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 | 4. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2, a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$, and $a_{n+1} a_{n+2} \neq 1$, then $S_{1999}=\sum_{n=1}^{1999} a_{n}=$ $\qquad$. | 3997 | 117 | 4 |
math | 5. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{2}=2, a_{n+1}-3 a_{n}+2 a_{n-1}=1\left(n \geqslant 2, n \in \mathbf{N}^{*}\right)$, then the general term formula of $\left\{a_{n}\right\}$ is $\qquad$ . | 2^{n}-n | 99 | 5 |
math | 4. In a football match, the winning team receives 3 points, and the losing team receives 0 points. If the match ends in a draw, each team receives 1 point. With these scoring rules, a tournament is organized in which four teams participate, each team playing two matches with each of the other three. At the end, a ranki... | 2 | 201 | 1 |
math | If the sides of an arbitrary triangle $ABC$ are unambiguously extended by their own lengths, then the triangle formed by the endpoints is seven times the original. | 7ABC\Delta | 34 | 4 |
math | Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle be... | 45^\circ | 114 | 4 |
math | 2. "Miki and I," said Philip, "can finish the assigned work in 20 days, but if I work with Ivan, we would finish the same work 5 days earlier." "I have a better combination," said Ivan, "If I worked with Miki, we would finish the work a fifth of the time earlier than when working with Philip."
How many days would each... | Miki:30 | 98 | 5 |
math | 9.8. A straight rod 2 meters long was sawn into $N$ sticks, the length of each of which is expressed as an integer number of centimeters. For what smallest $N$ can it be guaranteed that, using all the resulting sticks, one can, without breaking them, form the contour of some rectangle?
(A. Magazinov) | 102 | 75 | 3 |
math | We denote by $\mathbb{R}_{>0}$ the set of strictly positive real numbers. Find all functions $f: \mathbb{R}_{>0} \mapsto \mathbb{R}_{>0}$ such that
$$
f(x+f(x y))+y=f(x) f(y)+1
$$
for all strictly positive real numbers $x$ and $y$. | f: t \mapsto t + 1 | 82 | 10 |
math | 1. Find all pairs of real numbers ( $\mathrm{x}, \mathrm{y}$ ), for which the equality $\sqrt{x^{2}+y^{2}-1}=1-x-y$ holds. | (1,),(,1) | 42 | 7 |
math | For which $k$ do there exist $k$ pairwise distinct primes $p_{1}, p_{2}, \ldots, p_{k}$ such that
$$
p_{1}^{2}+p_{2}^{2}+\cdots+p_{k}^{2}=2010 ?
$$ | 7 | 67 | 1 |
math | 3. Let $n$ be a natural number, for any real numbers $x, y, z$ there is always $\left(x^{2}+y^{2}+z^{2}\right) \leqslant$ $n\left(x^{4}+y^{4}+z^{4}\right)$, then the minimum value of $n$ is $\qquad$ | 3 | 83 | 1 |
math | 2. Let $P$ be a regular 2006-gon. If an end of a diagonal of $P$ divides the boundary of $P$ into two parts, each containing an odd number of sides of $P$, then the diagonal is called a "good edge". It is stipulated that each side of $P$ is a good edge.
