Prove that for any prime $p,$ there exists a positive integer $n$ such that \[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\] [i]Robin Son[/i]

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

A $9\times9$ checkerboard is colored with 2 colors. If we choose any $3\times1$ region on the checkerboard we can paint all of the squares in that region with the color that is in the majority in that region. Show that with a finite number of these operations, we can paint the checkerboard all in the same color.

The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$ (A.Zaslavsky, A.Akopjan)

The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.

Let $ABC$ be an isosceles triangle with $\angle BAC = 108^o$. The angle bisector of the $\angle ABC$ intersects the circumcircle of a triangle $ABC$ at the point $D$. Let $E$ be a point on segment $CB$ such that $AB =BE$. Prove that the perpendicular bisector of $CD$ is tangent to circumcircle of triangle $ABE$ . (Bohdan Zheliabovskyi)

Consider triangle $ABC$ with side lengths $AB = 4$, $BC = 7$, and $AC = 8$. Let $M$ be the midpoint of segment $AB$, and let $N$ be the point on the interior of segment $AC$ that also lies on the circumcircle of triangle $MBC$. Compute $BN$.

Hexagon $ABCDEF$ has an inscribed circle $\Omega$ that is tangent to each of its sides. If $AB = 12$, $\angle FAB = 120^o$, and $\angle ABC = 150^o$, and if the radius of $\Omega$ can be written as $m +\sqrt{n}$ for positive integers $m, n$, find $m + n$.

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$ Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$

Let $H$ be a complex Hilbert space. Let $T:H\to H$ be a bounded linear operator such that $|(Tx,x)|\le\lVert x\rVert^2$ for each $x\in H$. Assume that $\mu\in\mathbb C$, $|\mu|=1$, is an eigenvalue with the corresponding eigenspace $E=\{\phi\in H:T\phi=\mu\phi\}$. Prove that the orthogonal complement $E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\}$ of $E$ is $T$-invariant, i.e., $T(E^\perp)\subseteq E^\perp$.

Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$

Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.

Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below. [img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img] After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy? [i]Proposed by Bogdan Rublov[/i]

Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

On side $AE$ of regular pentagon $ABCDE$ there is an equilateral triangle $AEF$, and on side $AB$ of the pentagon there is a square $ABHG$ as shown. Find the degree measure of angle $AFG$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/0d689d2665e67c9f9afdf193fb0a2db6dddb3d.png[/img]

Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$. Let line $g_1$ through $A_1$ be perpendicular to $A_1M$; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$, $g_3$ and $g_4$, resp. $g_4$ and $g_1$. Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$. [asy] import graph; size(15cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.773972777861085,xmax=12.231603726660566,ymin=-3.9255487671791487,ymax=7.37238601960895; pair M=(2.,2.), A_4=(-1.6391623316400197,1.2875505916864178), A_1=(3.068893183992864,-0.5728665455336459), A_2=(4.30385937824148,2.2922812065339455), A_3=(2.221541124684679,4.978916319940133), B_4=(-0.9482172571022687,-2.24176848577888), B_1=(4.5873184669543345,0.057960746374459436), B_2=(3.9796042717514277,4.848169684238838), B_3=(-2.4295496490492385,5.324816563638236); draw(circle(M,2.),linewidth(0.8)); draw(A_4--A_1,linewidth(0.8)); draw(A_1--A_2,linewidth(0.8)); draw(A_2--A_3,linewidth(0.8)); draw(A_3--A_4,linewidth(0.8)); draw(M--A_3,linewidth(0.8)+dotted); draw(M--A_2,linewidth(0.8)+dotted); draw(M--A_1,linewidth(0.8)+dotted); draw(M--A_4,linewidth(0.8)+dotted); draw((xmin,-0.07436970390935019*xmin+5.144131675605378)--(xmax,-0.07436970390935019*xmax+5.144131675605378),linewidth(0.8)); draw((xmin,-7.882338401302275*xmin+36.2167572574517)--(xmax,-7.882338401302275*xmax+36.2167572574517),linewidth(0.8)); draw((xmin,0.4154483588930812*xmin-1.847833182441644)--(xmax,0.4154483588930812*xmax-1.847833182441644),linewidth(0.8)); draw((xmin,-5.107958950031516*xmin-7.085223310768749)--(xmax,-5.107958950031516*xmax-7.085223310768749),linewidth(0.8)); dot(M,linewidth(3.pt)+ds); label("$M$",(2.0593440948136896,2.0872038897020024),NE*lsf); dot(A_4,linewidth(3.pt)+ds); label("$A_4$",(-2.6355449660387147,1.085078446888477),NE*lsf); dot(A_1,linewidth(3.pt)+ds); label("$A_1$",(3.1575637581709772,-1.2486383377457595),NE*lsf); dot(A_2,linewidth(3.pt)+ds); label("$A_2$",(4.502882845783654,2.30684782237346),NE*lsf); dot(A_3,linewidth(3.pt)+ds); label("$A_3$",(2.169166061149418,5.203402184478307),NE*lsf); label("$g_3$",(-9.691606303109287,5.354407388189934),NE*lsf); label("$g_2$",(3.0889250292111465,6.727181967386543),NE*lsf); label("$g_1$",(-4.763345563793459,-3.4725331560442676),NE*lsf); label("$g_4$",(-2.663000457622647,6.878187171098171),NE*lsf); dot(B_4,linewidth(3.pt)+ds); label("$B_4$",(-1.5647807942653595,-3.0332452907013523),NE*lsf); dot(B_1,linewidth(3.pt)+ds); label("$B_1$",(4.955898456918535,-0.6583452686912173),NE*lsf); dot(B_2,linewidth(3.pt)+ds); label("$B_2$",(4.104778217816637,5.0661247265586455),NE*lsf); dot(B_3,linewidth(3.pt)+ds); label("$B_3$",(-3.4454819677647146,5.656417795613188),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

In a circle with a radius of $1$ is an inscribed hexagon (convex). Prove that if the multiple of all diagonals that connects vertices of neighboring sides is equal to $27$ then all angles of hexagon are equals. [i]V. Petkov, I. Tonov[/i]

In the triangle $ABC$, the medians $BB_1$ and $CC_1$, which intersect at the point $M$, are drawn. Prove that a circle can be inscribed in the quadrilateral $AC_1MB_1$ if and only if $AB = AC$.

Find all positive integers $m$ such that $$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^\text{th}$-graders to $6^\text{th}$-graders is $5:3$, and the the ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $8:5$. What is the smallest number of students that could be participating in the project? $\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 79 \qquad \textbf{(E)}\ 89$

Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.