Let $A,\ B$ and $C$ be given points on a circumference $K$ such that the triangle $\triangle{ABC}$ is acute. Let $P$ be a point in the interior of $K$. $X,\ Y$ and $Z$ be the other intersection of $AP, BP$ and $CP$ with the circumference. Determine the position of $P$ such that $\triangle{XYZ}$ is equilateral.

Let $\vartriangle ABC$ be an acute triangle with circumcircle $\Omega$. A line passing through $A$ perpendicular to $BC$ meets $\Omega$ again at $D$. Draw two circles $\omega_b$, $\omega_c$ with $B, C$ as centers and $BD$, $CD$ as radii, respectively, and they intersect $AB$, $AC$ at $E, F,$ respectively. Let $K\ne A$ be the second intersection of $(AEF)$ and $\Omega$, and let $\omega_b$, $\omega_c$ intersect $KB$, $KC$ at $P, Q$, respectively. The circumcenter of triangle $DP Q$ is $O$, prove that $K, O, D$ are collinear. [i]proposed by Li4[/i]

Given a tetrahedron $PABC$, draw the height $PH$ from vertex $P$ to $ABC$. From point $H$, draw perpendiculars $HA’,HB’,HC’$ to the lines $PA,PB,PC$. Suppose the planes $ABC$ and $A’B’C’$ intersects at line $\ell$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $OH\perp \ell$.

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? $\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$,$ ...$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$

Consider a rectangular sheet of paper $ABCD$ such that the lengths of $AB$ and $AD$ are respectively $7$ and $3$ centimetres. Suppose that $B'$ and $D'$ are two points on $AB$ and $AD$ respectively such that if the paper is folded along $B'D'$ then $A$ falls on $A'$ on the side $DC$. Determine the maximum possible area of the triangle $AB'D'$.

Let $\triangle ABC$ satisfy $AB = 17, AC = \frac{70}{3}$ and $BC = 19$. Let $I$ be the incenter of $\triangle ABC$ and $E$ be the excenter of $\triangle ABC$ opposite $A$. (Note: this means that the circle tangent to ray $AB$ beyond $B$, ray $AC$ beyond $C$, and side $BC$ is centered at $E$.) Suppose the circle with diameter $IE$ intersects $AB$ beyond $B$ at $D$. If $BD = \frac{a}{b}$ where $a, b$ are coprime positive integers, find $a + b$.

