Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors \[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\] What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane? [i]Grigory Chelnokov, Russia[/i]

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$. Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$. Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$. [asy] size(200); pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C); draw(F--C--A--B--C^^A--D^^B--E); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("84", centroid(H, C, E), fontsize(9.5)); label("35", centroid(H, B, D), fontsize(9.5)); label("30", centroid(H, F, B), fontsize(9.5)); label("40", centroid(H, A, F), fontsize(9.5));[/asy]

A bug is walking on the surface of a Rubik's cube(cube $3 \times 3 \times 3$). It can go to the adjacent cell on the same face or on the adjacent face. One day the bug started walking from some cell and returned to it, and visited all other cells exactly once. Prove that he made an even amount of moves that changed the face he is on.

A triangle and a circle are in the same plane. Show that the area of the intersection of the triangle and the circle is at most one third of the area of the triangle plus one half of the area of the circle. [i]Krit Boonsiriseth[/i]

Given a non- isosceles triangle $ABC$. Let the points $B'$ and $C'$ be symmetric to the points $B$ and $C$ wrt $AC$ and $AB$ respectively. If the circles circumscribed around triangles $ABB'$ and $ACC'$ intersect at point $P$, prove that the line $AP$ passes through the center of the circumcircle of the triangle $ABC$.

A polyhedron $P$ is given in space. Find whether there exist three edges in $P$ which can be the sides of a triangle. Justify your answer! [i]Barbu Berceanu[/i]

Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

Let $ABC$ be a right triangle with $AB = BC = 2$. Let $ACD$ be a right triangle with angle $\angle DAC = 30$ degrees and $\angle DCA = 60$ degrees. Given that $ABC$ and $ACD$ do not overlap, what is the area of triangle $BCD$?

[b]p1.[/b] Given an equilateral triangle $ABC$ with area $16\sqrt3$, and an interior point $P$ with distances from vertices $|AP| = 4$ and $|BP| = 6$. (a) Find the length of each side. (b) Find the distance from point $P$ to the side $AB$. (c) Find the distance $|PC|$. [b]p2.[/b] Several players play the following game. They form a circle and each in turn tosses a fair coin. If the coin comes up heads, that player drops out of the game and the circle becomes smaller, if it comes up tails that player remains in the game until his or her next turn to toss. When only one player is left, he or she is the winner. For convenience let us name them $A$ (who tosses first), $B$ (second), $C$ (third, if there is a third), etc. (a) If there are only two players, what is the probability that $A$ (the first) wins? (b) If there are exactly $3$ players, what is the probability that $A$ (the first) wins? (c) If there are exactly $3$ players, what is the probability that $B$ (the second) wins? [b]p3.[/b] A circular castle of radius $r$ is surrounded by a circular moat of width $m$ ($m$ is the shortest distance from each point of the castle wall to its nearest point on shore outside the moat). Life guards are to be placed around the outer edge of the moat, so that at least one life guard can see anyone swimming in the moat. (a) If the radius $r$ is $140$ feet and there are only $3$ life guards available, what is the minimum possible width of moat they can watch? (b) Find the minimum number of life guards needed as a function of $r$ and $m$. [img]https://cdn.artofproblemsolving.com/attachments/a/8/d7ff0e1227f9dcf7e49fe770f7dae928581943.png[/img] [b]p4.[/b] (a)Find all linear (first degree or less) polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all linear polynomials $g(x)$. (b) Prove your answer to part (a). (c) Find all polynomials $f(x)$ with the property that $f(g(x)) = g(f(x))$ for all polynomials $g(x)$. (d) Prove your answer to part (c). [b]p5.[/b] A non-empty set $B$ of integers has the following two properties: i. each number $x$ in the set can be written as a sum $x = y+ z$ for some $y$ and $z$ in the set $B$. (Warning: $y$ and $z$ may or may not be distinct for a given $x$.) ii. the number $0$ can not be written as a sum $0 = y + z$ for any $y$ and $z$ in the set $B$. (a) Find such a set $B$ with exactly $6$ elements. (b) Find such a set $B$ with exactly $6$ elements, and such that the sum of all the $6$ elements is $1988$. (c) What is the smallest possible size of such a set $B$ ? (d) Prove your answer to part (c). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)? (T. Kazitsyna)

Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.

Let $ ABC$ be an obtuse inscribed in a circle of radius $ 1$. Prove that $ \triangle ABC$ can be covered by an isosceles right-angled triangle with hypotenuse of length $ \sqrt {2} \plus{} 1$.