Given 2003 non-intersecting diagonals inside $P$ that partition $... | 1003 | 118 | 4 |
math | 5. Calculate the maximum product of all natural numbers whose sum is 2019. | 3^{673} | 19 | 6 |
math | Example 6. Integrate the equation $y y^{\prime \prime}-y^{\prime 2}=0$. | C_{2}e^{C_{1}x} | 26 | 12 |
math | 20. Several 1s and 2s are arranged in a row:
$$
1,2,1,2,2,1,2,2,2,1,2, \cdots
$$
The rule is: The 1st number is 1, the 2nd number is 2, the 3rd number is 1. Generally, first write a row of 1s, then insert $k$ 2s between the $k$-th 1 and the $(k+1)$-th 1 ($k=1,2$, $\cdots$). Try to answer:
(1) Is the 2005th number 1 or... | 7789435 | 189 | 7 |
math | N4. Alice is given a rational number $r>1$ and a line with two points $B \not R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead ... | All\r=(b+1)/b\with\b=1,\ldots,1010 | 179 | 21 |
math | 3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=0, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$
If $(\tan A+\tan B) \tan C=m... | \frac{1}{2} | 127 | 7 |
math | Example 1 Find all non-square numbers $n \in \mathbf{N}_{+}$, such that $[\sqrt{n}]^{3} \mid n^{2}$. (1)
(3rd Northern Mathematical Olympiad Invitational Competition) | n=2,3,8,24 | 52 | 10 |
math | 92. The Meeting Problem. Two people agreed to meet at a certain place between 12 o'clock and 1 o'clock. According to the agreement, the one who arrives first waits for the other for 15 minutes, after which he leaves. What is the probability that these people will meet, if each of them chooses the moment of their arriva... | \frac{7}{16} | 94 | 8 |
math | 7. Let point $O$ be the origin, and the coordinates of points $A$ and $B$ be $(a, 0)$ and $(0, a)$, respectively, where $a$ is a positive constant. Point $P$ lies on the line segment $AB$, and $\overrightarrow{A P}=t \cdot \overrightarrow{A B}(0 \leqslant t \leqslant 1)$. Then the maximum value of $\overrightarrow{O A}... | ^2 | 116 | 2 |
math | 9.1. Append a digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (List all possible solutions.) | 3202220232 | 48 | 10 |
math | Given the function
$$
f(x)=\frac{(x+a)^{2}}{(a-b)(a-c)}+\frac{(x+b)^{2}}{(b-a)(b-c)}+\frac{(x+c)^{2}}{(c-a)(c-b)}
$$
where $a, b$, and $c$ are distinct real numbers. Determine the range of the function. | 1 | 81 | 1 |
math | What is the sum of the roots of the equation $10^{x}+10^{1-x}=10$? | 1 | 27 | 1 |
math | 4. Let $a, b, c$ be any real numbers, and satisfy: $a>b>c,(a-b)(b-c)(c-a)=-16$. Then the minimum value of $\frac{1}{a-b}+\frac{1}{b-c}-\frac{1}{c-a}$ is $\qquad$ | \frac{5}{4} | 70 | 7 |
math | 33 (1275). Find all prime numbers $p$ and $q$ for which $p^{2} - 2 q^{2}=1$. | p=3,q=2 | 35 | 6 |
math | The sides of a triangle form a geometric progression. What values can the ratio of the progression take?
Translation provided as requested, maintaining the original text's line breaks and format. | \frac{\sqrt{5}-1}{2}<q<\frac{\sqrt{5}+1}{2} | 36 | 25 |
math | Let $x=2 \cos \frac{2 \pi}{5}$ and $y=2 \cos \frac{4 \pi}{5}$. It is given that $x+y+1=0$. Calculate $x$ and $y$. | \frac{-1+\sqrt{5}}{2},\frac{-1-\sqrt{5}}{2} | 53 | 24 |
math | 24. (POL 5) For points $A_{1}, \ldots, A_{5}$ on the sphere of radius 1, what is the maximum value that $\min _{1 \leq i, j \leq 5} A_{i} A_{j}$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $\left\{A_{1}, \ldots, A_{5}\right\}$ for which this maximum ... | \sqrt{2} | 110 | 5 |
math | 14. Given the sequence $\left\{a_{n}\right\}$, the odd terms form an arithmetic sequence with the first term 1, and the even terms form a geometric sequence with the first term 2. The sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$ is $S_{n}$, and it satisfies $S_{5}=2 a_{4}+a_{5}, a_{9}=a_{3}+a_{4}$.... | =1,=2 | 238 | 5 |
math | 1.8. Find the cosine of the angle between the vectors with coordinates $\left(a_{1}, b_{1}, c_{1}\right)$ and $\left(a_{2}, b_{2}, c_{2}\right)$. | \frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}} | 49 | 81 |
math | 4A. In a basket, there are red, white, and yellow flowers. The number of yellow flowers is not less than the number of white flowers and is not greater than one third of the number of red flowers. The total number of white and yellow flowers is not less than 55. Find the smallest number of red flowers. | 84 | 69 | 2 |
math | Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$ | (3, 223, 9) | 44 | 12 |
math | ## 251. Math Puzzle $4 / 86$
The "Golden Rider" in Dresden almost magically draws the gaze of passersby, living up to its name. During the last restoration, it was overlaid with 140 g of gold leaf, which has a specific density of $19.3 \mathrm{~g} / \mathrm{cm}^{3}$.
