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 | Let $n$ be a natural composite number. For each proper divisor $d$ of $n$ we write the number $d + 1$ on the board. Determine all natural numbers $n$ for which the numbers written on the board are all the proper divisors of a natural number $m$. (The proper divisors of a natural number $a> 1$ are the positive divisors of $a$ different from $1$ and $a$.) | 4, 8 | 98 | 4 |
math | 3. Fill the numbers $1,2, \cdots, 36$ into a $6 \times 6$ grid, with each cell containing one number, such that the numbers in each row are in increasing order from left to right. The minimum value of the sum of the six numbers in the third column is $\qquad$
(2015, National Junior High School Mathematics League Competition) | 63 | 85 | 2 |
math | How many polynomials of degree exactly $5$ with real coefficients send the set $\{1, 2, 3, 4, 5, 6\}$ to a permutation of itself? | 718 | 42 | 3 |
math | Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations:
a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$;
b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.
| f(n) = 2n + 1 | 81 | 10 |
math | 2. Given $(a+b i)^{2}=3+4 i$, where $a, b \in \mathbf{R}, i$ is the imaginary unit, then the value of $a^{2}+b^{2}$ is $\qquad$ | 5 | 55 | 1 |
math | 3. Find the function $f: \mathbf{R} \rightarrow \mathbf{R}$ that satisfies the condition: $f\left(\frac{1+x}{x}\right)=\frac{x^{2}+1}{x^{2}}+\frac{1}{x}$. | f(x)=x^{2}-x+1 | 62 | 10 |
math | 15. Red and Black (from 8th grade, 3 points). The dealer ${ }^{2}$ deals one card at a time to the player from a well-shuffled standard deck of cards. At any moment, while the dealer still has cards left, the player can say "Stop". After this, the dealer reveals one more card. If it is of a red suit, the player wins; if it is black, the player loses.
Does there exist a strategy for the player that makes the probability of winning greater than $0.5 ?$[^6] | No | 118 | 1 |
math | 2. (6 points) Convert $\frac{15}{37}$ to a decimal, the 2016th digit from left to right in the decimal part is | 5 | 37 | 1 |
math | 10.064. The lateral side of an isosceles triangle is $10 \mathrm{~cm}$, and the base is $12 \mathrm{~cm}$. Tangents are drawn to the inscribed circle of the triangle, parallel to the height of the triangle, and cutting off two right triangles from the given triangle. Find the lengths of the sides of these triangles. | 3 | 84 | 1 |
math | 4. determine all natural numbers $n$ with exactly 100 different positive divisors, so that at least 10 of these divisors are consecutive numbers.
## Solution: | 45360 | 38 | 5 |
math | Example 8 Solve the system of equations in the set of real numbers
$$\left\{\begin{array}{l}
2 x+3 y+z=13 \\
4 x^{2}+9 y^{2}+z^{2}-2 x+15 y+3 z=82
\end{array}\right.$$ | x=3, y=1, z=4 | 72 | 11 |
math | 65. Find all integers $a$ such that the quadratic trinomial
$$
y=(x-a)(x-6)+1
$$
can be represented as the product $(x+b)(x+c)$, where $b$ and $c$ are integers. | 4or8 | 57 | 3 |
math | 5. In the country of Lemonia, coins in circulation have denominations of $2^{n}, 2^{n-1} \cdot 3, 2^{n-2} \cdot 3^{2}$, $2^{n-3} \cdot 3^{3}, \ldots, 2 \cdot 3^{n-1}, 3^{n}$ piastres, where $n$ is a natural number. A resident of the country went to the bank without any cash. What is the largest amount that the bank will not be able to give him? | 3^{n+1}-2^{n+2} | 124 | 12 |
math |
Problem 10.1. Find all values of the real positive parameter $a$ such that the inequality $a^{\cos 2 x}+a^{2 \sin ^{2} x} \leq 2$ holds for any real $x$.
