The sides of an equilateral triangle with sides of length $n$ have been divided into equal parts, each of length $1$, and lines have been drawn through the points of division parallel to the sides of the triangle, thus dividing the large triangle into many small triangles. Nils has a pile of rhombic tiles, each of side $1$ and angles $60^\circ$ and $120^\circ$, and wants to tile most of the triangle using these, so that each tile covers two small triangles with no overlap. In the picture, three tiles are placed somewhat arbitrarily as an illustration. How many tiles can Nils fit inside the triangle? [asy] /* original code by fedja: https://artofproblemsolving.com/community/c68h207503p1220868 modified by Klaus-Anton: https://artofproblemsolving.com/community/c2083h3267391_draw_me_a_grid_of_regular_triangles */ size(5cm); int n=6; pair A=(1,0), B=dir(60); path P=A--B--(0,0)--cycle; path Pp=A--shift(A)*B--B--cycle; /* label("$A$",A,S); label("$B$",B,dir(120)); label("$(0,0)$",(0,0),dir(210)); fill(shift(2*A-1+2*B-1)*P,yellow+white); fill(shift(2*A-1+2*B-0)*P,yellow+white); fill(shift(2*A-1+2*B+1)*P,yellow+white); fill(shift(2*A-1+2*B+2)*P,yellow+white); fill(shift(1*A-1+1*B)*P,blue+white); fill(shift(2*A-1+1*B)*P,blue+white); fill(shift(3*A-1+1*B)*P,blue+white); fill(shift(4*A-1+1*B)*P,blue+white); fill(shift(5*A-1+1*B)*P,blue+white); fill(shift(0*A+0*B)*P,green+white); fill(shift(0*A+1+0*B)*P,green+white); fill(shift(0*A+2+0*B)*P,green+white); fill(shift(0*A+3+0*B)*P,green+white); fill(shift(0*A+4+0*B)*P,green+white); fill(shift(0*A+5+0*B)*P,green+white); fill(shift(2*A-1+3*B-1)*P,magenta+white); fill(shift(3*A-1+3*B-1)*P,magenta+white); fill(shift(4*A-1+3*B-1)*P,magenta+white); fill(shift(5*A+5*B-5)*P,heavyred+white); fill(shift(4*A+4*B-4)*P,palered+white); fill(shift(4*A+4*B-3)*P,palered+white); fill(shift(0*A+0*B)*Pp,gray); fill(shift(0*A+1+0*B)*Pp,gray); fill(shift(0*A+2+0*B)*Pp,gray); fill(shift(0*A+3+0*B)*Pp,gray); fill(shift(0*A+4+0*B)*Pp,gray); fill(shift(1*A+1*B-1)*Pp,lightgray); fill(shift(1*A+1*B-0)*Pp,lightgray); fill(shift(1*A+1*B+1)*Pp,lightgray); fill(shift(1*A+1*B+2)*Pp,lightgray); fill(shift(2*A+2*B-2)*Pp,red); fill(shift(2*A+2*B-1)*Pp,red); fill(shift(2*A+2*B-0)*Pp,red); fill(shift(3*A+3*B-2)*Pp,blue); fill(shift(3*A+3*B-3)*Pp,blue); fill(shift(4*A+4*B-4)*Pp,cyan); fill(shift(0*A+1+0*B)*Pp,gray); fill(shift(0*A+2+0*B)*Pp,gray); fill(shift(0*A+3+0*B)*Pp,gray); fill(shift(0*A+4+0*B)*Pp,gray); */ fill(Pp, rgb(244, 215, 158)); fill(shift(dir(60))*P, rgb(244, 215, 158)); fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*Pp, rgb(244, 215, 158)); fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*P, rgb(244, 215, 158)); fill(shift(-.5,(-sqrt(3)/2))*shift(4*dir(60))*Pp, rgb(244, 215, 158)); fill(shift(.5,(-sqrt(3)/2))*shift(4*dir(60))*P, rgb(244, 215, 158)); for(int i=0;i<n;++i){ for(int j;j<n-i;++j) {draw(shift(i*A+j*B)*P);}} shipout(bbox(2mm,Fill(white))); [/asy]

Consider the sets $A = \{0,1,2\},$ and $B = \{1,2,3,4,5\}.$ Find the number of functions $f: A \to B$ such that $x + f(x) + xf(x)$ is odd for all $x.$ (A function $f:A \to B$ is a rule that assigns to every number in $A$ a number in $B.$) \[\mathrm a. ~15\qquad \mathrm b. ~27 \qquad \mathrm c. ~30 \qquad\mathrm d. ~42\qquad\mathrm e. ~45\]

We are given a polygon, a line $l$ and a point $P$ on $l$ in general position: all lines containing a side of the polygon meet $l$ at distinct points diering from $P$. We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the line $l$ on opposite sides of $P$. Show that $P$ lies inside the polygon if and only if on each side of $l$ there are an odd number of marked vertices. [i]O. Musin[/i]

In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 126 $

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

The sides of a convex pentagon are extended on both sides to form five triangles. If these triangles are congruent to one another, does it follow that the pentagon is regular?

Let $ABC$ be an acute triangle with $E, F$ being the reflections of $B,C$ about the line $AC, AB$ respectively. Point $D$ is the intersection of $BF$ and $CE$. If $K$ is the circumcircle of triangle $DEF$, prove that $AK$ is perpendicular to $BC$. Nguyen Minh Ha, College of Pedagogical University of Hanoi

Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$

Prove that $$\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}$$for all $n\in \mathbb{N}^*$

Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and \[f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.\]

Find all pairs of integers $ \left(m,n\right) $ such that $ \left(m,n+1\right) = 1 $ and $$ \sum\limits_{k=1}^{n}{\frac{m^{k+1}}{k+1}\binom{n}{k}} \in \mathbb{N} $$

In some city there are $2000000$ citizens. In every group of $2000$ citizens there are $3$ pairwise friends. Prove, that there are $4$ pairwise friends in city.

If $\alpha,\beta,\gamma$ are the angles of an acute-angled triangle, prove that \[\sin \alpha + \sin \beta > \cos \alpha + \cos\beta  + \cos\gamma.\]

Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.

For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

3. Show that there are infinitely many triples (x,y,z) of integers such that $x^3 + y^4 = z^{31}$.

Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.

Tim has a multiset of positive integers. Let $c_i$ be the number of occurrences of numbers that are [i]at least[/i] $i$ in the multiset. Let $m$ be the maximum element of the multiset. Tim calls a multiset [i]spicy[/i] if $c_1, \dots, c_m$ is a sequence of strictly decreasing powers of $3$. Tim calls the [i]hotness[/i] of a spicy multiset the sum of its elements. Find the sum of the hotness of all spicy multisets that satisfy $c_1 = 3^{2020}$. Give your answer $\pmod{1000}$. (Note: a multiset is an unordered set of numbers that can have repeats) [i]Proposed by Timothy Qian[/i]

Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$

A group $ G $ of order at least $ 4 $ has the property that there exists a natural number $ n\not\in\{ 1,|G| \} $ such that $ G $ admits exactly $ \binom{|G|-1}{n-1} $ subgroups of order $ n. $ Show that $ G $ is commutative. [i]Marius Tărnăuceanu[/i]

One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt. Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate. Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost. [i](4 points)[/i]

Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.