How thick is the average gold coating if the monum... | 0.18\mu\mathrm{} | 115 | 9 |
math | Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space.
Let $V$ be the cone obtained by rotating the triangle around the $x$-axis.
Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis. | \frac{8\pi}{3} | 93 | 9 |
math | 12. Given $f(x)=\ln (a x+b)+x^{2}(a \neq 0)$.
(1) If the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$ is $y=x$, find the values of $a$ and $b$;
(2) If $f(x) \leqslant x^{2}+x$ always holds, find the maximum value of $a b$. | \frac{\mathrm{e}}{2} | 102 | 10 |
math | Assume $A,B,C$ are three collinear points that $B \in [AC]$. Suppose $AA'$ and $BB'$
are to parrallel lines that $A'$, $B'$ and $C$ are not collinear. Suppose $O_1$ is circumcenter of circle passing through $A$, $A'$ and $C$. Also $O_2$ is circumcenter of circle passing through $B$, $B'$ and $C$. If area of $A'CB'$ is... | 30^\circ \text{ or } 150^\circ | 131 | 15 |
math | The 64th question: Let $S=\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ be a non-empty family of subsets $U$ with the property: if $A \in U, A \subseteq B$, then $B \in U$; and let $S$ be a non-empty family of subsets $V$ with the property: if $A \in V, A \supseteq B$, then $B \in V$. Find the maximum possible value of $... | \frac{1}{2^{n}} | 132 | 9 |
math | 【Question 20】
As shown, $A B C D$ is a square, and points $E$ and $F$ are the midpoints of sides $A B$ and $B C$, respectively. $D E$ and $D F$ intersect the diagonal $A C$ at points $M$ and $N$. If the area of square $A B C D$ is $48 \mathrm{~cm}^{2}$, find the area of pentagon $E B F N M$. | 16 | 109 | 2 |
math | 6. In a Cartesian coordinate system, there are 25 non-coincident horizontal and vertical lines, each dyed one of two colors: black or red. Then, the intersection points of black horizontal lines and black vertical lines are dyed black; the intersection points of red horizontal lines and red vertical lines are dyed red;... | 1:6 \text{ or } 6:1 | 128 | 12 |
math | 60. Let the number of positive divisors of 3600 be $m$, and the number of positive divisors of 36 be $n$, then $\frac{m}{n}=$ | 5 | 44 | 1 |
math | 36.
$$
\left\{\begin{array}{l}
x y=a^{2} \\
\lg ^{2} x+\lg ^{2} y=\frac{5}{2} \lg ^{2} a^{2}
\end{array}\right.
$$ | x_{1}=^{3},y_{1}=\frac{1}{},x_{2}=\frac{1}{},y_{2}=^{3} | 62 | 34 |
math | 1. A sequence of three numbers $a, b, c$ form an arithmetic sequence if the difference between successive terms in the sequence is the same. That is, when $b-a=c-b$.
(a) The sequence $2, b, 8$ forms an arithmetic sequence. Determine $b$.
(b) Given a sequence $a, b, c$, let $d_{1}$ be the non-negative number to increase... | 2d_{1}=d_{2} | 327 | 9 |
math | Example 2. Form a differential equation for which the functions $y_{1}(x)=e^{x^{2}}, y_{2}(x)=e^{-x^{2}}$ form a fundamental system of solutions. | xy^{\\}-y^{\}-4x^{3}0 | 45 | 14 |
math | 6. From the 11 positive integers $1,2,3, \cdots, 11$, if 3 different positive integers $a, b, c$ are randomly selected, then the probability that their product $a b c$ is divisible by 4 is $\qquad$. | \frac{20}{33} | 62 | 9 |
math | ## SUBIECTUL II
Sa se calculeze:
a) $\lim _{n \rightarrow \infty}\left(\sqrt[3]{n^{3}+9 n^{2}+1}+\sqrt[3]{n^{3}-9 n^{2}+1}-2 n\right)$
b) $\lim _{n \rightarrow \infty} n \cdot\left(\sqrt[3]{n^{3}+9 n^{2}+1}+\sqrt[3]{n^{3}-9 n^{2}+1}-2 n\right)$
| 0 | 127 | 1 |
math | $4 \cdot 210$ On the same route, there are four people: the first person is in a car, the second person is on a motorcycle, the third person is on a moped, and the fourth person is on a bicycle. The speeds of the vehicles are constant. The person in the car catches up with the person on the moped at 12 o'clock, meets t... | 15:20 | 158 | 5 |
math | Let $S_{n}$ denote the sum of the digits of the number $n$ in the decimal system. Determine all numbers $n$ for which $n=S^{2}(n)-S(n)+1$ | 1,13,43,91,157 | 44 | 14 |
math | 12.4. Let $I_{n}=\int_{1}^{n} \frac{[x]}{x^{2}+1} d x, n \in \mathbb{N}, n \geq 2$. Calculate: $\lim _{n \rightarrow \infty} \frac{I_{n}}{\ln n}$. | 1 | 77 | 1 |
math | 9.5. The older brother took identical uncolored cubes from Misha and used them to build a large cube. After that, he completely painted some (not all) faces of the large cube red. When the paint dried, Misha disassembled the large cube and found that exactly 343 small cubes had no red faces. How many faces of the large... | 3 | 90 | 1 |
math | One. (20 points) Given the parabola $y=a x^{2}+b x+c$ passes through the point $(b,-3)$, and $|a| c+b|c|=0$, and the inequality $a x^{2}+b x+c+3>0$ has no solution. Find all possible values of the triplet $(a, b, c)$.
---
Please note that the formatting and line breaks have been preserved as requested. | \left(-\frac{1}{2}, \frac{1}{2},-\frac{25}{8}\right) | 98 | 27 |
math | ## Task A-2.1.
Which number has more divisors in the set of natural numbers, $2013^{2}$ or 20480? | 2013^{2} | 37 | 7 |
math | 4B. Find all values of the real parameter $a \neq 0,1$ for which the equations $x^{2}-(2 a+1) x+a=0$ and $x^{2}+(a-4) x+a-1=0$ have real roots $x_{1}, x_{2}$ and $x_{3}, x_{4}$, respectively, such that the equality
$$
\frac{x_{1}}{x_{3}}+\frac{x_{4}}{x_{2}}=\frac{x_{1} x_{4}\left(x_{1}+x_{2}+x_{3}+x_{4}\right)}{a}
$$
... | -1-\sqrt{5},-1+\sqrt{5} | 152 | 14 |
math | Find the locus equation of points that are equidistant from the fixed line $1: x=-\frac{p}{2}$ and the fixed point $F\left(\frac{p}{2}, 0\right)$. | y^{2}=2 p x | 48 | 7 |
math | 6. For $n$ an integer, evaluate
$$
\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^{2}-0^{2}}}+\frac{1}{\sqrt{n^{2}-1^{2}}}+\cdots+\frac{1}{\sqrt{n^{2}-(n-1)^{2}}}\right)
$$ | \frac{\pi}{2} | 81 | 7 |
math | In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the $n$ -th switch the $n$ -th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from th... | 1024 | 218 | 4 |
math | 2. Find the maximum value of the quantity $x^{2}+y^{2}+z^{2}$, given that
$$
x^{2}+y^{2}+z^{2}=3 x+8 y+z
$$ | 74 | 52 | 2 |
math | Three. (20 points) Let $a, b$ be integers. How many solutions does the system of equations
$$
\left\{\begin{array}{l}
{[x]+2 y=a,} \\
{[y]+2 x=b}
\end{array}\right.
$$
(here $[x]$ denotes the greatest integer less than or equal to $x$) have? | 2 \text{ or } 1 | 85 | 8 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(3-4 n)^{2}}{(n-3)^{3}-(n+3)^{3}}$ | -\frac{8}{9} | 51 | 7 |
math | Example 7 Given that $a$ is a real number, and makes the quadratic equation in $x$, $x^{2}+a^{2} x+a=0$, have real roots. Find the maximum value that the root $x$ of the equation can take.