[u]Part 1[/u] [b]p1.[/b] Two kids $A$ and $B$ play a game as follows: From a box containing $n$ marbles ($n > 1$), they alternately take some marbles for themselves, such that: 1. $A$ goes first. 2. The number of marbles taken by $A$ in his first turn, denoted by $k$, must be between $1$ and $n$, inclusive. 3. The number of marbles taken in a turn by any player must be between $1$ and $k$, inclusive. The winner is the one who takes the last marble. What is the sum of all $n$ for which $B$ has a winning strategy? [b]p2.[/b] How many ways can your rearrange the letters of "Alejandro" such that it contains exactly one pair of adjacent vowels? [b]p3.[/b] Assuming real values for $p, q, r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $q + 6i$, and the product of the other two roots is $3 - 4i$. Find the smallest value of $q$. [b]p4.[/b] Lisa has a $3$D box that is $48$ units long, $140$ units high, and $126$ units wide. She shines a laser beam into the box through one of the corners, at a $45^o$ angle with respect to all of the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the eight corners of the box. [u]Part 2[/u] [b]p5.[/b] How many ways can you divide a heptagon into five non-overlapping triangles such that the vertices of the triangles are vertices of the heptagon? [b]p6.[/b] Let $a$ be the greatest root of $y = x^3 + 7x^2 - 14x - 48$. Let $b$ be the number of ways to pick a group of $a$ people out of a collection of $a^2$ people. Find $\frac{b}{2}$ . [b]p7.[/b] Consider the equation $$1 -\frac{1}{d}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ with $a, b, c$, and $d$ being positive integers. What is the largest value for $d$? [b]p8.[/b] The number of non-negative integers $x_1, x_2,..., x_{12}$ such that $$x_1 + x_2 + ... + x_{12} \le 17$$ can be expressed in the form ${a \choose b}$ , where $2b \le a$. Find $a + b$. [u]Part 3[/u] [b]p9.[/b] In the diagram below, $AB$ is tangent to circle $O$. Given that $AC = 15$, $AB = 27/2$, and $BD = 243/34$, compute the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/f/b403e5e188916ac4fb1b0ba74adb7f1e50e86a.png[/img] [b]p10.[/b] If $$\left[2^{\log x}\right]^{[x^{\log 2}]^{[2^{\log x}]...}}= 2, $$ where $\log x$ is the base-$10$ logarithm of $x$, then it follows that $x =\sqrt{n}$. Compute $n^2$. [b]p11.[/b] [b]p12.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5, $$ where $n$ is an integer less than $170$. [u]Part 4[/u] [b]p13.[/b] Let $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Define $f(n)$ as the number of distinct two-digit integers that can be formed from digits in $n$. For example, $f(15) = 4$ because the integers $11$, $15$, $51$, $55$ can be formed from digits of $15$. Let $w$ be such that $f(3xz - w) = w$. Find $w$. [b]p14.[/b] Let $w$ be the answer to number $13$ and $z$ be the answer to number $16$. Let $x$ be such that the coefficient of $a^xb^x$ in $(a + b)^{2x}$ is $5z^2 + 2w - 1$. Find $x$. [b]p15.[/b] Let $w$ be the answer to number $13$, $x$ be the answer to number $14$, and $z$ be the answer to number $16$. Let $A$, $B$, $C$, $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. Now, let $AE = 3x$, $ED = w^2 - w + 1$, and $AD = 2z$. If $FG = y$, find $y$. [b]p16.[/b] Let $w$ be the answer to number $13$, and $x$ be the answer to number $16$. Let $z$ be the number of integers $n$ in the set $S = \{w,w + 1, ... ,16x - 1, 16x\}$ such that $n^2 + n^3$ is a perfect square. Find $z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that on a coordinate plane it is impossible to draw a closed broken line such that [list][*] coordinates of each vertex are rational, [*] the length of its every edge is equal to $1$, [*] the line has an odd number of vertices.[/list]

Let $P$ and $Q$ be points inside a triangle $ABC$ such that $\angle PAC = \angle QAB$ and $\angle PBC = \angle QBA$. Let $D$ and $E$ be the feet of the perpendiculars from $P$ to the lines $BC$ and $AC$, and $F$ be the foot of perpendicular from $Q$ to the line $AB$. Let $M$ be intersection of the lines $DE$ and $AB$. Prove that $MP \perp CF$

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)

A tetrahedron is a polyhedron composed of four triangular faces. Faces $ABC$ and $BCD$ of a tetrahedron $ABCD$ meet at an angle of $\pi/6$. The area of triangle $\triangle ABC$ is $120$. The area of triangle $\triangle BCD$ is $80$, and $BC = 10$. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$. [asy] size(200); pathpen=black+linewidth(0.65);pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.

Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that (1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]