Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

Let $\triangle ABC$ be an acute triangle. Point $Z$ is on $A$ altitude and points $X$ and $Y$ are on the $B$ and $C$ altitudes out of the triangle respectively, such that: $\angle AYB=\angle BZC=\angle CXA=90$ Prove that $X$,$Y$ and $Z$ are collinear, if and only if the length of the tangent drawn from $A$ to the nine point circle of $\triangle ABC$ is equal with the sum of the lengths of the tangents drawn from $B$ and $C$ to the nine point circle of $\triangle ABC$.

The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.

Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that: $ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.) $ \displaystyle \textbf{(A)} \ \frac {1}{2} \sqrt {3} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ \sqrt {2} \qquad \textbf{(D)} \ \frac {3}{2} \qquad \textbf{(E)} \ 2$

Let be a triangle $ ABC $ with $ \angle BAC = 90^{\circ } . $ On the perpendicular of $ BC $ through $ B, $ consider $ D $ such that $ AD=BC. $ Find $ \angle BAD. $

Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$. Let $P$ a point on $\Gamma$. Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$. Prove that when $P$ varies on $\Gamma$, the line $AB$ is tangent to a fixed circunference.

We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$ Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.

Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting. Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.

Anne and Lisa enter a park that has two concentric circular paths joined by two radial paths, one of which is at the point where they enter. Anne goes in to the inner circle along the first radial path, around by the shorter way to the second radial path and out along it to the exit. Walking at a the same rate, Lisa goes around the outer circle to the exit, taking the shorter of the two directions around the park. They arrive at the exit at the same time. The radial paths meet at the center of the park; what is the angle between them?

Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

In a rectangular box $ABCDEFGH$ with edge lengths $AB = AD = 6$ and $AE = 49$, a plane slices through point $A$ and intersects edges $BF$, $FG$, $GH$, $HD$ at points $P$, $Q$, $R$, $S$ respectively. Given that $AP = AS$ and $PQ = QR = RS$, find the area of pentagon $APQRS$.

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.

In triangle $ABC$ let $N, M, P$ be the midpoints of the sides $BC, CA, AB$ and $G$ be the centroid of this triangle. Let the circle circumscribed to $BGP$ intersect the line $MP$ in point $K$, $P \neq K$, and the circle circumscribed to $CGN$ intersect the line $MN$ in point $L$, $N \neq L$. Prove that $ \angle BAK = \angle CAL $.

Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed line? [asy] unitsize(48); for (int a=0; a<3; ++a) { fill((2a+1,1)--(2a+.8,1)--(2a+.8,.8)--(2a+1,.8)--cycle,black); } draw((.8,1)--(0,1)--(0,0)--(1,0)--(1,.8)); draw((2.8,1)--(2,1)--(2,0)--(3,0)--(3,.8)); draw((4.8,1)--(4,1)--(4,0)--(5,0)--(5,.8)); draw((.2,.4)--(.6,.8),linewidth(1)); draw((.4,.6)--(.8,.2),linewidth(1)); draw((2.4,.8)--(2.8,.4),linewidth(1)); draw((2.6,.6)--(2.2,.2),linewidth(1)); draw((4.4,.2)--(4.8,.6),linewidth(1)); draw((4.6,.4)--(4.2,.8),linewidth(1)); draw((7,.2)--(7,1)--(6,1)--(6,0)--(6.8,0)); fill((6.8,0)--(7,0)--(7,.2)--(6.8,.2)--cycle,black); draw((6.2,.6)--(6.6,.2),linewidth(1)); draw((6.4,.4)--(6.8,.8),linewidth(1)); draw((8,.8)--(8,0)--(9,0)--(9,1)--(8.2,1)); fill((8,1)--(8,.8)--(8.2,.8)--(8.2,1)--cycle,black); draw((8.4,.8)--(8.8,.8),linewidth(1)); draw((8.6,.8)--(8.6,.2),linewidth(1)); draw((6,1.2)--(6,1.4)); draw((6,1.6)--(6,1.8)); draw((6,2)--(6,2.2)); draw((6,2.4)--(6,2.6)); draw((6.4,2.2)--(6.4,1.4)--(7.4,1.4)--(7.4,2.4)--(6.6,2.4)); fill((6.4,2.4)--(6.4,2.2)--(6.6,2.2)--(6.6,2.4)--cycle,black); draw((6.6,1.8)--(7,2.2),linewidth(1)); draw((6.8,2)--(7.2,1.6),linewidth(1)); label("(A)",(0,1),W); label("(B)",(2,1),W); label("(C)",(4,1),W); label("(D)",(6,1),W); label("(E)",(8,1),W); [/asy]

A convex solid is formed in four-dimensional Euclidean space with vertices at the $24$ possible permutations of $\{1, 2, 3, 4\}$ (so $(1, 2, 3, 4)$, $(1, 2, 4, 3)$, etc.). What is the product of the number of faces and edges of this solid?

Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube.

In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.

Consider a container of a hollow cube $ABGCDEPF$ (where $ABGC$, $DEPF$ are squares and $AD\parallel BE\parallel GP\parallel CF$). The cube is placed on a table in a way that the space diagonal $AP=1$ is perpendicular to the table. Then, water is poured into the cube. Denote $x$ the length of part of $AP$ submerged in water. Determine the volume of water $y$ in terms of $x$ when a) $0 < x \leq\frac13$, b) $\frac13 < x \leq\frac12$.

[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order? [b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$. [b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.) [b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour? [b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$. [b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make? [b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$? [b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$? [b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions: (a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$, (b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and (c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$. [b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$. [b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up. [b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients. [b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy $$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$ where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$. [b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place. [b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{ 107^2}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.