(1994, Beijing Junior High School Mathematics Competition). | \frac{\sqrt[3]{2}}{2} | 70 | 12 |
math | (15) (50 points) Let integers $a, b, c$ satisfy $1 \leqslant a \leqslant b \leqslant c$, and $a \mid b+c+1$, $b \mid c+a+1$, $c \mid a+b+1$. Find all triples $(a, b, c)$. | (1,1,1),(1,2,2),(3,4,4),(1,1,3),(2,2,5),(1,2,4),(2,3,6),(4,5,10),(1,4,6),(2,6,9),(3,8,12),(6,14,21) | 79 | 77 |
math | [ Geometric progression $]$ $[$ Classical combinatorics (other). ]
A frog jumps from vertex to vertex of a hexagon $A B C D E F$, each time moving to one of the adjacent vertices.
a) In how many ways can it get from $A$ to $C$ in $n$ jumps?
b) The same question, but with the condition that it cannot jump to $D$?
##... | )\frac{1}{3}(4^n-1)forevenn;cannotreachforoddn.\quadb)3^{n/2-1}forevenn;cannotreachforoddn.\quad)(3/4)^{k-1}forn=2k-1forn | 150 | 63 |
math | A quadrilateral is drawn on a sheet of transparent paper. What is the minimum number of times the sheet needs to be folded to ensure that it is a square?
# | 2 | 34 | 1 |
math | 4. Determine all triples $(x, y, z)$ of natural numbers such that
$$
x y z + x y + y z + z x + x + y + z = 243
$$ | (1,1,60),(1,60,1),(60,1,1) | 45 | 22 |
math | 17. Find all positive integers $(x, y, z, n)$, such that
$$
x^{3}+y^{3}+z^{3}=n x^{2} y^{2} z^{2}
$$ | (1,1,1,3),(1,2,3,1),(2,1,3,1),(1,3,2,1),(3,1,2,1),(3,2,1,1),(2,3,1,1) | 50 | 57 |
math | 12. find all natural numbers that are in the form
$$
\frac{(a+b+c)^{2}}{a b c}
$$
where $a, b$ and $c$ are natural numbers. | 1,2,3,4,5,6,8,9 | 46 | 15 |
math | Example 1 Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}=$ 5. If $s=x^{2}+y^{2}$, then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}=$ $\qquad$ .
(1993. National High School Mathematics Competition) | \frac{8}{5} | 86 | 7 |
math | Question 4 Let $P$ be a polynomial of degree $3n$, such that
$$
\begin{array}{l}
P(0)=P(3)=\cdots=P(3 n)=2, \\
P(1)=P(4)=\cdots=P(3 n-2)=1, \\
P(2)=P(5)=\cdots=P(3 n-1)=0 .
\end{array}
$$
$$
\text { If } P(3 n+1)=730 \text {, find } n \text {. }
$$ | 4 | 124 | 1 |
math | ## Task A-2.3.
Determine all ordered triples $(m, n, p)$, where $m$ and $n$ are natural numbers, and $p$ is a prime number, for which
$$
(2 m+3)(4 n+1)=p m n
$$ | (5,1,13),(15,1,11),(3,3,13) | 63 | 23 |
math | $C$ is a point on the extension of diameter $A B, C D$ is a tangent, angle $A D C$ is $110^{\circ}$. Find the angular measure of arc $B D$.
# | 40 | 50 | 2 |
math | Example 1 In $\triangle A B C$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. Given $a+c=2 b, A-C=\frac{\pi}{3}$, find the value of $\sin B$. | \frac{\sqrt{39}}{8} | 58 | 11 |
math | $7 \cdot 74$ There are several locks, and now six people each hold a part of the keys. It is known that any two people trying to open the locks together will have exactly one lock that they cannot open, while any three people can open all the locks. How many locks are there at least?
