Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

A charged particle with charge $q$ and mass $m$ is given an initial kinetic energy $K_0$ at the middle of a uniformly charged spherical region of total charge $Q$ and radius $R$. $q$ and $Q$ have opposite signs. The spherically charged region is not free to move. Throughout this problem consider electrostatic forces only. [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); size(100); filldraw(circle((0,0),1),gray(.8)); draw((0,0)--(0.5,sqrt(3)/2),EndArrow); label("$R$",(0.25,sqrt(3)/4),SE); [/asy] (a) Find the value of $K_0$ such that the particle will just reach the boundary of the spherically charged region. (b) How much time does it take for the particle to reach the boundary of the region if it starts with the kinetic energy $K_0$ found in part (a)?

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

Let $ABC$ be triangle with $A$-excenter $J$. The reflection of $J$ in $BC$ is $K$. The points $E$ and $F$ are on $BJ, CJ$ such that $\angle EAB=\angle CAF=90^{\circ}$. Prove that $\angle FKE+\angle FJE=180^{\circ}$.

A needle (a segment) lies on a plane. One can rotate it $45^{\circ}$ round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?

Denote by $F_0(x)$, $F_1(x)$, $\ldots$ the sequence of Fibonacci polynomials, which satisfy the recurrence $F_0(x)=1$, $F_1(x)=x$, and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$ for all $n\geq 2$. It is given that there exist unique integers $\lambda_0$, $\lambda_1$, $\ldots$, $\lambda_{1000}$ such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real $x$. For which integer $k$ is $|\lambda_k|$ maximized?

Metadieu, Tercieu and Quartieu are three bodybuilder warriors who fight against an $n$-headed monster. Each of them can attack the monster according to the following rules: (1) Metadieu's attack consists of cutting off half of the monster's heads, then cutting off one more head. If the monster's number of heads is odd, Metadieu cannot attack; (2) Tercieu's attack consists of cutting off a third of the monster's heads, then cutting off two more heads. If the monster's number of heads is not a multiple of 3, Tercieu cannot attack; (3) Quartieu's attack consists of cutting off a quarter of the monster's heads, then cutting off three more heads. If the monster's number of heads is not a multiple of 4, Quartieu cannot attack; If none of the three warriors can attack the monster at some point, then it will devour our three heroes. The objective of the three warriors is to defeat the monster, and for that they need to cut off all its heads, one warrior attacking at a time. For what positive integer values of $n$ is it possible for the three warriors to combine a sequence of attacks in order to defeat the monster?

Decide if the vertices of a regular $30$-gon can be numbered by numbers $1, 2,.., 30$ in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.

Let $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Determine the number of positive integers $2\leq n\leq 50$ such that all coefficients of the polynomial \[ \left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1\leq k\leq n\\\gcd(k,n) = 1}}(x-k) \] are divisible by $n$.

Starting from a positive integer $n$, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of $n$, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times. There is a nice solution to this...

Two positive whole numbers differ by $3$. The sum of their squares is $117$. Find the larger of the two numbers.

Let $ABC$ be a triangle contained in one of the halfplanes determined by a line $r$. Points $A',B',C'$ are the reflections of $A,B,C$ in $r,$ respectively. Consider the line through $A'$ parallel to $BC$, the line through $B'$ parallel to $AC$ and the line through $C'$ parallel to $AB$. Show that these three lines have a common point.

Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride? $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$

Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying \[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\] Show that for any real number $\alpha$, we have \[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\] Here $[\beta]$ denotes the greastest integer not larger than $\beta$.

In an isosceles triangle \(ABC\) with \(AB = AC\), \(BK\) is the altitude and \(H\) is the orthocenter. On the side \(AB\), a point \(N\) is chosen such that \(AN = HN\). Prove that the circumcircles of triangles \(BCK\) and \(ABH\) have a common point on the line \(KN\). [i]Proposed by Fedir Yudin[/i]

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]

Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral. Nguyễn Văn Linh

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

$n$ volleyball teams have competed to each other (each $2$ teams have competed exactly $1$ time.). For every $2$ distinct teams like $A,B$, there exist exactly $t$ teams which have lost their match with $A,B$. Prove that $n=4t+3$. (Notabene that in volleyball, there doesn’t exist tie!)

A sphere $S$ with center $O$ and radius $R$ is given. Let $P$ be a fixed point on this sphere. Points $A,B,C$ move on the sphere $S$ such that we have $\angle APB = \angle BPC = \angle CPA = 90^\circ.$ Prove that the plane of triangle $ABC$ passes through a fixed point.

Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$. Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$. Proposed by [i]Alireza Dadgarnia[/i]

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that $$a_{n+1}=1+\frac{n}{a_n}$$ Find $n$ such that $2020\le a_n <2021$