Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.

An unfair coin, which has the probability of $a\ \left(0<a<\frac 12\right)$ for showing $Heads$ and $1-a$ for showing $Tails$, is flipped $n\geq 2$ times. After $n$-th trial, denote by $A_n$ the event that heads are showing on at least two times and by$B_n$ the event that are not showing in the order of $tails\rightarrow heads$, until the trials $T_1,\ T_2,\ \cdots ,\ T_n$ will be finished . Answer the following questions: (1) Find the probabilities $P(A_n),\ P(B_n)$. (2) Find the probability $P(A_n\cap B_n )$. (3) Find the limit $\lim_{n\to\infty} \frac{P(A_n) P(B_n)}{P(A_n\cap B_n )}.$ You may use $\lim_{n\to\infty} nr^n=0\ (0<r<1).$

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$

Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[f(\alpha f(x)+f(y))=\beta x+f(y)\] holds for all real $x$ and $y$. [i](Walther Janous)[/i]

We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals. [i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]

Are there $10$ consecutive positive integers, such that if we consider the digits that appear in the decimal representations of those numbers as a multiset, every digit appears the same number of times in this multiset?

A single segment contains several non-intersecting red segments, the total length of which is greater than $0.5$. Are there necessarily two red dots at the distance: a) $1/99$ b) $1/100$ ?

Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties: (a) $0$ is not in $S$; (b) for each $s_1,s_2$ in $S$, the rational number $s_1/s_2$ is in $S$; (c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$. Prove that if $x$ belongs to $S$, then there exists elements $y,z$ in $S$ such that $x=y+z$.

Prove that the triangle ABC is right-angled if it holds: \[ \sin^2 A+\sin^2 B+\sin^2 C = 2 \]

Let $E$ be the midpoint of the side $[BC]$ of $\triangle ABC$ with $|AB|=7$, $|BC|=6$, and $|AC|=5$. The line, which passes through $E$ and is perpendicular to the angle bisector of $\angle A$, intersects $AB$ at $D$. What is $|AD|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ \frac 92 \qquad\textbf{(D)}\ 3\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$.

Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$, and let $\ell_1$ and $\ell_2$ be two lines through $T$. The lines $\ell_1$ and $\ell_2$ meet again $\gamma_1$ at points $A$ and $B$, respectively, and $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let further $X$ be a point in the complement of $\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2$. The circles $ATX$ and $BTX$ meet again $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that the lines $TX$, $A_1B_2$ and $A_2B_1$ are concurrent. [i]***[/i]

The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square. [asy] size(130); real r = sqrt(2)/2; defaultpen(linewidth(0.8)); draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r)); [/asy]

The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter

How many ordered sequences of $36$ digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from $0$ to $9$.)

Let $ANC$, $CLB$ and $BKA$ be triangles erected on the outside of the triangle $ABC$ such that $\angle NAC = \angle KBA = \angle LCB$ and $\angle NCA = \angle KAB = \angle LBC$. Let $D$, $E$, $G$ and $H$ be the midpoints of $AB$, $LK$, $CA$ and $NA$ respectively. Prove that $DEGH$ is a parallelogram.

(a) In this drawing, there are three squares on each side of the square. Place a natural number in each of the boxes so that the sum of the numbers of two adjacent boxes is always odd. [img]https://cdn.artofproblemsolving.com/attachments/e/6/75517b7d49857abd3f8f0430a70ae5b0eb1554.gif[/img] (b) In this drawing, there are now four squares on each side of the triangle. Justify why a natural number cannot be placed in each box so that the sum of the numbers in two adjacent boxes is always odd. [img] https://cdn.artofproblemsolving.com/attachments/c/8/061895b9c1cdcb132f7d37087873b7de3fb5f3.gif[/img] (c) If you now draw a polygon with$ 51$ sides and on each side you place $50$ boxes, taking care that there is a box at each vertex. Can you place a natural number in each box so that the sum of the numbers in two adjacent boxes is always odd? Why?

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$. Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.