---
Note: The translation keeps t... | 15 | 100 | 2 |
math | 2. Determine all pairs of positive integers $(m, n)$ for which
$$
3 \cdot 2^{n}+1=m^{2} .
$$ | (,n)=(5,3)(,n)=(7,4) | 34 | 15 |
math | 5. (5 points) The decimal counting method is to carry 1 when reaching 10, such as $24_{10}=2 \times 10+4 \times 1,36510=3 \times 10^{2}+6 \times 10+5 \times 1$; computers use the binary counting method, which is to carry 1 when reaching 2, such as $710=1 \times 2^{2}+1 \times 2+1 \times 1=111_{2}$, $12_{10}=1 \times 2^... | 11,9 | 263 | 4 |
math | 8. In $\mathrm{Rt} \triangle A B C$, there is a point $M$ on the right-angle side $A B$ and a point $P$ on the hypotenuse $B C$. It is known that $M P \perp B C$, the area of $\triangle B M P$ is
equal to half the area of quadrilateral MPCA, $B P$
$=2$ cm, $P C=3$
cm. Then the area of $\mathrm{Rt} \triangle A B C$ i... | \sqrt{39} | 122 | 6 |
math | Example. Solve the Dirichlet boundary value problem for Poisson's equation in a sphere
$$
\begin{gathered}
\Delta u=14 x y, \quad 0 \leqslant r<2 \\
\left.u\right|_{r=2}=14
\end{gathered}
$$ | u(x,y,z)=(x^{2}+y^{2}+z^{2}-4)xy+14 | 70 | 25 |
math | Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, h... | (1, 50) | 198 | 8 |
math | 7. (4 points) Solve the equation:
$$
\sin x + \sin 3x + \ldots + \sin 2013x = \cos x + \cos 3x + \ldots + \cos 2013x
$$ | \frac{\pik}{1007},k\in\mathbb{Z},k | 59 | 21 |
math | ## Task 1.
Let $a=0.365$. The sequence $a_{1}, a_{2}, \ldots, a_{2020}$ is defined by the formulas
$$
a_{1}=a, \quad a_{n+1}=a^{a_{n}} \quad \text { for } \quad n=1, \ldots, 2019
$$
Arrange the numbers $a_{1}, a_{2}, \ldots, a_{2020}$ from the smallest to the largest. | a_{1}<a_{3}<\cdots<a_{2k-1}<a_{2k}<\cdots<a_{2}<1 | 120 | 31 |
math | ## Task 34/74
Choose a natural number $n$ with at least two digits, whose decimal representation contains no zero. By arbitrarily swapping the digits in it, a second natural number $n^{\prime}$ is obtained.
If one digit is erased from the difference $n-n^{\prime}$, the erased digit can be determined from the sum of t... | 9m-S | 85 | 3 |
math | 4- 204 A car transports daily necessities from Town A to Village B, traveling 20 kilometers uphill, 14 kilometers downhill, and 5 kilometers on flat road. It then transports grain from Village B back to Town A. The time difference for the round trip is 10 minutes. It is known that the speed ratio of the car when going ... | 1\frac{2}{3} | 144 | 8 |
math | 29. In 5 lottery tickets, there is 1 winning ticket. 5 people draw 1 ticket each in a predetermined order to decide who gets the winning ticket. Would the probability of drawing the winning ticket be the same for the first drawer and the later drawers (later drawers do not know the results of the earlier drawers)? | \frac{1}{5} | 68 | 7 |
math | 21st Putnam 1960 Problem B1 Find all pairs of unequal integers m, n such that m n = n m . Solution | (2,4),(4,2),(-2,-4),(-4,-2) | 31 | 19 |
math | 21. Let $x_{1}$ and $x_{2}$ be two real numbers that satisfy $x_{1} x_{2}=2013$. What is the minimum value of $\left(x_{1}+x_{2}\right)^{2}$ ? | 8052 | 58 | 4 |
math | 4. In $\triangle A B C$, it is known that the three interior angles $\angle A, \angle B, \angle C$ form an arithmetic sequence, with their opposite sides being $a, b, c$ respectively, and $c-a$ equals the height $h$ from vertex $A$ to side $AC$. Then $\sin \frac{C-A}{2}=$ | \frac{1}{2} | 81 | 7 |
math | What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$? | 1 | 51 | 1 |
math | Through the midpoint of the hypotenuse of a right triangle, a perpendicular is drawn to it. The segment of this perpendicular, enclosed within the triangle, is equal to c, and the segment enclosed between one leg and the extension of the other is equal to 3c. Find the hypotenuse.
