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 | A totó requires tipping 12 matches, in each case with the symbols $\gg 1 \ll, » 2 \ll$ or $\gg \mathrm{x} \ll$. What is the expected number of correct tips if we fill out the ticket randomly (e.g., by drawing lots) so that each of the three possibilities has an equal chance for every match? | 4 | 77 | 1 |
math | 9. (18 points) A square piece of paper, if folded 2 times, folded into a small square, and a notch is cut on each side of this small square, when the paper is unfolded, there are 4 small holes in the middle of the paper (notches on the edges do not count). A square piece of paper, if folded 6 times into a small square,... | 112 | 118 | 3 |
math | Twelve $1$'s and ten $-1$'s are written on a chalkboard. You select 10 of the numbers and compute their product, then add up these products for every way of choosing 10 numbers from the 22 that are written on the chalkboard. What sum do you get?
| -42 | 70 | 3 |
math | 2. a) $1+3+5+\ldots+2013=\frac{2014 \cdot 1007}{2}=1007^{2}$
$1+3+5+\ldots+105=\frac{106 \cdot 53}{2}=53^{2}$
The equation is equivalent to: $\frac{1007^{2}}{x}=53^{2} \Rightarrow x=\left(\frac{1007}{53}\right)^{2} \Rightarrow x=19^{2} \Rightarrow x=361$
b) $x=361 \Rightarrow(361-1): 10=y^{2} \Rightarrow y^{2}=36$
... | {-6,6} | 180 | 5 |
math | 6 - 123 Let the complex number $z=x+i y(x, y \in R)$ satisfy $|z-i| \leqslant 1$. Find the maximum and minimum values of $A$ $=x\left(|z-i|^{2}-1\right)$ and the corresponding $z$ values. | A_{max}=\frac{2\sqrt{3}}{9}whenz=-\frac{\sqrt{3}}{3}+i,A_{}=-\frac{2\sqrt{3}}{9}whenz=\frac{\sqrt{3}}{3}+i | 70 | 61 |
math | Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 2 | 67 | 1 |
math | 5. Given $a_{1}=2$,
$$
a_{n+1}=\frac{2^{n+1} a_{n}}{\left(n+\frac{1}{2}\right) a_{n}+2^{n}}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{n^{3}+2 n^{2}+2 n+2}{n(n+1)\left(n^{2}+1\right) a_{n}}$, find the sum of the first $n... | 2-\frac{(n+2)^{2}}{2^{n+1}(n+1)} | 180 | 22 |
math | Solve the following equation in the set of real numbers:
$$
(\sqrt{5}+2)^{x}+(\sqrt{5}-2)^{x}=18
$$ | x_{1}=2x_{2}=-2 | 39 | 11 |
math | 13.308. What two-digit number is less than the sum of the squares of its digits by 11 and greater than their doubled product by 5? | 95or15 | 36 | 5 |
math | 1. The solution to the equation $\frac{16}{3} x^{4}+\frac{1}{6 x^{2}}=\sin \pi x$ is | \frac{1}{2} | 36 | 7 |
math | 9. (10 points) A positive integer, its double has exactly 2 more divisors than itself, and its triple has exactly 3 more divisors than itself. Then, this positive integer is $\qquad$ | 12 | 46 | 2 |
math | 1. Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0,1,2, \ldots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll? | \frac{11}{200} | 70 | 10 |
math | 10.2 For the function $f(x)=2013-a+12 x^{2}-\cos 2 \pi x-8 x^{3}-16 x$ find the number of integer values of $a$, for each of which the equation
$$
\underbrace{f(f(\ldots f}_{2013 \text { times }}(x) \ldots))=2 x-1
$$
on the interval $[50 ; 51]$ has a unique solution. | 60015 | 111 | 5 |
math | 6. (20 points) Calculate the value of the expression:
$$
1 \cdot 2 \cdot(1+2)-2 \cdot 3 \cdot(2+3)+3 \cdot 4 \cdot(3+4)-\cdots+2019 \cdot 2020 \cdot(2019+2020)
$$ | 8242405980 | 81 | 10 |
math | ## Task 1 - 250811
a) Let $b$ be the number obtained when the number 30 is increased by $50 \%$.