| \in[1,\frac{-1+\sqrt{5}}{2}] | 58 | 16 |
math | 8.5. Given a convex quadrilateral $A B C D$, where $A B=A D=1, \angle A=80^{\circ}$, $\angle C=140^{\circ}$. Find the length of the diagonal $A C$. | 1 | 57 | 1 |
math | 18. Let $A$ denote the set of integer solutions $\left(x_{1}, x_{2}, \cdots, x_{2 n}\right)$ to the equation $x_{1}+x_{2}+\cdots+x_{2 n}=n$ that satisfy the constraints: $x_{1}+x_{2}+\cdots+x_{j}<\frac{1}{2} j(1 \leqslant j \leqslant 2 n-1)$ and $0 \leqslant x_{j} \leqslant 1(1 \leqslant j \leqslant 2 n)$. Find $|A|$. | \frac{1}{n}C_{2n-2}^{n-1} | 149 | 19 |
math | Example 5 If for any positive real numbers, $\frac{a^{2}}{\sqrt{a^{4}+3 b^{4}+3 c^{4}}}+\frac{k}{a^{3}} \cdot\left(\frac{c^{4}}{b}+\frac{b^{4}}{c}\right) \geqslant \frac{2 \sqrt{2}}{3}$. Always holds, find the minimum value of the real number $k$.
| \frac{1}{\sqrt[4]{24}} | 103 | 13 |
math | following relations:
$$
\sum_{i=1}^{5} i x_{i}=a, \quad \sum_{i=1}^{5} i^{3} x_{i}=a^{2}, \quad \sum_{i=1}^{5} i^{5} x_{i}=a^{3} .
$$
What are the possible values of $a$ ? | {0,1,4,9,16,25} | 83 | 15 |
math | 13. Draw two perpendicular chords $O A, O B$ through the vertex of the parabola $y^{2}=4 p x(p>0)$, find the locus of the projection $M$ of the vertex $O$ of the parabola on the line $A B$. | (x-2p)^{2}+y^{2}=4p^{2} | 62 | 18 |
math | 7. The center of a unit square coincides with the center of a circle, and the square is inside the circle. If a point is randomly selected on the circle, the probability that this point can see two complete sides of the square is $\frac{1}{2}$, then the radius of the circle is . $\qquad$ | \frac{\sqrt{4+2\sqrt{2}}}{2} | 69 | 16 |
math |
3. For a rational number $r$, its period is the length of the smallest repeating block in its decimal expansion. For example, the number $r=0.123123123 \cdots$ has period 3 . If $S$ denotes the set of all rational numbers $r$ of the form $r=0 . \overline{a b c d e f g h}$ having period 8 , find the sum of all the elements of $S$.
| 49995000 | 105 | 8 |
math | 13.138. Two workers, the second of whom started working 1.5 days later than the first, independently wallpapered several rooms in 7 days, counting from the moment the first worker started. If this work had been assigned to each separately, the first would have needed 3 days more than the second to complete it. How many days would each of them take to complete the same work individually? | 14 | 86 | 2 |
math | 1. In the set of real numbers, solve the equation:
$$
\frac{5 x}{x^{2}+3 x+6}+\frac{7 x}{x^{2}+7 x+6}=1
$$ | x_1=6,x_2=1,x_3=-2,x_4=-3 | 50 | 20 |
math | Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$? | 5 | 55 | 3 |
math | 7. Suppose that $(1+\sec \theta)(1+\csc \theta)=6$. Determine the value of $(1+\tan \theta)(1+\cot \theta)$. | \frac{49}{12} | 38 | 9 |
math | 732. Given the sides of a triangle expressed in numbers. How can one determine with a rather simple calculation whether this triangle is right-angled, acute-angled, or obtuse-angled? Provide numerical examples. | ^{2}=b^{2}+^{2} | 45 | 11 |
math | Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$. | 865 | 57 | 3 |
math | 5. In a school there are 300 boys and 300 girls, divided into 5 classes, each with the same number of students. It is known that there are at least 33 boys and 33 girls in each class. A boy and a girl from the same class may form a group to enter a contest, and each student may only belong to one group. What is the maximum number of groups that can be guaranteed to form?
