An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

Inside a $2\times 2$ square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a $1\times 2$ rectangle.

In a triangle $ABC$, $\angle B = 70^o$, $\angle C = 50^o$. A point $M$ is taken on the side $AB$ such that $\angle MCB = 40^o$ , and a point $N$ is taken on the side $AC$ such that $\angle NBC = 50^o$. Find $\angle NMC$.

Let $\Gamma$ be the circumscribed circle of a triangle $ABC$ and let $D$ be a point at line segment $BC$. The circle passing through $B$ and $D$ tangent to $\Gamma$ and the circle passing through $C $and $D$ tangent to $\Gamma$ intersect at a point $E \ne D$. The line $DE$ intersects $\Gamma$ at two points $X$ and $Y$ . Prove that $|EX| = |EY|$.

Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$. (a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi. (b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$ [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=286[/img]

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

Let $ABC$ be an equilateral triangle. Let $Q$ be a random point on $BC$, and let $P$ be the meeting point of $AQ$ and the circumscribed circle of $\triangle ABC$. Prove that $\frac{1}{PQ}=\frac{1}{PB}+\frac{1}{PC}$.

Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle. If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.

Meghal is playing a game with $2016$ rounds $1, 2, ..., 201$6. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2\pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?

Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$. If the $(EFGH) = 1$. Find: $(ABCD) \times (XYZW)$ Note $(P)$ denote the area of the polygon $P$.

$ABC$ is an acute triangle. Points $M$ and $K$ on side $AC$ are such that $\angle ABM = \angle CBK$. Prove that the circumcenters of triangles $ABM$, $ABK$, $CBM$ and $CBK$ are concyclic. (Author: T. Emelyanova)

[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].

Excircle of $ABC$ is tangent to the side $BC$ at point $K$ and is tangent to the extension of $AB$ at point $L$. Another excircle is tangent to extensions of sides $AB$ and $BC$ at points $M$ and $N$. Lines $KL$ and $MN$ intersect at point $X$. Prove that $CX$ is the bisector of angle $ACN$. [I]Proposed by S. Berlov[/i]

Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$

Let $M, N, P$ be the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Let $X$ be a fixed point inside the triangle $MNP$. The lines $L_1, L_2, L_3$ that pass through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2$; $L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2$; $L_3$ intersects segment $CA$ at point $B_1$ and segment $CB$ at point $A_2$. Indicates how to construct the lines $L_1, L_2, L_3$ in such a way that the sum of the areas of the triangles $A_1A_2X, B_1B_2X$ and $C_1C_2X$ is a minimum.

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.

A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

Let ABCD be a quadrilateral; X and Y be the midpoints of AC and BD respectively and lines through X and Y respectively parallel to BD, AC meet in O. Let P,Q,R,S be the midpoints of AB, BC, CD, DA respectively. Prove that (A) APOS and APXS have the same area (B) APOS, BQOP, CROQ, DSOR have the same area.

$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that: a) $\widehat{ZDE}=45^0$ b) Points $E, M, K$ lie on a line. c) $BM//EC$

By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

Two circles $\gamma $ and $\gamma'$ cross one another at points $A$ and $B$ . The tangent to $\gamma'$ at $A$ meets $\gamma$ again at $C$ , the tangent to $\gamma$ at $A$ meets $\gamma'$ again at $C'$ , and the line $CC'$ separates the points $A$ and $B$ . Let $\Gamma$ be the circle externally tangent to $\gamma$ , externally tangent to $\gamma'$ , tangent to the line $CC'$, and lying on the same side of $CC'$ as $B$ . Show that the circles $\gamma$ and $\gamma'$ intercept equal segments on one of the tangents to $\Gamma$ through $A$ .