Let $A_1, A_2, \ldots, A_n (n \geq 4)$ be $n$ convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.

Let $ABC$ be an acute triangle with circumcenter $O$. Let the parallel to $BC$ through $A$ intersect line $BO$ at $B_A$ and $CO$ at $C_A$. Lines $B_AC$ and $BC_A$ intersect at $A'$. Define $B'$ and $C'$ similarly. (a) Prove that the the perpendicular from $A'$ to $BC$, the perpendicular from $B'$ to $AC$, and $C'$ to $AB$ are concurrent. (b) Prove that likes $AA'$, $BB'$, and $CC'$ are concurrent.

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O_1$. The diagonals $AC$ and $BD$ meet at point $P$. Suppose the four incentres of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$. Prove that $P, O_1, O_2$ are collinear. [i]Proposed by Shantanu Nene[/i]

Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus? (Sharygin Igor)

Consider the function $f$ defined by \[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\] for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that (a) $f(n+1)>f(n)$ for infinitely many $n$. (b) $f(n+1)<f(n)$ for infinitely many $n$.

Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$. Prove that the lines $AB$ and $CF$ are perpendicular.

Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$. [i]Proposed by Nikola Velov[/i]

Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that: [b]a)[/b] $ abc=xyz $ [b]b)[/b] $ ab+bc+ca=xy+yz+zx $ [i]Bogdan Enescu[/i] and [i]Marcel Chiriță[/i]

A cross with length $p$ (or [i]p-cross[/i] for short) will be called the figure formed by a unit square and 4 rectangles $p-1$ x $1$ on its sides. What’s the least amount of colors one has to use to color the cells of an infinite table, so that each [i]p-cross[/i] on it covers cells, no two of which are in the same color?

Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now?

Given an arbitrary tetrahedron. Prove that its six edges can be divided into two triplets so that from each triple it was possible to form a triangle.

Let $p \ge 5$ be a prime number. Prove that there exists an integer $a$ with $1 \le a \le p-2$ such that neither $a^{p-1} -1$ nor $(a+1)^{p-1} -1$ is divisible by $p^2$.

Let $x,y$ be real numbers with $x \geqslant \sqrt{2021}$ such that \[ \sqrt[3]{x+\sqrt{2021}}+\sqrt[3]{x-\sqrt{2021}} = \sqrt[3]{y}\] Determine the set of all possible values of $y/x$.

Given positive numbers $a,b,c,d$. Prove that the set of inequalities $$a+b<c+d$$ $$(a+b)(c+d)<ab+cd$$ $$(a+b)cd<ab(c+d)$$ contain at least one wrong.

Let $(x_n)$ define by $x_1\in \left(0;\dfrac{1}{2}\right)$ and $x_{n+1}=3x_n^2-2nx_n^3$ for all $n\ge 1$. a) Prove that $(x_n)$ convergence to $0$. b) For each $n\ge 1$, let $y_n=x_1+2x_2+\cdots+n x_n$. Prove that $(y_n)$ has a limit.

Let $ s$ be the limiting sum of the geometric series $ 4\minus{} \frac83 \plus{} \frac{16}{9} \minus{} \dots$, as the number of terms increases without bound. Then $ s$ equals: $ \textbf{(A)}\ \text{a number between 0 and 1} \qquad \textbf{(B)}\ 2.4 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 3.6 \qquad \textbf{(E)}\ 12$

Let $d(n)$ be the number of positive divisors of a positive integer $n$. Let $\mathbb{N}$ be the set of all positive integers. Say that a function $F$ from $\mathbb{N}$ to $\mathbb{N}$ is [i]divisor-respecting[/i] if $d(F(mn)) = d(F(m)) d(F(n))$ for all positive integers $m$ and $n$, and $d(F(n)) \le d(n)$ for all positive integers $n$. Find all divisor-respecting functions. Justify your answer.

Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles? [i]Proposed by Michael Tang[/i]

If $x,y,z\in\mathbb N$ and $xy=z^2+1$ prove that there exists integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.

The graph shows velocity as a function of time for a car. What was the acceleration at time = $90$ seconds? [asy] size(275); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2)); draw((0,3)--(6,3)); draw((0,4)--(6,4)); draw((0,0)--(0,4)); draw((1,0)--(1,4)); draw((2,0)--(2,4)); draw((3,0)--(3,4)); draw((4,0)--(4,4)); draw((5,0)--(5,4)); draw((6,0)--(6,4)); label("$0$",(0,0),S); label("$30$",(1,0),S); label("$60$",(2,0),S); label("$90$",(3,0),S); label("$120$",(4,0),S); label("$150$",(5,0),S); label("$180$",(6,0),S); label("$0$",(0,0),W); label("$10$",(0,1),W); label("$20$",(0,2),W); label("$30$",(0,3),W); label("$40$",(0,4),W); draw((0,0.6)--(0.1,0.55)--(0.8,0.55)--(1.2,0.65)--(1.9,1)--(2.2,1.2)--(3,2)--(4,3)--(4.45,3.4)--(4.5,3.5)--(4.75,3.7)--(5,3.7)--(5.5,3.45)--(6,3)); label("Time (s)", (7.5,0),S); label("Velocity (m/s)",(-1,3),W); [/asy] $ \textbf{(A)}\ 0.2\text{ m/s}^2\qquad\textbf{(B)}\ 0.33\text{ m/s}^2\qquad\textbf{(C)}\ 1.0\text{ m/s}^2\qquad\textbf{(D)}\ 9.8\text{ m/s}^2\qquad\textbf{(E)}\ 30\text{ m/s}^2 $

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively. If $\Gamma $ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma $.