In $\triangle ABC$, $\angle A=55^{\circ}$, $\angle C=75^{\circ}$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle BED=$ [asy] size((100)); draw((0,0)--(10,0)--(8,10)--cycle); draw((4,5)--(9.2,4)); dot((0,0)); dot((10,0)); dot((8,10)); dot((4,5)); dot((9.2,4)); label("A", (0,0), SW); label("B", (8,10), N); label("C", (10,0), SE); label("D", (4,5), NW); label("E", (9.2,4), E); label("$55^{\circ}$", (.5,0), NE); label("$75^{\circ}$", (9.8,0), NW); [/asy] $ \textbf{(A)}\ 50^{\circ} \qquad\textbf{(B)}\ 55^{\circ} \qquad\textbf{(C)}\ 60^{\circ} \qquad\textbf{(D)}\ 65^{\circ} \qquad\textbf{(E)}\ 70^{\circ} $

Consider a $2n\times 2n$ chessboard with all the $4n^2$ cells being white to start with. The following operation is allowed to be performed any number of times: "Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)." Find all possible values of $n\ge 2$ for which using the above operation one can obtain the normal chess coloring of the given board.

All ordinary proper irreducible fractions whose numerators are two-digit numbers were ordered in ascending order. Between what two consecutive fractions is the number $\frac58$ located?

In a chess tournament, each player played at most one game against each other, and the number of games played by each player is not less than the set natural number $ n $. Prove that it is possible to divide players into two groups $ A $ and $ B $ in such a way that the number of games played by each player of group $ A $ with players of group $ B $ is not less than $ n/2 $ and at the same time the number of games played by each player of the $ B $ group with players of the $ A $ group was not less than $ n/2 $.

Let $n$ be a positive integer. Find as many as possible zeros as last digits the following expression: $1^n + 2^n + 3^n + 4^n$.

Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

Let $0\leq p_i \leq 1$ for $i=1,2, \dots, n.$ Show that $$\sum_{i=1}^{n} \frac{1}{|x-p_i|} \leq 8n(1+1/3+1/5+\dots +\frac{1}{2n-1})$$ for some $x$ satisfying $0\leq x \leq 1.$

Let $n\geqslant 3$ be an integer. Ion draws a regular $n$-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the $n$-gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.

Find the value of $$\frac{1}{\sqrt{2}^1} + \frac{4}{\sqrt{2}^2} + \frac{9}{\sqrt{2}^3} + \cdots$$

Find all primes $p$ such that for any integer $k$, there exist two integers $x$ and $y$ such that $$x^3+2023xy+y^3 \equiv k \pmod p$$ [i]Proposed by Tristan Chaang Tze Shen[/i]

In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. [b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas. [b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.

In a convex quadrilateral $PQRS$, the areas of triangles $PQS$, $QRS$ and $PQR$ are in the ratio $3:4:1$. A line through $Q$ cuts $PR$ at $A$ and $RS$ at $B$ such that $PA:PR=RB:RS$. Prove that $A$ is the midpoint of $PR$ and $B$ is the midpoint of $RS$.

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?

In a given community of people, each person has at least two friends within the community. Whenever some people from this community sit on a round table such that each adjacent pair of people are friends, it happens that no non-adjacent pair of people are friends. Prove that there exist two people in this community such that each has exactly two friends and they have at least one common friend.

Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

Which of the following answer choices is equivalent to $\sqrt{a^3b^2c}$? $\textbf{(A) }ab\sqrt{ac}\qquad\textbf{(B) }bc\sqrt{ac}\qquad\textbf{(C) }b\sqrt{ac}\qquad\textbf{(D) }abc\sqrt{ab}\qquad\textbf{(E) }a\sqrt{bc}$

$100$ Queens are placed on a $100 \times 100$ chessboard so that no two attack each other. Prove that each of four $50 \times 50$ corners of the board contains at least one Queen.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]