(1 mark)
某校有男生和女生各 300 名，他們被分成 5 班，每班人數相同。已知每班均最少有男生和女生各 33 名。同班的一名男生和一名女生可組隊參加一項比賽, 而每名學生只可隸屬一隊。保證能夠組成的隊伍數目的最大值是多少？
(1 分) | 192 | 191 | 3 |
math | Example 1-17 5 girls and 7 boys are to form a group of 5 people, with the requirement that boy A and girl B cannot be in the group at the same time. How many schemes are there? | 672 | 48 | 3 |
math | 10. If each element in set $A$ can be expressed as the product of two different numbers from 1, $2, \cdots, 9$, then the maximum number of elements in set $A$ is $\qquad$. | 31 | 51 | 2 |
math | 10.61 Find a two-digit number that is equal to twice the product of its digits.
(Kyiv Mathematical Olympiad, 1952) | 36 | 34 | 2 |
math | 1. An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9, 11 is an arithmetic sequence with five terms.
(a) The first four terms of an arithmetic sequence are $20,13,6,-1$. What are the next two terms in the sequence?
(b) The numbers $2, a, b, c, 14$ in that order form an arithmetic sequence with five terms. What are the values of $a, b$, and $c$ ?
(c) The numbers 7, 15 and $t$, arranged in some unknown order, form an arithmetic sequence with three terms. Determine all possible values of $t$.
(d) The numbers $r, s, w, x, y, z$ in that order form an arithmetic sequence with six terms. The sequence includes the numbers 4 and 20 , with the number 4 appearing before the number 20 in the sequence. Determine the largest possible value of $z-r$ and the smallest possible value of $z-r$. | 80,16 | 240 | 5 |
math | 6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village. | 2019 | 122 | 4 |
math | Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$. | n \le 5 | 76 | 6 |
math | Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | 9 | 128 | 1 |
math | 2. Fishermen caught 80 fish in a pond with a net, marked them, and released them back into the pond. The next day, they caught 150 fish, 5 of which were marked. How many fish are there in the pond in total?
Note. The marked fish are evenly distributed among the others. | 2400 | 69 | 4 |
math | 5. In $\triangle A B C$, one side is 5, and the other two sides are exactly the two roots of the equation $2 x^{2}-12 x+m=0$. Then the range of values for $m$ is | \frac{11}{2}<m \leqslant 18 | 51 | 17 |
math | 1. Let the circumcenter, incenter, and orthocenter of non-isosceles $\triangle ABC$ be $O$, $I$, and $H$, respectively, with the circumradius being $1$ and $\angle A=60^{\circ}$. Then the circumradius of $\triangle OIH$ is $\qquad$. | 1 | 72 | 1 |
math | Example 6 (2001 National High School Mathematics League Question) Let $\left\{a_{n}\right\}$ be an arithmetic sequence, and $\left\{b_{n}\right\}$ be a geometric sequence, and $b_{1}=a_{1}^{2}, b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}\left(a_{1}<a_{2}\right)$, and $\lim \left(b_{1}+b_{2}+\cdots+b_{n}\right)=1+\sqrt{2}$. Try to find the first term and common difference of the sequence $\left\{a_{n}\right\}$. | a_{1}=-\sqrt{2},=2\sqrt{2}-2 | 150 | 18 |
math | Let $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$ for all real $x$.