# | 4c | 63 | 2 |
math | How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$. | 30 | 39 | 2 |
math | 9. (16 points) Let $a, b, c > 0$, and $a+b+c=3$. Find the maximum value of $a^{2} b+b^{2} c+c^{2} a+a b c$.
| 4 | 52 | 1 |
math | ## 1. Solve the system of equations
$$
\begin{aligned}
& x^{m}=y^{n} \\
& \log _{a} \frac{x}{y}=\frac{\log _{a} x}{\log _{a} y .}
\end{aligned}
$$ | (^{\frac{n^{2}}{(n-)}},^{\frac{n}{n-}}) | 65 | 21 |
math | 6. Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Determine all functions $f$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$,
$$
f(x) f(y)=f(y) f(x f(y))+\frac{1}{x y} .
$$
(Pavel Calábek) | f(x)=1+\frac{1}{x} | 101 | 11 |
math | Let $p$ be a prime number. Find all possible values of the remainder when $p^{2}-1$ is divided by 12 . | 3,8,0 | 31 | 5 |
math | Example 6 (6th "Hope Cup" Competition Question) Given that the equation $x^{2}-a x+a^{2}-4=0$ has two distinct real roots and one positive root, the range of values for $a$ is $\qquad$ . | -2\leqslant\leqslant2 | 56 | 13 |
math | VI OM - II - Task 3
What should be the angle at the vertex of an isosceles triangle so that a triangle can be constructed with sides equal to the height, base, and one of the remaining sides of this isosceles triangle? | 106 | 54 | 3 |
math | 5. Given $\odot O: x^{2}+y^{2}=1$ with a moving chord $A B=\sqrt{3}$, and a moving point $P$ on $\odot C:(x-2)^{2}+(y-3)^{2}=2$. Then the minimum value of $\sqrt{\overrightarrow{P A} \cdot \overrightarrow{P B}+\frac{3}{4}}$ is $\qquad$. | \sqrt{13}-\frac{1}{2}-\sqrt{2} | 98 | 18 |
math | Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$. | 757 | 56 | 3 |
math | 7.2 In the glass, there was a solution in which water made up $99 \%$. The glass with the solution was weighed, and the weight turned out to be 500 gr. After that, part of the water evaporated, so that in the end, the proportion of water was $98 \%$. What will be the weight of the glass with the resulting solution, if ... | 400 | 95 | 3 |
math | 15. Let $f_{1}(x)=\frac{2}{1+x}, f_{n+1}(x)=f_{1}\left(f_{n}(x)\right)$, and $a_{n}=\frac{f_{n}(0)-1}{f_{n}(0)+2}$. Then $a_{2014}=$ $\qquad$ . | -\left(\frac{1}{2}\right)^{2015} | 82 | 17 |
math | Question 32, Given a strictly monotonic function $f(x)$ defined on $\mathrm{R}$ that satisfies $f(x+y)=f(x)+f(y)$, and $f(1)=2$,
(1) Prove that $f(x)$ is an odd function;
(2) When $t>2$, the inequality $f\left(k \cdot \log _{2} t\right)+f\left(\log _{2} t-\left(\log _{2} t\right)^{2}-2\right)<0$ always holds, find the ... | (-\infty,2\sqrt{2}-1) | 132 | 13 |
math | Sure, here is the translated text:
```
I. Fill-in-the-Blanks (8 points per question, total 64 points)
1. Given the sets
$$
A=\{x \mid 5 x-a \leqslant 0\}, B=\{x \mid 6 x-b>0\}
$$
where $a, b \in \mathbf{N}_{+}$.
If $A \cap B \cap \mathbf{N}=\{2,3,4\}$, then the number of integer pairs $(a, b)$ is . $\qquad$
``` | 30 | 131 | 2 |
math | [ Auxiliary similar triangles ]
Given an equilateral triangle $A B C$. Point $K$ is the midpoint of side $A B$, point $M$ lies on side $B C$, and $B M: M C=1: 3$. A point $P$ is chosen on side $A C$ such that the perimeter of triangle $P K M$ is the smallest possible. In what ratio does point $P$ divide side $A C$? | 2:3 | 96 | 3 |
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