[b]p1.[/b] Let $ABCD$ be a square with side length $1$, $\Gamma_1$ be a circle centered at $B$ with radius 1, $\Gamma_2$ be a circle centered at $D$ with radius $1$, $E$ be a point on the segment $AB$ with $|AE| = x$ $(0 < x \le 1)$, and $\Gamma_3$ be a circle centered at $A$ with radius $|AE|$. $\Gamma_3$ intersects $\Gamma_1$ and $\Gamma_2$ inside the square at $G$ and $F$, respectively. Let region $I$ be the region bounded by the segment $GC$ and the minor arc $GC$ of $\Gamma_1$, and region II be the region bounded by the segment $FG$ and the minor arc $FG$ of $\Gamma_3$, as illustrated in the graph below. Let $r(x)$ be the ratio of the area of region I to the area of region II. (i) Find $r(1)$. Justify your answer. (ii) Find an explicit formula of $r(x)$ in terms of $x$ $(0 < x \le 1)$. Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/0/bd2379a1390a578d78dc7e9f4cde756d5f4723.png[/img] [b]p2.[/b] We call a [i]party [/i] any set of people $V$ . If $v_1 \in V$ knows $v_2 \in V$ in a party, we always assume that $v_2$ also knows $v_1$. For a person $v \in V$ in some party, the degree of v, denoted by $deg\,\,(v)$, is the number of people $v$ knows in the party. (i) Suppose that a party has four people with $V = \{v_1, v_2, v_3, v_4\}$, and that $deg\,\,(v_i) = i$ for $i = 1, 2, 3$ show that $deg\,\,(v_4) = 2$. (ii) Suppose that a party is attended by $n = 4k$ ($k \ge 1$) people with $V = \{v_1, v_2, ..., v_{4k}\}$, and that $deg\,\,(v_i) = i$ for $1 \le i \le n - 1$. Show that $deg\,\,(v_n) = \frac{n}{2}$ . [b]p3.[/b] Let $a, b$ be two real number parameters and consider the function $f(x) =\frac{b + \sin x}{a + \cos x}$. (i) Find an example of $(a, b)$ such that $f(x) \ge 2$ for all real numbers $x$. Justify your answer. (ii) If $a > 1$ and the range of the function $f(x)$ (when x varies over the set of all real numbers) is $[-1, 1]$, find the values of $a$ and $b$. Justify your answer. [b]p4.[/b] Let $f$ be the function that assigns to each positive multiple $x$ of $8$ the number of ways in which $x$ can be written as a difference of squares of positive odd integers. (For example, $f(8) = 1$, because $8 = 3^2 -1^2$, and $f(24) = 2$, because $24 = 5^2 - 1^2 = 7^2 - 5^2$.) (a) Determine with proof the value of $f(120)$. (b) Determine with proof the smallest value $x$ for which $f(x) = 8$. (c) Show that the range of this function is the set of all positive integers. [b]p5.[/b] Consider the binomial coefficients $C_{n,r} ={n \choose r}= \frac{n!}{r!(n - r)!}$, for $n \ge 2$. Prove that $C_{n,r}$ are even, for all $1 \le r \le n - 1$, if and only if $n = 2^m$, for some counting number $m$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The tangent to $\omega$ through $B$ cuts the parallel to $BC$ through $A$ at $P$. The line $CP$ cuts the circumcircle of $\triangle ABP$ again in $Q$ and line $AQ$ cuts $\omega$ at $R$. Prove that $BQCR$ is parallelogram if and only if $AC=BC$.

Points $K$ and $L$ are on side $AB$ of triangle $ABC$ such that $KL=BC$ and $AK=LB$. Let $M$ be a midpoint of $AC$. Prove that $\angle KML = 90^{\circ}$

Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.

Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector? (Problem from Leningrad)

Three unit vectors $a,b,c$ are given on the plane. Prove that one can choose the signs in the expression $x=\pm a\pm b\pm c$ so as to obtain a vector $x$ with $|x|\le\sqrt2$.

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

In the regular pentagon $ABCDE$, the perpendicular at $C$ to $CD$ meets $AB$ at $F$. Prove that $AE+AF=BE$. [i]Proposed by Alireza Cheraghi[/i]

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

Quadrilateral $ABCD$ is inscribed in a circle such that the midpoints of its sides also lie on a (different) circle. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{CD}$ respectively, and let $P$ be the foot of the perpendicular from the intersection of $\overline{AC}$ and $\overline{BD}$ onto $\overline{BC}$. If the side lengths of $ABCD$ are $1$, $3$, $\sqrt 2$, and $2\sqrt 2$ in some order, compute the greatest possible area of the circumcircle of triangle $MNP$. [i]Proposed by Connor Gordon[/i]

Let $n$ unit circles be given on a plane. Prove that on one of the circles there is an arc of length at least $\frac{2\pi}n$ not intersecting any other circle.

A line through a vertex of a non-degenerate triangle cuts it in two similar triangles with $\sqrt{3}$ as the ratio between correspondent sides. Find the angles of the given triangle.

Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Let $ U$ be an $ n \times n$ orthogonal matrix. Prove that for any $ n \times n$ matrix $ A$, the matrices \[ A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j\] converge entrywise as $ m \rightarrow \infty.$ [i]L. Kovacs[/i]

Given an $ n\times n$ chessboard. a) Find the number of rectangles on the chessboard. b) Assume there exists an $ r\times r$ square (label $ B$) with $ r<n$ which is located on the upper left corner of the board. Define "inner border" of $ A$ as the border of $ A$ which is not the border of the chessboard. How many rectangles in $ B$ that touch exactly one inner border of $ B$?

$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.