Find the least natural number $n$ such that $f(n\pi + x)= f(x)$ for all real $x$. | 60 | 61 | 2 |
math | 9. (15 points) Find all values of the parameter $a$ for which the equation
$$
3 x^{2}-4(3 a-2) x+a^{2}+2 a=0
$$
has roots $x_{1}$ and $x_{2}$, satisfying the condition $x_{1}<a<x_{2}$. | (-\infty;0)\cup(1,25;+\infty) | 76 | 18 |
math | 8.1. In the wagon, several kilograms of apple jam were loaded, of which $20 \%$ was good and $80 \%$ was bad. Every day, half of the existing bad jam rotted, and it was thrown away. After several days, it turned out that $20 \%$ of the jam in the wagon was bad and $80 \%$ was good. How many days have passed since the loading? | 4 | 92 | 1 |
math | 3. Let a tangent line of the circle $x^{2}+y^{2}=1$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Then the minimum value of $|AB|$ is $\qquad$ . | 2 | 60 | 1 |
math | Task A-3.4. (8 points)
If for the lengths $a, b, c$ of the sides of a triangle it holds that $(a+b+c)(a+b-c)=3 a b$, determine the angle opposite side $c$. | 60 | 52 | 2 |
math | ## 11. Polar Expedition
A polar explorer set out from one point to another on a sled pulled by a team of five dogs. But after 24 hours, two of the dogs died, causing the explorer's speed to decrease to $3 / 5$ of the original speed, and he was delayed by two days.
- Oh! - exclaimed the polar explorer. - If the two dogs that died had run another 120 km, I would have been delayed by only one day.
What distance did the polar explorer travel? | 320 | 112 | 3 |
math | 6.26 For a positive integer $k$, there exist positive integers $n$ and $m$ such that $\frac{1}{n^{2}}+\frac{1}{m^{2}}=\frac{k}{n^{2}+m^{2}}$. Find all positive integers $k$.
(Hungarian Mathematical Olympiad, 1984) | 4 | 77 | 1 |
math | II. (50 points) Let $x_{i} \in \mathbf{R}^{+}(i=1,2, \cdots, n)$, and $x_{1} x_{2} \cdots x_{n}=1, n$ be a given positive integer. Try to find the smallest positive number $\lambda$, such that the inequality $\frac{1}{\sqrt{1+2 x_{1}}}+\frac{1}{\sqrt{1+2 x_{2}}}+\cdots+\frac{1}{\sqrt{1+2 x_{n}}} \leqslant \lambda$ always holds. | \lambda={\begin{pmatrix}\frac{\sqrt{3}n}{3}(n=1,2),\\n-1(n\geqslant3)0\end{pmatrix}.} | 137 | 45 |
math | For which natural numbers $n$ is the value of the following expression a perfect square?
$$
n^{5}-n^{4}-2 n^{3}+2 n^{2}+n-1
$$ | k^{2}+1 | 45 | 6 |
math | For $0<x<1,$ let $f(x)=\int_0^x \frac{dt}{\sqrt{1-t^2}}\ dt$
(1) Find $\frac{d}{dx} f(\sqrt{1-x^2})$
(2) Find $f\left(\frac{1}{\sqrt{2}}\right)$
(3) Prove that $f(x)+f(\sqrt{1-x^2})=\frac{\pi}{2}$ | \frac{\pi}{2} | 105 | 7 |
math | $4 \cdot 2$ Find all positive values of $a$ that make the equation $a^{2} x^{2}+a x+1-7 a^{2}=0$ have two integer roots. | 1,\frac{1}{2},\frac{1}{3} | 46 | 15 |
math | 1. The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
x y+y z+z x=1, \\
5 x+8 y+9 z=12
\end{array}\right.
