z-1.45|=1.05$. Compute the real part of $z$. | \frac{20}{29} | 41 | 9 |
math | 11. For a $2n$-digit number, if the sum of the first $n$ digits and the last $n$ digits, each considered as an $n$-digit number, squared equals the original $2n$-digit number, then this $2n$-digit number is called a Kabulek strange number. For example, $(30+25)^{2}=3025$, so 3025 is a Kabulek strange number. What are t... | 2025,3025,9801 | 118 | 14 |
math | ## Task 11/89
We are looking for all pairs $(n ; k)$ of natural numbers $n$ and $k$ for which the following holds: The sum of the squares of $n$ and its $k$ immediate predecessors is equal to the sum of the squares of the $k$ immediate successors of $n$. | 2k(k+1) | 71 | 6 |
math | $2$ darts are thrown randomly at a circular board with center $O$, such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked $A$ and $B$. What is the probability that $\angle AOB$ is acute? | \frac{1}{2} | 61 | 7 |
math | (Czech-Polish-Slovak Match 2018)(M-D) Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+x y\right)=f(x) f(y)+y f(x)+x f(x+y)
$$ | f(x)=1-xorf(x)=-xorf(x)=0 | 70 | 14 |
math | Problem 2. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+y+f(y)\right)=2 y+(f(x))^{2}
$$
for any $x, y \in \mathbb{R}$. | f(x)\equivx | 69 | 5 |
math | 4. (24 points)(1) If the sum of the volumes of $n\left(n \in \mathbf{N}_{+}\right)$ cubes with edge lengths as positive integers equals 2005, find the minimum value of $n$ and explain the reason;
(2) If the sum of the volumes of $n\left(n \in \mathbf{N}_{+}\right)$ cubes with edge lengths as positive integers equals $2... | 4 | 120 | 1 |
math | 2. (10 points) Xiao Long's mother is 3 years younger than his father, his mother's age this year is 9 times Xiao Long's age this year, and his father's age next year will be 8 times Xiao Long's age next year. What is the father's age this year? | 39 | 65 | 2 |
math | ## Problem B1
A $\square\{1,2,3, \ldots, 49\}$ does not contain six consecutive integers. Find the largest possible value of |A|. How many such subsets are there (of the maximum size)?
## Answer
$\max =41$; no. ways 495
| 495 | 71 | 3 |
math | 3. In the Cartesian coordinate system, $\vec{e}$ is a unit vector, and vector $\vec{a}$ satisfies $\vec{a} \cdot \vec{e}=2$, and $|\vec{a}|^{2} \leq 5|\vec{a}+t \vec{e}|$ for any real number $t$, then the range of $|\vec{a}|$ is $\qquad$ . | [\sqrt{5},2\sqrt{5}] | 92 | 11 |
math | Given a cube $A B C D A 1 B 1 C 1 D 1$. A sphere touches the edges $A D, D D 1, C D$ and the line $B C 1$. Find the radius of the sphere, if the edge of the cube is 1. | 2\sqrt{2}-\sqrt{5} | 64 | 11 |
math | ## Task 2
Calculate by dictation! (Duration 3 to 4 minutes)
$$
\begin{array}{l|l|l|l}
6 \cdot 7 & 9 \cdot 0 & 26+17 \\
45: 5 & 73-19 & 6 \text{~cm}=\ldots \text{mm} & \begin{array}{l}
28: 4 \\
70 \text{dm}=\ldots \text{m}
\end{array}
\end{array}
$$ | 42,0,43,7,9,54,60 | 123 | 17 |
math | 4. Given $z_{1}, z_{2} \in \mathbf{C}$, and $\left|z_{1}\right|=3,\left|z_{2}\right|=5,\left|z_{1}-z_{2}\right|=7$, then $\frac{z_{1}}{z_{2}}=$ | \frac{3}{5}(-\frac{1}{2}\\frac{\sqrt{3}}{2}\mathrm{i}) | 71 | 27 |
math | Example 9. Solve the inequality
$$
\left|x^{3}-x\right| \geqslant 1-x
$$ | (-\infty;\frac{-1-\sqrt{5}}{2}]\cup[\frac{\sqrt{5}-1}{2};+\infty) | 30 | 33 |
math | Example 4 Find all integer pairs $(x, y)$ that satisfy the equation $y^{4}+2 x^{4}+1=4 x^{2} y$. | (1, 1), (-1, 1) | 37 | 12 |
math | 11. (5 points) In a basketball game, a shot made from outside the three-point line scores 3 points, a shot made from inside the three-point line scores 2 points, and a free throw scores 1 point. A team scored a total of 65 points by making 32 shots in one game. It is known that the number of two-point shots is 3 more t... | 4 | 110 | 1 |
math | find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have
$$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$. | \{0\} | 85 | 6 |
math | 75. Solve the system:
$$
\begin{aligned}
& x+y=2(z+u) \\
& x+z=3(y+u) \\
& x+u=4(y+z)
\end{aligned}
$$ | 73,7,17,u=23 | 49 | 11 |
math | XXVIII - II - Task 3
In a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum? | 127 | 82 | 3 |
math | $1 \cdot 13$ If a natural number $N$ is appended to the right of every natural number (for example, appending 2 to 35 gives 352), and if the new number is always divisible by $N$, then $N$ is called a magic number. Among the natural numbers less than 130, how many magic numbers are there?
(China Junior High School Math... | 9 | 95 | 1 |
math | Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$. Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$. If $BC'=30$, find the sum of all possible side lengths of $\triangle ABC$.
[i]Proposed by Connor Gor... | 130 | 94 | 3 |
math | 4. Two students, A and B, are selecting courses from five subjects. They have exactly one course in common, and A selects more courses than B. The number of ways A and B can select courses to meet the above conditions is $\qquad$
翻译完成，保留了原文的换行和格式。 | 155 | 64 | 3 |
math | 10. (20 points) Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n(n \geqslant 1) \text {, and } \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n}=0(n \geqslant 2) \text {. }
$$
Find all possible values of $a_{2021}$. | 2021 | 110 | 4 |
math | Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$. | 4 - 2\sqrt{2} | 87 | 9 |
math | 6. Let $M=\{1,2,3, \cdots, 1995\}, A$ 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 $\qquad$ . | 1870 | 71 | 4 |
math | 21. Find the sum $\sum_{k=1}^{19} k\binom{19}{k}$. | 19\cdot2^{18} | 28 | 9 |
math | Example 5. Find the integral $\int \frac{d x}{\sqrt{5-4 x-x^{2}}}$. | \arcsin\frac{x+2}{3}+C | 27 | 14 |
math | (12) Let the sequence $\left\{a_{n}\right\}$ satisfy: $a_{1}=1, a_{2}=2, \frac{a_{n+2}}{a_{n}}=\frac{a_{n+1}^{2}+1}{a_{n}^{2}+1}(n \geqslant 1)$.
(1) Find the recursive relation between $a_{n+1}$ and $a_{n}$, i.e., $a_{n+1}=f\left(a_{n}\right)$;
(2) Prove: $63<a_{2008}<78$. | 63<a_{2008}<78 | 143 | 11 |
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