By what percent must this number $b$ be reduced to obtain the number 30 again?
b) Check whether the statement found for the number 30 also holds for any positive number $a$ under the same problem statement! | 33\frac{1}{3} | 86 | 9 |
math | Example: Let $P_{1}, P_{2}, \cdots, P_{n}$ be any $n$ points given in the plane, try to find a point $A$ such that $A P_{1}^{2}+A P_{2}^{2}+\cdots+A P_{n}^{2}$ is minimized. | \sum_{k=1}^{n}|z_{k}|^{2}-\frac{1}{n}|\sum_{k=1}^{n}z_{k}|^{2} | 73 | 41 |
math | 5. Determine all solutions $(p, n)$ of the equation
$$
n^{3}=p^{2}-p-1
$$
where $p$ is a prime number and $n$ is an integer. | (p,n)=(2,1)(p,n)=(37,11) | 46 | 16 |
math | 7.1. (16 points) Vasya's parents allowed him to buy himself two toys. In the store, there are 7 different remote-controlled cars and 5 different construction sets. In how many ways can he choose his gift? | 66 | 51 | 2 |
math | G2.2 If there are $K$ integers that satisfy the equation $\left(x^{2}-3 x+2\right)^{2}-3\left(x^{2}-3 x\right)-4=0$, find the value of $K$ | 2 | 54 | 1 |
math | Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is $6$. | \frac{5}{108} | 26 | 9 |
math | Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: [list][*] $f(2)=2$, [*] $f(mn)=f(m)f(n)$, [*] $f(n+1)>f(n)$. [/list] | f(n) = n | 75 | 6 |
math | 7. In the tetrahedron $A-B C D$, it is known that the lateral edges $A B$, $A C$, and $A D$ are pairwise perpendicular, and the areas of $\triangle A B C$, $\triangle A C D$, and $\triangle A D B$ are $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$, and $\frac{\sqrt{6}}{2}$, respectively. Then the volume of the circumscribe... | \sqrt{6} \pi | 124 | 7 |
math | 10. (ROM 4) Consider two segments of length $a, b(a>b)$ and a segment of length $c=\sqrt{a b}$.
(a) For what values of $a / b$ can these segments be sides of a triangle?
(b) For what values of $a / b$ is this triangle right-angled, obtuse-angled, or acute-angled? | 1<k<\frac{3+\sqrt{5}}{2},k=\frac{1+\sqrt{5}}{2},1<k<\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}<k<\frac{3+\sqrt{5}}{2} | 82 | 69 |
math | For every positive integer $1 \leqq k \leqq 100$, let $a_{k}$ denote the sum $\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}$. Calculate the value of
$$
a_{1}+a_{1}^{2}+a_{2}^{2}+\ldots+a_{100}^{2}
$$
the sum. | 200 | 100 | 3 |
math | 5. Given that $D, F$ are points on the sides $A B, A C$ of $\triangle A B C$ respectively, and $A D: D B=C F: F A=2: 3$, connect $D F$ to intersect the extension of side $B C$ at point $E$, find $E F: F D$. | EF:FD=2:1 | 76 | 7 |
math | 1. Let real numbers $a, b, c, d, e \geqslant -1$, and $a+b+c+d+e=5$. Find
$$
S=(a+b)(b+c)(c+d)(d+e)(e+a)
$$
the minimum and maximum values.
(Xiong Bin, problem contributor) | 288 | 72 | 3 |
math | 778*. Find the smallest natural number that can be represented in exactly two ways as $3 x+4 y$, where $x$ and $y$ are natural numbers. | 19 | 37 | 2 |
math | ## Subject III. (20 points)
The natural numbers $1, 2, 3, \ldots, 2013$ are written on cards, with the written side facing down. We say that a card is "lucky" if the number written on it is divisible by 20 or 13. What is the smallest number of cards that must be turned over, without looking, to be sure that at least o... | 1767 | 121 | 4 |
math | # Task 2.
Find the set of values of the parameter $a$ for which the discriminant of the equation $a x^{2}+2 x+1=0$ is 9 times the square of the difference of its two distinct roots. | \in{-3} | 53 | 5 |
math | 1. Find the smallest positive integer $k$, such that there exist positive integers $m, n$, satisfying $k=19^{n}-5^{m}$. | 14 | 35 | 2 |
math | Problem 1. The arithmetic mean of four numbers is 20. After adding one more number, the arithmetic mean of the five numbers will be 18. What is the added number? | 10 | 40 | 2 |
math | 7. Let $f(m)$ be the product of the digits of the positive integer $m$. Find the positive integer solutions to the equation $f(m)=m^{2}-10 m-36$. | 13 | 43 | 2 |
math | 1. [5 points] The altitudes $C F$ and $A E$ of an acute triangle $A B C$ intersect at point $H$. Points $M$ and $N$ are the midpoints of segments $A H$ and $C H$ respectively. It is known that $F M=1, E N=4$, and $F M \| E N$. Find $\angle A B C$, the area of triangle $A B C$, and the radius of the circumscribed circle... | \angleABC=60,S_{\triangleABC}=18\sqrt{3},R=2\sqrt{7} | 147 | 27 |
math | 11.6. Solve the system of equations
\[
\left\{\begin{aligned}
x+2 y+3 z & =2 \\
\frac{1}{x}+\frac{1}{2 y}+\frac{1}{3 z} & =\frac{5}{6} \\
x y z & =-1
\end{aligned}\right.
\] | (1,-1,1),(1,\frac{3}{2},-\frac{2}{3}),(-2,\frac{1}{2},1),(-2,\frac{3}{2},\frac{1}{3}),(3,-1,\frac{1}{3}),(3,\frac{1}{2},-\frac{2} | 82 | 74 |
math | Example 3. In an equilateral $\triangle ABC$, take a point $D$ inside such that $DA = DB$; also take a point $E$ outside $\triangle ABC$ such that $\angle DBE = \angle DBC$, and $BE = BA$. Find $\angle BED$. (1992, Sichuan Province Junior High School Mathematics League) | 30^{\circ} | 78 | 6 |
math | 4. (USS) Find four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ such that the sum of any of the numbers and the product of the other three is equal to 2. | (1,1,1,1),(-1,-1,-1,3) | 49 | 18 |
math | 19. Given that $x, y$ are positive integers, and satisfy
$$
x y-(x+y)=2 p+q \text{, }
$$
where $p, q$ are the greatest common divisor and the least common multiple of $x$ and $y$, respectively. Find all such pairs $(x, y)(x \geqslant y)$. | (x, y) = (5, 5) | 80 | 11 |
math | 3. Given the set $M=\{(a, b) \mid a \leqslant-1, b \leqslant m\}$. If for any $(a, b) \in M$, it always holds that $a \cdot 2^{b}-b-3 a \geqslant 0$, then the maximum value of the real number $m$ is $\qquad$. | 1 | 87 | 1 |
math | 4. In $\triangle A B C$, if $\sin A=2 \sin C$, and the three sides $a, b, c$ form a geometric sequence, then the value of $\cos A$ is $\qquad$ . | -\frac{\sqrt{2}}{4} | 49 | 10 |
math | 4. (USA) Let $a, b, c$ be given positive constants.
Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a+b+c, \\
4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c
\end{array}\right.
$$
for all positive real numbers $x, y, z$. | (x, y, z)=\left(\frac{1}{2}(b+c), \frac{1}{2}(c+a), \frac{1}{2}(a+b)\right) | 94 | 40 |
math | 1. (8 points) The calculation result of the expression $101 \times 2012 \times 121 \div 1111 \div 503$ is | 44 | 44 | 2 |
math | Example 5 In the border desert area, patrol vehicles travel 200 kilometers per day, and each patrol vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complete their mission, and then return along the original route to the base. To allow three... | 1800 | 152 | 4 |
math | How many quadratic residues are there modulo $p$? | \frac{p-1}{2} | 11 | 9 |
math | 16. Several workers load and unload a batch of goods, with each worker having the same loading and unloading speed. If these workers work simultaneously, it will take $10 \mathrm{~h}$ to complete the loading and unloading. Now, the loading and unloading method is changed, starting with one person working, and then addi... | 16 | 173 | 2 |
math | 6. (8 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=75 \\
y^{2}+y z+z^{2}=64 \\
z^{2}+x z+x^{2}=139
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 80 | 102 | 2 |
math | 4A. Calculate the sum
$$
2\binom{2007}{2}+4\binom{2007}{4}+\ldots+2006\left({ }_{2006}^{2007}\right)
$$ | 2007\cdot2^{2005} | 61 | 13 |
math | 2 [Rectangles and Squares. Properties and Characteristics]
Auto: Dorricheno $C$.