$$
all real solutions $(x, y, z)$ are | (1,\frac{1}{2},\frac{1}{3}) | 68 | 16 |
math | Question 1. The sum of $\mathrm{n}$ positive numbers is 1983. When is their product maximized, and what is the maximum value? | \left(\frac{1983}{n}\right)^n | 34 | 15 |
math | 24. Let $f(x)=x^{3}+3 x+1$, where $x$ is a real number. Given that the inverse function of $f$ exists and is given by
$$
f^{-1}(x)=\left(\frac{x-a+\sqrt{x^{2}-b x+c}}{2}\right)^{1 / 3}+\left(\frac{x-a-\sqrt{x^{2}-b x+c}}{2}\right)^{1 / 3}
$$
where $a, b$ and $c$ are positive constants, find the value of $a+10 b+100 c$. | 521 | 136 | 3 |
math | 3-ча 1. Find all positive rational solutions of the equation \(x^{y}=y^{x}(x \neq y)\). | (\frac{p+1}{p})^{p},(\frac{p+1}{p})^{p+1} | 30 | 26 |
math | Determine the largest possible radius of a circle that is tangent to both the $x$-axis and $y$-axis, and passes through the point $(9,2)$.
(A circle in the $x y$-plane with centre $(a, b)$ and radius $r$ has equation $(x-a)^{2}+(y-b)^{2}=r^{2}$.) | 17 | 81 | 2 |
math | Find the sum of the coefficients of the even powers in the polynomial that results from the expression $f(x)=(x^{3} - x + 1)^{100}$ after expanding the brackets and combining like terms.
# | 1 | 47 | 1 |
math | 7. Let the set $S=\left\{2^{0}, 2^{1}, \cdots, 2^{10}\right\}$, then the sum of the absolute values of the differences of any two distinct elements in $S$ is equal to | 16398 | 56 | 5 |
math | 2. For a positive integer of three digits, we can multiply the three digits with each other. The result of this we call the digit product of that number. So, 123 has a digit product of $1 \times 2 \times 3=6$ and 524 has a digit product of $5 \times 2 \times 4=40$. A number cannot start with the digit 0. Determine the three-digit number that is exactly five times as large as its own digit product. | 175 | 109 | 3 |
math | ## Subject IV. (20 points)
Amaya and Keiko are two girls from Osaka. They live on the same floor in a building with two staircases, each with 5 apartments per floor. At the ground floor of the building are shops.
The apartments are numbered in ascending order, starting from the first floor. Amaya lives in apartment 28, and Keiko in apartment 164. How many floors does the building have?
Prof. Sorin Borodi, "Alexandru Papiu Ilarian" Theoretical High School, Dej
All subjects are mandatory. 10 points are awarded by default.
Effective working time - 2 hours.
## Grading Scale for Grade 5 (OLM 2013 - Local Stage)
## Of. $10 \mathrm{p}$ | 27 | 171 | 2 |
math | \section*{Problem 4 - 161044}
Determine all integer pairs \((x ; y)\) that satisfy the following equation!
\[
x y + 3 x - 2 y - 3 = 0
\] | (-1,-2),(1,0),(3,-6),(5,-4) | 55 | 17 |
math | As usual, let $n$ ! denote the product of the integers from 1 to $n$ inclusive. Determine the largest integer $m$ such that $m$ ! divides $100 !+99 !+98 !$. | 98 | 51 | 2 |
math | If there are $2 k(k \geqslant 3)$ points on a plane, where no three points are collinear. Draw a line segment between any two points, and color each line segment red or blue. A triangle with three sides of the same color is called a monochromatic triangle, and the number of monochromatic triangles is denoted as $S$. For all possible coloring methods, find the minimum value of $S$. | \frac{k(k-1)(k-2)}{3} | 92 | 14 |
math | 4. (8 points) Tongtong and Linlin each have several cards: If Tongtong gives 6 cards to Linlin, Linlin will have 3 times as many as Tongtong; if Linlin gives 2 cards to Tongtong, then Linlin will have 2 times as many as Tongtong. How many cards did Linlin originally have. | 66 | 79 | 2 |
math |
4. Find all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ such that for all non-zero numbers $x, y$,
$$
x \cdot f(x y)+f(-y)=x \cdot f(x) .
