In a triangle $ABC$, $BC=a$, $AC=b$, $\angle B=\beta$ and $\angle C=\gamma$. Prove that the bisector of the angle at $A$ is equal to the altitude from $B$ if and only if $b=a\cos\frac{\beta-\gamma}2$.

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that \[a^2+b^2+c^2 \geq 2(ab+bc+ca)\]

Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element? Please prove your conclusion.

Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions: $f(0, 0) = 0$; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$, $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$. Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different? b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

Given is an acute angled triangle $ ABC$. Determine all points $ P$ inside the triangle with \[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\]

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i] (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

We consider the proposition $ p(n)$: $ n^2\plus{}1$ divides $ n!$, for positive integers $ n$. Prove that there are infinite values of $ n$ for which $ p(n)$ is true, and infinite values of $ n$ for which $ p(n)$ is false.

Prove that every positive integer has a unique representation in the form \[ \sum_{i=0}^k d_i 2^i \, , \] where $k$ is a nonnegative integer and each $d_i$ is either 1 or 2. (This representation is similar to usual binary notation except that the digits are 1 and 2, not 0 and 1.)

It is well known that the mathematical constant $e$ can be written in the form $e = \tfrac{1}{0!}+\tfrac{1}{1!}+\tfrac{1}{2!}+\cdots$. With this in mind, determine the value of \[\sum_{j=3}^\infty\dfrac{j}{\lfloor\frac j2\rfloor!}.\] Express your answer in terms of $e$.

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]

For each permutation $ a_1, a_2, a_3, \ldots,a_{10}$ of the integers $ 1,2,3,\ldots,10,$ form the sum \[ |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|.\] The average value of all such sums can be written in the form $ p/q,$ where $ p$ and $ q$ are relatively prime positive integers. Find $ p \plus{} q.$

The first row of a table of size $ 2005\times5$ is filled with 1,2,3,4,5 so that every two neighbouring cells contain distinct numbers. Prove that it is possible to fill four other rows with 1,2,3,4,5 so that any neighbouring cells in them will contain distinct numbers as well as any cells of the same column will contain pairwise distinct numbers.

2) all elements in {0,1,2}; B[n] = number of rows with no 2 sequent 0's; A[n] with no 3 sequent elements the same; prove |A[n+1]|=3.|B[n]|

Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.

Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

Say that a (nondegenerate) triangle is [i]funny[/i] if it satisfies the following condition: the altitude, median, and angle bisector drawn from one of the vertices divide the triangle into 4 non-overlapping triangles whose areas form (in some order) a 4-term arithmetic sequence. (One of these 4 triangles is allowed to be degenerate.) Find with proof all funny triangles.

In the non-isosceles triangle $ABC$,$D$ is the midpoint of side $BC$,$E$ is the midpoint of side $CA$,$F$ is the midpoint of side $AB$.The line(different from line $BC$) that is tangent to the inscribed circle of triangle $ABC$ and passing through point $D$ intersect line $EF$ at $X$.Define $Y,Z$ similarly.Prove that $X,Y,Z$ are collinear.

We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.

(a) The vertices of a regular $10$-gon are painted in turn black and white. Two people play the following game . Each in turn draws a diagonal connecting two vertices of the same colour . These diagonals must not intersect . The winner is the player who is able to make the last move. Who will win if both players adopt the best strategy? (b) Answer the same question for the regular $12$-gon . (V.G. Ivanov)

Find the maximum value of $M$, if you choose $10$ different real numbers randomly in $[1,M]$, there must be $3$ numbers $a<b<c$, satisfy $ax^2+bx+c=0$ has no real root.

Map the interior of the unit disc conformally onto the first quadrant.