Which are more numerous: rectangles with integer sides and a perimeter of 1996, or rectangles with integer sides and a perimeter of 1998?
(Rectangles $a \times b$ and $b \times a$ are considered the same.) | 499 | 75 | 3 |
math | ## 16. ESERCIZIO DIMOSTRATIVO
Sia $k \geq 1$ un numero naturale. Determinare in funzione di $k$ il numero di interi positivi $n$ con le seguenti proprietà:
(a) in base dieci si scrivono con $k$ cifre, tutte dispari;
(b) sono divisibili per 5 , e il quoziente $\frac{n}{5}$, scritto in base dieci, ha ancora $k$ cifre,... | 3^{k-1} | 118 | 6 |
math | 2. Find the smallest possible value of the function
$$
f(x)=|x+1|+|x+2|+\ldots+|x+100|
$$
$(25$ points. $)$ | 2500 | 48 | 4 |
math | Find all $n \in \mathbb{N}^{*}$ such that $n$ divides $2^{n}-1$. | 1 | 28 | 1 |
math | 1. In the increasing sequence $1,3,4,9,10,12,13, \cdots$, it includes all positive integers that can be written as the sum of distinct powers of 3. What is the value of the 100th term in this sequence? | 981 | 63 | 3 |
math | 2. $\sin ^{2} 100^{\circ}-\sin 50^{\circ} \sin 70^{\circ}=$ | \frac{1}{4} | 35 | 7 |
math | 11. (3 points) The Young Pioneers plan to make some lucky stars to give to the children in the kindergarten. If each person makes 10, they will still be 6 short of completing the plan; if 4 of them each make 8, and the rest each make 12, the plan will be exactly completed. How many lucky stars are planned to be made in... | 116 | 86 | 3 |
math | Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.... | 0 | 134 | 1 |
math | 31. [20] Compute
$$
\sum_{k=1}^{1007}\left(\cos \left(\frac{\pi k}{1007}\right)\right)^{2014} .
$$ | \frac{2014(1+\binom{2013}{1007})}{2^{2014}} | 52 | 31 |
math | Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first prime, the remainder is $1$. When she divides $N$ by the second prime, the remainder is $2$. When she divides $N$ by the third prime, the remainder is $3$. When she divides $N$ by the fourth pr... | 53 | 93 | 2 |
math | 2. (3 points) Solve the equation:
$$
\left[\frac{5+6 x}{8}\right]=\frac{15 x-7}{5}
$$ | \frac{7}{15};\frac{4}{5} | 38 | 15 |
math | B3. If you were to calculate the result of
$$
\underbrace{999 \ldots 99}_{2014 \text { nines }} \times \underbrace{444 \ldots 44}_{2014 \text { fours }}
$$
and then add up all the digits of the result, what would the outcome be? | 18126 | 82 | 5 |
math | Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors.
[i]Proposed by Ankan Bhattacharya [/i] | n = 1, 2, 4, 6, 8, 16, 32 | 41 | 25 |
math | 1. Given the hyperbola $C: \left(1-a^{2}\right) x^{2}+a^{2} y^{2}=a^{2}(a>1)$, let the vertex of the upper branch of the hyperbola be $A$, and the upper branch intersects the line $y=-x$ at point $P$. A parabola with focus at $A$, vertex at $M(0, m)$, and opening downwards passes through point $P$, and the slope of $P ... | \frac{12}{7}\leqslant\leqslant4 | 153 | 18 |
math | (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$. | 2 | 81 | 1 |
math | 10. (10 points) In a grade, Class One and Class Two participated in a tree planting activity. The total number of trees planted by both classes is the same, and it is between 205 and 300. In both classes, one person did not plant trees but provided water for everyone. In Class One, the rest of the students each planted... | 62 | 109 | 2 |
math | 2.2.7* Find the acute angle $x$ that satisfies $2 \sin ^{2} x+\sin x-\sin 2 x=3 \cos x$ | \frac{\pi}{3} | 38 | 7 |
math | Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2... | 24 | 181 | 2 |
math | 1.28 Given $x, y \in N$, find the largest $y$ value such that there exists a unique $x$ value satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15} \text {. }
$$
(Wuhan, Hubei Province, China Mathematical Summer Camp, 1987) | 112 | 85 | 3 |
math | Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three. | (1, 2, 3, 6), (1, 2, 6, 9), (1, 3, 8, 12), (1, 4, 5, 10), (1, 6, 14, 21), (2, 3, 10, 15) | 29 | 79 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{3 ; 1 ; 3\}$