$$
(Pavel Calábek)
| f(x)=(1+\frac{1}{x}) | 72 | 11 |
math | 3. (8 points) In another 12 days, it will be 2016, Hao Hao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9, so in 2016, Hao Hao is $\qquad$ years old. | 9 | 69 | 1 |
math | Example 2 Find the value of the sum $\sum_{k=1}^{n} k^{2} C_{n}^{k}=C_{n}^{1}+2^{2} C_{n}^{2}+\cdots+n^{2} C_{n}^{n}$. | n(n+1)\cdot2^{n-2} | 64 | 12 |
math | 11. Determine the largest even positive integer which cannot be expressed as the sum of two composite odd positive integers. | 38 | 23 | 2 |
math | Example 6 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: x^{2}+2 x y+y^{2}+3 x+y=$ 0 with respect to the line $l: 2 x-y-1=0$. | x^{2}+14xy+49y^{2}-21x+103y+54=0 | 58 | 28 |
math | 6. A point on the coordinate plane whose both horizontal and vertical coordinates are integers is called an integer point. The number of integer points in the region enclosed by the parabola $y=x^{2}+1$ and the line $2 x-y+81=0$ is $\qquad$ . | 988 | 64 | 3 |
math | 4. Let the function $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$, and satisfy:
(1) $\operatorname{gcd}(f(1), f(2), \cdots, f(n), \cdots)=1$;
(2) For sufficiently large $n, f(n) \neq 1$, for any positive integers $a, b$ and sufficiently large $n$ we always have
$$
(f(a))^{n} \mid\left((f(a+b))^{a^{n-1}}-(f(b))^{a^{n-1}}\right) \text {. }
$$
Find $f$. | f(x)=x | 146 | 4 |
math | 8. The function $y=f(t)$ is such that the sum of the roots of the equation $f(\sin x)=0$ on the interval $[3 \pi / 2, 2 \pi]$ is $33 \pi$, and the sum of the roots of the equation $f(\cos x)=0$ on the interval $[\pi, 3 \pi / 2]$ is $23 \pi$. What is the sum of the roots of the second equation on the interval $[\pi / 2, \pi]$? | 17\pi | 116 | 4 |
math | 1. $x, y$ are real numbers, and $\left(x+\sqrt{x^{2}+1}\right)(y+$ $\left.\sqrt{y^{2}+1}\right)=1$. Then $x+y=$ $\qquad$ . | 0 | 54 | 1 |
math | 19.6. In the institute, there are truth-lovers, who always tell the truth, and liars, who always lie. One day, each employee made two statements.
1) There are not even ten people in the institute who work more than I do.
2) At least a hundred people in the institute earn more than I do.
It is known that the workload of all employees is different, and so are their salaries. How many people work in the institute? | 110 | 98 | 3 |
math | Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i] | \left\lfloor \frac{n^2}{2} \right\rfloor - 1 | 92 | 22 |
math | PROBLEM 3. Let triangle $ABC$ and $M$ be the midpoint of side [ $BC$ ]. If $N$ is the midpoint of segment $[AM]$ and $BN \cap AC=\{P\}$, determine the ratio between the area of quadrilateral $MCPN$ and the area of triangle $ABC$. | \frac{5}{12} | 71 | 8 |
math | 4. 7 (CMO17) For four distinct points $P_{1}, P_{2}, P_{3}, P_{4}$ in the plane, find the minimum value of the ratio $\frac{\sum_{1 \leqslant i<j \leqslant 4} P_{i} P_{j}}{\min _{1 \leqslant j \leqslant 4} P_{i} P_{j}}$. | 5+\sqrt{3} | 99 | 6 |
math | 1. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$ the following holds
$$
(x+1) f(x f(y))=x f(y(x+1))
$$ | f(x)=0f(x)=x | 59 | 8 |
math | Let's determine the value of $\sqrt{3(7+4 \sqrt{3})}$ as the sum of two simple square roots! | 2\sqrt{3}+3 | 29 | 8 |
math | 3. In the arithmetic progression $\left(a_{n}\right) a_{1000}=150, d=0.5$.