Shaft section of a circular cone with vertex $P$ is an isosceles right triangle. $A$ is a point on the circle of the bottom surface, while $B$ is a point inside the circle, $O$ is the center of the circle. If $AB\perp OB$ at $B$, $OH\perp PB$ at $H$, $PA=4$, $C$ is the midpoint of $PA$, then when the volume of triangular pyramid $O-HPC$ takes its maximum value, the length of $OB$ is $\text{(A)}\frac{\sqrt5}{3}\qquad\text{(B)}\frac{2\sqrt5}{3}\qquad\text{(C)}\frac{\sqrt6}{3}\qquad\text{(D)}\frac{2\sqrt6}{3}\qquad$

[b]p1.[/b] Warren interrogates the $25$ members of his cabinet, each of whom always lies or always tells the truth. He asks them all, “How many of you always lie?” He receives every integer answer from $1$ to $25$ exactly once. Find the actual number of liars in his cabinet. [b]p2.[/b] Abraham thinks of distinct nonzero digits $E$, $M$, and $C$ such that $E +M = \overline{CC}$. Help him evaluate the sum of the two digit numbers $\overline{EC}$ and $\overline{MC}$. (Note that $\overline{CC}$, $\overline{EC}$, and $\overline{MC}$ are read as two-digit numbers.) [b]p3.[/b] Let $\omega$, $\Omega$, $\Gamma$ be concentric circles such that $\Gamma$ is inside $\Omega$ and $\Omega$ is inside $\omega$. Points $A,B,C$ on $\omega$ and $D,E$ on $\Omega$ are chosen such that line $AB$ is tangent to $\Omega$, line $AC$ is tangent to $\Gamma$, and line $DE$ is tangent to $\Gamma$. If $AB = 21$ and $AC = 29$, find $DE$. [b]p4.[/b] Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes. [b]p5.[/b] At Stillwells Ice Cream, customers must choose one type of scoop and two different types of toppings. There are currently $630$ different combinations a customer could order. If another topping is added to the menu, there would be $840$ different combinations. If, instead, another type of scoop were added to the menu, compute the number of different combinations there would be. [b]p6.[/b] Eleanor the ant takes a path from $(0, 0)$ to $(20, 24)$, traveling either one unit right or one unit up each second. She records every lattice point she passes through, including the starting and ending point. If the sum of all the $x$-coordinates she records is $271$, compute the sum of all the $y$-coordinates. (A lattice point is a point with integer coordinates.) [b]p7.[/b] Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them. [b]p8.[/b] Michelle is playing Spot-It with a magical deck of $10$ cards. Each card has $10$ distinct symbols on it, and every pair of cards shares exactly $1$ symbol. Find the minimum number of distinct symbols on all of the cards in total. [b]p9.[/b] Define the function $f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + ...$ for integers $n \ge 2$. Find $$f(2) + f(4) + f(6) + ... .$$ [b]p10.[/b] There are $9$ indistinguishable ants standing on a $3\times 3$ square grid. Each ant is standing on exactly one square. Compute the number of different ways the ants can stand so that no column or row contains more than $3$ ants. [b]p11.[/b] Let $s(N)$ denote the sum of the digits of $N$. Compute the sum of all two-digit positive integers $N$ for which $s(N^2) = s(N)^2$. [b]p12.[/b] Martha has two square sheets of paper, $A$ and $B$. With each sheet, she repeats the following process four times: fold bottom side to top side, fold right side to left side. With sheet $A$, she then makes a cut from the top left corner to the bottom right. With sheet $B$, she makes a cut from the bottom left corner to the top right. Find the total number of pieces of paper yielded from sheets $A$ and sheets $B$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/ff3a459a135562002aa2c95067f3f01441d626.png[/img] [b]p13.[/b] Let $x$ and $y$ be positive integers such that gcd $(x^y, y^x) = 2^{28}$. Find the sum of all possible values of min $(x, y)$. [b]p14.[/b] Convex hexagon $TRUMAN$ has opposite sides parallel. If each side has length $3$ and the area of this hexagon is $5$, compute $$TU \cdot RM \cdot UA \cdot MN \cdot AT \cdot NR.$$ [b]p15.[/b] Let $x$, $y$, and $z$ be positive real numbers satisfying the system $$\begin{cases} x^2 + xy + y^2 = 25\\ y^2 + yz + z^2 = 36 \\ z^2 + zx + x^2 = 49 \end{cases}$$ Compute $x^2 + y^2 + z^2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ ABCDE$ be a regular pentagon of side length $ 1$. Let $ F$ be the midpoint of $ AB$ and let $ G$ and $ H$ be the points on sides $ CD$ and $ DE$ respectively $ \angle GFD \equal{} \angle HFD \equal{} 30^{\circ}$. Show that the triangle $ GFH$ is equilateral. A square of side $ a$ is inscribed in $ \triangle GFH$ with one side of the square along $ GH$. Prove that: $ FG \equal{} t \equal{} \frac {2 \cos 18^{\circ} \cos^2 36^{\circ}}{\cos 6^{\circ}}$ and $ a \equal{} \frac {t \sqrt {3}}{2 \plus{} \sqrt {3}}$.

Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.