In $\vartriangle ABC$, $AB = 13$, $BC = 14,$ and $C A = 15$. Let $E$ and $F$ be the feet of the altitudes from $B$ onto $C A$, and $C$ onto $AB$, respectively. A line $\ell$ is parallel to $EF$ and tangent to the circumcircle of $ABC$ on minor arc $BC$. Let $X$ and $Y$ be the intersections of $\ell$ with $AB$ and $AC$ respectively. Find $X Y$ .

Let $R$ be the rectangle in the coordinate plane with corners $(0, 0)$, $(20, 0)$, $(20, 22)$, and $(0, 22)$, and partition $R$ into a $20\times 22$ grid of unit squares. For a given line in the coordinate plane, let its [i]pixelation[/i] be the set of grid squares in $R$ that contain part of the line in their interior. If $P$ is a point chosen uniformly at random in $R$, then compute the expected number of sets of grid squares that are pixelations of some line through $P$.

Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.

Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .

The Order "For Faithful Service" of the $7$th degree in shape is a combination of a semicircle with a diameter $AB = 2$ and a triangle $AM B$. The sides$ AM$ and $BM$ intersect the semicircle (the border of the semicircle). The part of the circle outside the triangle and the part of the triangle outside the circle are made of pure copper. What should the side of the triangle be equal to in order for the area of the copper part to be the smallest?

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6$. The number of possible sets of $6$ cards that can be drawn from the deck is $6$ times the number of possible sets of $3$ cards that can be drawn. Find $n$.

Let $ABCD$ be a convex quadrilateral such that $\angle{BAD} = 90^{\circ}$ and its diagonals $AC$ and $BD$ are perpendicular. Let $M$ be the midpoint of side $CD$, and $E$ be the intersection of $BM$ and $AC$. Let $F$ be a point on side $AD$ such that $BM$ and $EF$ are perpendicular. If $CE = AF\sqrt{2}$ and $FD = CE\sqrt{2}$, show that $ABCD$ is a square.

The evil Dr. Doom seeks to destroy his enemy, the Intergalactic Federation, and has devised a plan to despin the Federation's space station. The hoop-shaped space station of mass $M$ and radius $R$ rotates once every $T$ hours to maintain artificial gravity equal to that on IPhOO. Dr. Doom plans to mount two thruster rockets, one rocket on opposite sides of the space station, to stop its rotation. Dr. Doom must accomplish his crime within a time $t$ to avoid getting caught. How much force should each rocket deliver in order to despin the Federation's space station in $t$? Express your answer in terms of $M$, $R$, $T$, $t$, and/or constants, as necessary. [i]Problem proposed by Kimberly Geddes[/i]

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