Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

Let $ABCD$ be a trapezium inscribed in a circle $\Gamma$ with diameter $AB$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . The circle with center $B$ and radius $BE$ meets $\Gamma$ at the points $K$ and $L$ (where $K$ is on the same side of $AB$ as $C$). The line perpendicular to $BD$ at $E$ intersects $CD$ at $M$. Prove that $KM$ is perpendicular to $DL$. [i]Greece - Silouanos Brazitikos[/i]

Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$. $\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$

$n(n>3)$ ping-pong players have played a few ping-pong games. The set of players that player A has played with is $A$, The set of players that player B has played with is $B$. for any two players, $A\neq B$. Prove that we can delete a player, so that this character remains.

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city. We call a set $D$ of towns [i]dominant[/i] if every town not in $D$ is connected by a road to a town in $D$. Suppose that each dominant set consists of at least $k$ towns. Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.

[b]8.1 / 7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ? [b]8.2 / 7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]8.3 / 7.6[/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. [b]8.4[/b] Prove that the sum of all divisors of the number $n^2$ is odd. [b]8.5[/b] A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle. [b]8.6[/b] Numbers $x_1, x_2, . . . $ are constructed according to the following rule: $$x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...$$ Prove that no matter how much we continued this construction, all the resulting numbers will be no less $1/5$ and no more than $2$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$. [i]2016 CCA Math Bonanza #10[/i]

The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $T$. Let $X$ be a point on circle $\omega_1$. Line $l_1$ is tangent to circle $\omega_1$ and $X$, and line $l$ intersects circle $\omega_2$ at $A$ and $B$. Line $XT$ meets circle $\omega$ at $S$. Point $C$ lies on arc $TS$ (of circle $\omega_2$, not containing points $A$ and $B$). Point $Y$ lies on circle $\omega_1$ and line $YC$ is tangent to circle $\omega_1$. Let $I$ be the intersection of lines $XY$ ad $SC$. Prove that... a) points $C$, $T$, $Y$, $I$ lie on a circle (B) $I$ is an excenter of triangle $ABC$.

(a) $n$ is a natural number. $d_1,\dots,d_n,r_1,\dots ,r_n$ are natural numbers such that for each $i,j$ that $1\leq i < j \leq n$ we have $(d_i,d_j)=1$ and $d_i\geq 2$. Prove that there exist an $x$ such that (i) $1 \leq x \leq 3^n$ (ii)For each $1 \leq i \leq n$ \[x \overset{d_i}{\not{\equiv}} r_i\] (b) For each $\epsilon >0$ prove that there exists natural $N$ such that for each $n>N$ and each $d_1,\dots,d_n,r_1,\dots ,r_n$ satisfying the conditions above there exists an $x$ satisfying (ii) such that $1\leq x \leq (2+\epsilon )^n$. Time allowed for this exam was 75 minutes.

$f$ be a function satisfying $2f(x)+3f(-x)=x^2+5x$. Find $f(7)$ [list=1] [*] $-\frac{105}{4}$ [*] $-\frac{126}{5}$ [*] $-\frac{120}{7}$ [*] $-\frac{132}{7}$ [/list]

Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by $ a_1 \equal{} a$ and $ a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)$ where $ \pi(x)$ is the product of the digits of $ x$ (written in base ten) Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms

The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

Find all positive integers $n$ such that: \[ \dfrac{n^3+3}{n^2+7} \] is a positive integer.

Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.

How many four digit perfect square numbers are there in the form $AABB$ where $A,B \in \{1,2,\dots, 9\}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None of above} $

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

If a convex set of points in the line has at least two diameters, say $AB$ and $CD$, prove that $AB$ and $CD$ have a common point.

In a square of sidelength $7$, $51$ points are given. Show that there's a disk of radius $1$ covering at least $3$ of these points.

For dessert, Melinda eats a spherical scoop of ice cream with diameter $2$ inches. She prefers to eat her ice cream in cube-like shapes, however. She has a special machine which, given a sphere placed in space, cuts it through the planes $x = n$, $y = n$, and $z = n$ for every integer $n$ (not necessarily positive). Melinda centers the scoop of ice cream uniformly at random inside the cube $0 \le x, y,z \le 1$, and then cuts it into pieces using her machine. What is the expected number of pieces she cuts the ice cream into?

If $ (a,b)$ and $ (c,d)$ are two points on the line whose equation is $ y\equal{}mx\plus{}k$, then the distance between $ (a,b)$ and $ (c,d)$, in terms of $ a$, $ c$, and $ m$, is $ \textbf{(A)}\ |a\minus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(B)}\ |a\plus{}c|\sqrt{1\plus{}m^2} \qquad \textbf{(C)}\ \frac{|a\minus{}c|}{\sqrt{1\plus{}m^2}} \qquad$ $ \textbf{(D)}\ |a\minus{}c|(1\plus{}m^2) \qquad \textbf{(E)}\ |a\minus{}c|$ $ |m|$

Suppose that $a,b,c$ are real numbers in the interval $[-1,1]$ such that $1 + 2abc \geq a^2+b^2+c^2$. Prove that $1+2(abc)^n \geq a^{2n} + b^{2n} + c^{2n}$ for all positive integers $n$.