How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

A regular $n$-gon is inscribed in a circle radius $1$. Let $X$ be the set of all arcs $PQ$, where $P, Q$ are distinct vertices of the $n$-gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$.

A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$. Restore the triangle.

ABC  is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0). (a) Let P  be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P. (b) Let Q be the point on line BC  for which PQ is perpendicular to AB. Determine the length of AQ. (c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.

A triangle has altitudes of length $15$, $21$, and $35$. Find its area.

While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.

Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$. (a) Prove that $I$ lies on ray $CV$. (b) Prove that line $XI$ bisects $\overline{UV}$.

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.

Given an isosceles triangle $ABC$ with base $BC$. On the sides $BC$, $AC$ and $AB$ points $X,Y$ and $Z$ are chosen respectively such that triangles $ABC$ and $YXZ$ are similar. Point $W$ is symmetric to point $X$ with respect to the midpoint of $BC$. Prove that points $X,Y,Z$ and $W$ lie on a circle.

Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.

In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.

Let $\Omega$ and $O$ be the circumcircle of acute triangle $ABC$ and its center, respectively. $M\ne O$ is an arbitrary point in the interior of $ABC$ such that $AM$, $BM$, and $CM$ intersect $\Omega$ at $A_{1}$, $B_{1}$, and $C_{1}$, respectiuvely. Let $A_{2}$, $B_{2}$, and $C_{2}$ be the circumcenters of $MBC$, $MCA$, and $MAB$, respectively. It is to be proven that $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C{2}$ concur.

In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality: $\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square? (C.Pohoata, A.Zaslavsky)

There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a [i]congruent [/i] pair of staight lines, otherwise it is called a [i]separated[/i] pair of straight lines. If the number of [i]congruent [/i] pairs of straight line among $n$ straight lines is $2012$ more than the number of [i]separated[/i] pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).

Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10\times 11$ rectangle, we can find a domino that has an inner point above it. How many stories is the lowest [i]stable[/i] tower?

Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.

[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].

In a triangle $ABC$ let $K$ be a point on the median $BM$ such that $CK=CM$. It appears that $\angle CBM = 2 \angle ABM$. Prove that $BC=MK$.

Given an acute triangle $ABC$, $(c)$ its circumcircle with center $O$ and $H$ the orthocenter of the triangle $ABC$. The line $AO$ intersects $(c)$ at the point $D$. Let $D_1, D_2$ and $H_2, H_3$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $AB, AC$ respectively. Let $(c_1)$ be the circumcircle of the triangle $AD_1D_2$. Suppose that the line $AH$ intersects again $(c_1)$ at the point $U$, the line $H_2H_3$ intersects the segment $D_1D_2$ at the point $K_1$ and the line $DH_3$ intersects the segment $UD_2$ at the point $L_1$. Prove that one of the intersection points of the circumcircles of the triangles $D_1K_1H_2$ and $UDL_1$ lies on the line $K_1L_1$.

What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).