Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$. [i]Proposed by Andrew Wen[/i]

Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have \[ \left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K. \] [i]2015 CCA Math Bonanza Lightning Round #3.4[/i]

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$. (Spain)

Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$ ($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits). [b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table: the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matches. Then they take turns making moves. In a move, a player may take any nonzero number of matches FROM ONE PILE. The player who takes the last match from the table loses the game. a) The player who makes the first move can win the game. What is the winning first move? b) How can he win? (Describe his strategy.) [b]p3.[/b] The planet Naboo is under attack by the imperial forces. Three rebellion camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle with a defensive field. What is the maximal area that they may need to cover? [b]p4.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$ 1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back. [b]p5.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $\angle BAC> 90 ^ \circ$ and $D$ a point on $BC$. Let $E$ and $F$be the reflections of the point $D$ about $AB$ and $AC$, respectively. Suppose that $BE$ and $CF$ intersect at $P$. Show that $AP$ passes through the circumcenter of triangle $ABC$.

Let $ABC$ be an acute triangle with $AB < AC$, and let $H$ be its orthocenter. Let $D$, $E$, $F$ and $M$ be the midpoints of $BC$, $AC$, and $AH$, respectively. Prove that the circumcircles of triangles $AHD$, $BMC$, and $DEF$ pass through a common point.

Two equal parallel chords are drawn $ 8$ inches apart in a circle of radius $ 8$ inches. The area of that part of the circle that lies between the chords is: $ \textbf{(A)}\ 21\frac {1}{3}\pi \minus{} 32\sqrt {3} \qquad\textbf{(B)}\ 32\sqrt {3} \plus{} 21\frac {1}{3}\pi \qquad\textbf{(C)}\ 32\sqrt {3} \plus{} 42\frac {2}{3}\pi$ $ \textbf{(D)}\ 16\sqrt {3} \plus{} 42\frac {2}{3}\pi \qquad\textbf{(E)}\ 42\frac {2}{3}\pi$

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.

Given a circle $\gamma$ and two points $A$ and $B$ in its plane. By $B$ passes a variable secant that intersects $\gamma$ at two points $M$ and $N$. Determine the locus of the centers of the circles circumscribed to the triangle $AMN$.

Let $ABC$ be a triangle with $\angle B > 90^o$ such that there is a point $H$ on side $AC$ with $AH = BH$ and BH perpendicular to $BC$. Let $D$ and $E$ be the midpoints of $AB$ and $BC$ respectively. A line through $H$ parallel to $AB$ cuts $DE$ at $F$. Prove that $\angle BCF = \angle ACD$.

Let $ABCD$ be a regular tetrahedron with side length $2$. The plane parallel to edges $AB$ and $CD$ and lying halfway between them cuts $ABCD$ into two pieces. Find the surface area of one of these pieces.

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

Prove that any uncountable subset of the Euclidean $ n$-space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points $ P_1 \not\equal{} P_2$ and $ Q_1\not\equal{} Q_2$ of this subset, $ \overline{P_1P_2}\equal{}\overline{Q_1Q_2}$ implies either $ P_1\equal{}Q_1$ and $ P_2\equal{}Q_2$, or $ P_1\equal{}Q_2$ and $ P_2\equal{}Q_1$). Show that a similar statement is not valid if the Euclidean $ n$-space is replaced with a (separable) Hilbert space.

Let $AD$ be the common chord of two circles $\Gamma_1$ and $\Gamma_2$. A line through $D$ intersects $\Gamma_1$ at $B$ and $\Gamma_2$ at $C$. Let $E$ be a point on the segment $AD$, different from $A$ and $D$. The line $CE$ intersect $\Gamma_1$ at $P$ and $Q$. The line $BE$ intersects $\Gamma_2$ at $M$ and $N$. (i) Prove that $P,Q,M,N$ lie on the circumference of a circle $\Gamma_3$. (ii) If the centre of $\Gamma_3$ is $O$, prove that $OD$ is perpendicular to $BC$.

Let $AA_1, BB_1, CC_1$ be the altitudes of an acute-angled triangle $ABC$; $I_a$ be its excenter corresponding to $A$; $I_a'$ be the reflection of $I_a$ about the line $AA_1$. Points $I_b', I_c'$ are defined similarily. Prove that lines $A_1I_a', B_1I_b', C_1I_c'$ concur.

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. [img]https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png[/img] They are going to make two straight cuts on the cake, thus obtaining $4$ portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$. [i]Proposed by Michael Tang[/i]