Calculate: $99 \cdot 100 \cdot\left(\frac{1}{a_{1580} \cdot a_{1581}}+\frac{1}{a_{1581} \cdot a_{1582}}+\ldots+\frac{1}{a_{2019} \cdot a_{2020}}\right)$. | 15 | 117 | 2 |
math | 10.4. Let x be some natural number. Among the statements:
$2 \times$ greater than 70
x less than 100
$3 x$ greater than 25
x not less than 10
x greater than 5
three are true and two are false. What is x | 9 | 70 | 1 |
math | The angles of a certain triangle form an arithmetic progression. The smallest side is half of the largest one. The angles need to be calculated! | 30,60,90 | 28 | 8 |
math | 3. What can the value of the expression $p^{4}-3 p^{3}-5 p^{2}+16 p+2015$ be if $p$ is a root of the equation $x^{3}-5 x+1=0$?
Answer: 2018 | 2018 | 65 | 4 |
math | ## Task 2 - 330512
At a birthday party, a game is played with blue game tokens and an equal number of red game tokens.
After some time, each child had received 12 blue and 15 red game tokens, and there were still 48 blue and 15 red game tokens left.
How many children played this game? | 11 | 79 | 2 |
math | Example 9. Find the integral $\int \sqrt{a^{2}-x^{2}} d x(a>0)$. | -\frac{^{2}}{2}\arccos\frac{x}{}+\frac{x}{2}\sqrt{^{2}-x^{2}}+C | 27 | 34 |
math | 2. In a certain kingdom, there are 32 knights, some of whom are servants to other knights. Each servant can have at most one master, and each master must be richer than any of his servants. If a knight has at least four servants, he is ennobled as a noble. If it is stipulated that a servant of $A$'s servant is not a servant of $A$, then the maximum possible number of nobles is $\qquad$ | 7 | 98 | 1 |
math | Example 3. In $\triangle A B C$, $D$ is a point on side $B C$. It is known that $A B=13, A D=12, A C=15, B D=5$. What is $D C$? (3rd Zu Chongzhi Cup Junior High School Mathematics Invitational Competition) | 9 | 75 | 1 |
math | Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive.
[i]Proposed by Isabella Grabski[/i] | \frac{1}{2} | 92 | 7 |
math | Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 | 58 | 3 |
math | 19.1 .37 Give methods to find all pairs of positive integers, the sum and product of each pair being square numbers. In particular, find all such pairs of positive integers less than 100. | 2,2;5,20;8,8;10,90;18,18;20,80;9,16;32,32;50,50;72,72;2,98;98,98;36,64 | 45 | 70 |
math | Find all integers $n \geqslant 1$ such that $n$ divides $3^{n}+1$ and $11^{n}+1$. | n=1n=2 | 37 | 6 |
math | 3. Given that $x$ and $y$ are positive integers, and $xy + x + y$ $=23, x^2y + xy^2=120$. Then $x^2 + y^2=$ $\qquad$ | 34 | 55 | 2 |
math | 9.1. Find the maximum value of the expression $(\sqrt{8-4 \sqrt{3}} \sin x-3 \sqrt{2(1+\cos 2 x)}-2) \cdot(3+2 \sqrt{11-\sqrt{3}} \cos y-\cos 2 y)$. If the answer is not an integer, round it to the nearest integer. | 33 | 84 | 2 |
math | 6. In triangle $A B C$, it is known that $A B=4, A C=6, \angle B A C=60^{\circ}$. The extension of the angle bisector $A A_{1}$ intersects the circumcircle of triangle $A B C$ at point $A_{2}$. Find the areas of triangles $O A_{2} C$ and $A_{1} A_{2} C$. ( $O$ - the center of the circumcircle of triangle $\left.A B C\right)$ | S_{OA_{2}C}=\frac{7}{\sqrt{3}};S_{A_{1}A_{2}C}=\frac{7\sqrt{3}}{5} | 115 | 43 |
math | 5. In a regular quadrilateral pyramid $P-ABCD$, the side faces are equilateral triangles with a side length of 1. $M, N$ are the midpoints of edges $AB, BC$ respectively. The distance between the skew lines $MN$ and $PC$ is $\qquad$. | \frac{\sqrt{2}}{4} | 65 | 10 |
math | 7.016. If $\log _{a} 27=b$, then what is $\log _{\sqrt{3}} \sqrt[6]{a}$? | \frac{1}{b} | 37 | 7 |
math | Evin and Jerry are playing a game with a pile of marbles. On each players' turn, they can remove $2$, $3$, $7$, or $8$
marbles. If they can’t make a move, because there's $0$ or $1$ marble left, they lose the game. Given that Evin goes first and both players play optimally, for how many values of $n$ from $1$ to $1434$ does Evin lose the game?
