In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

Find the set of solutions for $x$ in the inequality $\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}$ when $x\neq 2$, $x\neq -\dfrac{9}{2}$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality \[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \] [i](N. Alexandrov)[/i]

A number of flowers are distributed between $n$ persons so that the first of them, Andreas, gets one flower, the other gets two flowers, the third gets three flowers, etc., to $n$-th person who gets $n$ flowers. Andreas then walks around shaking hands with each other of the others, in any order. In order to do so, he receives a flower from everyone which he hangs on to and which has more flowers than himself at the moment they shake hands. Which is the smallest number of flowers Andreas can have after shaking hands with everyone?

In an inscribed quadrilateral $ABCD$ we have $BC = CD$. Prove that the area of the quadrilateral is equal to $\frac{(AC)^2 \sin A}{2}$ (D. Fomin, Leningrad)

A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.

There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too. Is it possible that the entire class buys tickets to the concert? (Assume that friendship is mutual; if student $A$ is close friends with student $B$, then $B$ is close friends with $A$.)

[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $ [b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

Let $a_1=1$ and $a_{n+1}=2/(2+a_n)$ for all $n\geqslant 1$. Similarly, $b_1=1$ and $b_{n+1}=3/(3+b_n)$ for all $n\geqslant 1$. Which is greater between $a_{2022}$ and $b_{2022}$? [i]Proposed by P. Kozhevnikov[/i]

There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.

Source: 2018 Canadian Open Math Challenge Part A Problem 1 ----- Suppose $x$ is a real number such that $x(x+3)=154.$ Determine the value of $(x+1)(x+2)$.

In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED = \angle ABC$. by Morteza Saghafian

Let $n$ be a positive integer. Initially, a bishop is placed in each square of the top row of a $2^n \times 2^n$ chessboard; those bishops are numbered from $1$ to $2^n$ from left to right. A [i]jump[/i] is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row. Find the total number of permutations $\sigma$ of the numbers $1, 2, \ldots, 2^n$ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order $\sigma(1), \sigma(2), \ldots, \sigma(2^n)$, from left to right. [i]Israel[/i]

A unit square is cut into $m$ quadrilaterals $Q_1,\ldots ,Q_m$. For each $i=1,\ldots ,m$ let $S_i$ be the sum of the squares of the four sides of $Q_i$. Prove that \[S_1+\ldots +S_m\ge 4\]

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

Let $M$ be the midpoint of cathetus $AB$ of triangle $ABC$ with right angle $A$. Point $D$ lies on the median $AN$ of triangle $AMC$ in such a way that the angles $ACD$ and $BCM$ are equal. Prove that the angle $DBC$ is also equal to these angles.