$p=\{2 ; 1 ; 0\}$
$q=\{1 ; 0 ; 1\}$
$r=\{4 ; 2 ; 1\}$ | -3p+q+2r | 75 | 8 |
math | Suppose that $a, b, c, d$ and $e$ are consecutive positive integers with $a<b<c<d<e$. If $a^{2}+b^{2}+c^{2}=d^{2}+e^{2}$, what is the value of $a$ ? | 10 | 64 | 2 |
math | 1. Find all five-digit numbers where the second digit is five times the first, and the product of all five digits is 1000. | 15855,15585,15558 | 31 | 17 |
math | 3. Let $n$ be a natural number. For any real numbers $x, y, z$, it always holds that $\left(x^{2}+y^{2}+z^{2}\right)$ $\leqslant n\left(x^{4}+y^{4}+z^{4}\right)$. Then the minimum value of $n$ is $\qquad$. | 3 | 82 | 1 |
math | Example 2. There are $n$ points on a plane, where any three points can be covered by a circle of radius 1, but there are always three points that cannot be covered by any circle of radius less than 1. Find the minimum radius of a circle that can cover all $n$ points. | 1 | 65 | 1 |
math | 5 married couples gather at a party. As they come in and greet each other, various people exchange handshakes - but, of course, people never shake hands with themselves or with their own respective spouses. At the end of the party, one woman goes around asking people how many hands they shook, and she gets nine differe... | 4 | 76 | 1 |
math | Example 10 The sports meet lasted for $n(n>1)$ consecutive days, and a total of $m$ medals were awarded. On the first day, 1 medal and $\frac{1}{7}$ of the remaining $(m-1)$ medals were awarded. On the second day, 2 medals and $\frac{1}{7}$ of the remaining medals were awarded, and this pattern continued for subsequent... | n=6, m=36 | 130 | 8 |
math | Find the maximum number of natural numbers $x_{1}, x_{2}, \ldots, x_{m}$ satisfying the conditions:
a) No $x_{i}-x_{j}, 1 \leq i<j \leq m$ is divisible by 11 ; and
b) The sum $x_{2} x_{3} \ldots x_{m}+x_{1} x_{3} \ldots x_{m}+\cdots+x_{1} x_{2} \ldots x_{m-1}$ is divisible by 11. | 10 | 124 | 2 |
math | Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for t... | a = 1 | 182 | 5 |
math | In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominato... | 49 | 81 | 4 |
math | 7. Let $x_{1}, x_{2}$ be the two real roots of the quadratic equation $x^{2} + a x + a = 2$. Then the maximum value of $\left(x_{1}-2 x_{2}\right)\left(x_{2}-2 x_{1}\right)$ is $\qquad$ . | -\frac{63}{8} | 71 | 8 |
math | $p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that
$$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$
[i]Propo... | (x_1, x_2, \cdots, x_k) \in \{0, 1\}^k | 174 | 29 |
math | ## 43. Rue Saint-Nicaise
On December 24, 1800, First Consul Bonaparte was heading to the Opera along Rue Saint-Nicaise. A bomb exploded on his route with a delay of a few seconds. Many were killed and wounded. Bonaparte accused the Republicans of the plot; 98 of them were exiled to the Seychelles and Guiana. Several p... | 9 | 198 | 1 |
math | A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks. | \text{all multiples of } 12 | 45 | 11 |
math | Example 5 Find the largest positive integer $x$, such that for every positive integer $y$, we have $x \mid\left(7^{y}+12 y-1\right)$.
| 18 | 43 | 2 |
math | Lines intersect at point $P$, draw perpendiculars to the tangents through points $A$ and $B$, and their intersection is point $Q$. Find the expression for the coordinates of point $Q$ in terms of the coordinates of point $P$, and answer: (1) When the y-coordinate of point $P$ remains constant and the x-coordinate chang... | y_{Q}=2y_{P}x_{Q}-2y_{P}(2y_{P}^{2}+1) | 115 | 29 |
math | 7. If $a_{1}=a_{2}=1, a_{n}=4 a_{n-1}-a_{n-2}$, then $a_{n} a_{n-2}-a_{n-1}^{2}=$ | 2 | 54 | 1 |
math | 10,11
Construct a rational parametrization of the circle $x^{2}+y^{2}=1$ by drawing lines through the point $(1,0)$. | (\frac{^{2}-1}{^{2}+1},\frac{-2}{^{2}+1}) | 39 | 25 |
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 | 4. Given the set $S=\{1,2, \cdots, 2005\}, A \subseteq S, A$ such that the sum of any two numbers in $A$ is not divisible by 117, find the maximum value of $|A|$.