[i]Proposed by Evin Liang[/i]
[hide=Solution][i]Solution.[/i] $\boxed{573}$
Observe that no matter how many marbles a one of them removes, the next player can always remove marbles such
that the total number of marbles removed is $10$. Thus, when the number of marbles is a multiple of $10$, the first player loses the game. We analyse this game based on the number of marbles modulo $10$:
If the number of marbles is $0$ modulo $10$, the first player loses the game
If the number of marbles is $2$, $3$, $7$, or $8$ modulo $10$, the first player wins the game by moving to $0$ modulo 10
If the number of marbles is $5$ modulo $10$, the first player loses the game because every move leads to $2$, $3$, $7$, or $8$ modulo $10$
In summary, the first player loses if it is $0$ mod 5, and wins if it is $2$ or $3$ mod $5$. Now we solve the remaining cases by induction. The first player loses when it is $1$ modulo $5$ and wins when it is $4$ modulo $5$. The base case is when there is $1$ marble, where the first player loses because there is no move. When it is $4$ modulo $5$, then the first player can always remove $3$ marbles and win by the inductive hypothesis. When it is $1$ modulo $5$, every move results in $3$ or $4$ modulo $5$, which allows the other player to win by the inductive hypothesis.
Thus, Evin loses the game if n is $0$ or $1$ modulo $5$. There are $\boxed{573}$ such values of $n$ from $1$ to $1434$.[/hide] | 573 | 533 | 3 |
math | 36. [25] How many numbers less than $1,000,000$ are the product of exactly 2 distinct primes? You will receive $\left\lfloor 25-50 \cdot\left|\frac{N}{A}-1\right|\right\rfloor$ points, if you submit $N$ and the correct answer is $A$. | 209867 | 83 | 6 |
math | Santa Claus arrived at Arnaldo and Bernaldo's house carrying ten distinct toys numbered from 1 to 10 and said to them: "the toy number 1 is for you, Arnaldo and the toy number 2 is for you, Bernaldo. But this year, you can choose to keep more toys as long as you leave at least one for me". In how many ways can Arnaldo and Bernaldo divide the remaining toys between them? | 6305 | 95 | 4 |
math | 2. Given the complex numbers $z_{1}, z_{2}$ satisfy
$$
\left|z_{1}-z_{2}\right|=\sqrt{3}, z_{1}^{2}+z_{1}+q=0, z_{2}^{2}+z_{2}+q=0 \text {. }
$$
then the real number $q=$ $\qquad$ | 1 | 87 | 1 |
math | ## Task $4 / 62$
What remainder does the number $2^{n}$ leave when divided by 3? | The2^{n}leavesremainderof1whendivided3ifniseven,remainderof2ifnisodd | 26 | 26 |
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