Find the greatest possible integer value of the side length of an equilateral triangle whose vertices belong to the interior region of a square with side length $100$.

The [i]subnumbers[/i] of an integer $n$ are the numbers that can be formed by using a contiguous subsequence of the digits. For example, the subnumbers of 135 are 1, 3, 5, 13, 35, and 135. Compute the number of primes less than 1,000,000,000 that have no non-prime subnumbers. One such number is 37, because 3, 7, and 37 are prime, but 135 is not one, because the subnumbers 1, 35, and 135 are not prime. [i]Proposed by Lewis Chen[/i]

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$, with $AB=6$, $BC=7$, $CD=8$. Find $AD$.

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$. The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$, and $F$ is the midpoint of $TN$. Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$, respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$, and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$, respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

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 $\triangle AED$.