untranslated text remains the same as requested. However, if you need any further assistance or a different translation, feel free to let me know! | 1003 | 90 | 4 |
math | Find all integers $ (x,y,z)$, satisfying equality:
$ x^2(y \minus{} z) \plus{} y^2(z \minus{} x) \plus{} z^2(x \minus{} y) \equal{} 2$ | (x, y, z) = (k + 1, k, k - 1) | 55 | 22 |
math | Question 78: Given $x, y, z > 0$, find the minimum value of $2 \sqrt{(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)}-\sqrt{\left(1+\frac{x}{y}\right)\left(1+\frac{y}{z}\right)}$. | 2\sqrt{2}+1 | 81 | 8 |
math | 10. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $M$ and $N$ lie on segments $A B$ and $B B_{1}$ (excluding the endpoints of the segments), and $A M=B_{1} N$. Then the range of the angle formed by $A_{1} M$ and $C_{1} N$ is $\qquad$ | (\frac{\pi}{3},\frac{\pi}{2}) | 96 | 14 |
math | The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties:
$AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$.
Find the length of the segment $ED$. | 74 | 79 | 2 |
math | (Corée 2012)
Find all triplets of strictly positive integers $(m, n, p)$ with $p$ prime, such that $2^{m} p^{2} +$ $1=n^{5}$. | (1,3,11) | 50 | 8 |
math | 10. Among $1,2,3,4, \cdots, 1000$, the numbers that can be written in the form $a^{2}-b^{2}+1(a, b \in \mathbf{N})$ and are not divisible by 3 are $\qquad$ in number. | 501 | 70 | 3 |
math | Two natural numbers form a pair when they have the same number of digits and their sum only contains the digit 9. For example, 225 and 774 form a pair, because both have three digits and \(225 + 774 = 999\).
a) What is the number that forms a pair with 2010?
b) How many pairs are formed by two-digit numbers?
Special... | 7989 | 171 | 4 |
math | 2. For any two points on the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$, if the perpendicular bisector of the line segment joining these two points intersects the $x$-axis at point $P\left(x_{0}, 0\right)$, then the range of $x_{0}$ is $\qquad$ | (-3,3) | 81 | 5 |
math | Example 10. Find the fractional-linear function that maps the points $z_{1}=1, z_{2}=i, z_{3}=-1$ to the points $w_{1}=-1, w_{2}=0, w_{3}=1$. | i\frac{i-z}{i+z} | 57 | 9 |
math | 3.
3 red birds for 4 days eat 36 grams of seed, 5 blue birds for 3 days eat 60 gram of seed.
For how many days could be feed 2 red birds and 4 blue birds with 88 gr seed? | 4 \text{ days} | 56 | 6 |
math | 2. Given a positive geometric sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{6}+a_{5}+a_{4}-a_{3}-a_{2}-a_{1}=49 \text {. }
$$
Then the minimum value of $a_{9}+a_{8}+a_{7}$ is $\qquad$ | 196 | 81 | 3 |
math | Example 4 Given that the three sides of $\triangle A B C$ are $a$, $b$, and $c$, and they satisfy
$$
a b c=2(a-1)(b-1)(c-1) .
$$
Does there exist a $\triangle A B C$ with all sides being integers? If it exists, find the lengths of the three sides; if not, explain the reason. | 4,5,6 \text{ or } 3,7,8 | 87 | 16 |
math | 2. Find all four-digit numbers $m$ for which it is known that:
i) $m$ is a square of a natural number;
ii) the first and second digits of $m$ are the same; and
iii) the third and fourth digits of $m$ are the same. | 7744 | 62 | 4 |
math | Let $P$ be a polynomial such that the remainder of the Euclidean division of $P$ by $X-1$ is 2 and the remainder of the division of $P$ by $X-2$ is 1. What is the remainder of the division of $P$ by $(X-1)(X-2)$? | R(x)=3-x | 71 | 5 |
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