If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $$(a - d)^2 \ge 4d + 8$$

Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$. Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.

Given triangle $ ABC$ with base $ AB$ fixed in length and position. As the vertex $ C$ moves on a straight line, the intersection point of the three medians moves on: $ \textbf{(A)}\ \text{a circle} \qquad \textbf{(B)}\ \text{a parabola} \qquad \textbf{(C)}\ \text{an ellipse} \qquad \textbf{(D)}\ \text{a straight line} \qquad \textbf{(E)}\ \text{a curve here not listed}$

In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.

Two boys are given a bag of potatoes, each bag containing $150$ tubers. They take turns transferring the potatoes, where in each turn they transfer a non-zero tubers from their bag to the other boy's bag. Their moves must satisfy the following condition: In each move, a boy must move more tubers than he had in his bag before any of his previous moves (if there were such moves). So, with his first move, a boy can move any non-zero quantity, and with his fifth move, a boy can move $200$ tubers, if before his first, second, third and fourth move, the numbers of tubers in his bag was less than $200$. What is the maximal total number of moves the two boys can do? [i]Proposed by E. Molchanov[/i]

Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$

In $\triangle ABC$ with integer side lengths, \[ \cos A=\frac{11}{16}, \qquad \cos B= \frac{7}{8}, \qquad \text{and} \qquad\cos C=-\frac{1}{4}. \] What is the least possible perimeter for $\triangle ABC$? $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 23 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 44$

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$. (V.Yassinsky)

We say that the rank of a group $ G$ is at most $ r$ if every subgroup of $ G$ can be generated by at most $ r$ elements. Prove that here exists an integer $ s$ such that for every finite group $ G$ of rank $ 2$ the commutator series of $ G$ has length less than $ s$. [i]J. Erdos[/i]

The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$ $\textbf{(A) } 20 \qquad \textbf{(B) } 21 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 24 $

Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,

On a line $n+1$ segments are marked such that one of the points of the line is contained in all of them. Prove that one can find $2$ distinct segments $I, J$ which intersect at a segment of length at least $\frac{n-1}{n}d$, where $d$ is the length of the segment $I$.

Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.

Compute the number of ordered pairs $(m,n)$ of [i]odd[/i] positive integers both less than $80$ such that $$\gcd(4^m+2^m+1, 4^n+2^n+1)>1.$$

Let $ABCD$ be a quadrilateral with all the internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4$ denote the areas of the shaded triangles as shown. Prove that \[\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.\] [asy] //made from sweat and hardwork by SatisfiedMagma import olympiad; import geometry; size(250); pair A = (-3,0); pair B = (0,2); pair C = (5.88,0.44); pair D = (0.96, -1.86); pair H = B + rotate(90)*(C-B); pair G = C + rotate(270)*(B-C); pair J = C + rotate(90)*(D-C); pair I = D + rotate(270)*(C-D); pair L = D + rotate(90)*(A-D); pair K = A + rotate(270)*(D-A); pair F = A + rotate(90)*(B-A); pair E = B + rotate(270)*(A-B); draw(B--H--G--C--B, blue); draw(C--J--I--D--C, red); draw(B--E--F--A--B, orange); draw(D--L--K--A--D, magenta); draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia); pen lightFuchsia = deepgreen + 0.5*white; fill(D--L--I--cycle, lightFuchsia); fill(A--K--F--cycle, lightFuchsia); fill(E--B--H--cycle, lightFuchsia); fill(C--J--G--cycle, lightFuchsia); label("$\triangle_2$", (E+B+H)/3); label("$\triangle_4$", (D+L+I)/3); label("$\triangle_3$", (C+G+J)/3); label("$\triangle_1$", (A+F+K)/3); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, S); dot("$D$", D, N); dot("$H$", H, dir(H)); dot("$G$", G, dir(G)); dot("$J$", J, dir(J)); dot("$I$", I, dir(I)); dot("$L$", L, dir(L)); dot("$K$", K, dir(K)); dot("$F$", F, dir(F)); dot("$E$", E, dir(E)); [/asy]

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A\cup B=S,\ A\cap B=\emptyset,$ and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_{S}$ be the number of linear partitions of $S.$ For each positive integer $n,$ find the maximum of $L_{S}$ over all sets $S$ of $n$ points.

Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$

Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.

Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent friendships and points represent people. [img]https://cdn.artofproblemsolving.com/attachments/5/5/7265067f585e3dfe77ba94ac6261b4462cd015.png[/img] If Sue drew the first graph, how many friends does she have?

A quadratic polynomial $p(x)$ with integer coefficients satisfies $p(41) = 42$. For some integers $a, b > 41$, $p(a) = 13$ and $p(b) = 73$. Compute the value of $p(1)$. [i]Proposed by Aaron Lin[/i]

The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

For positive $a,b,c$, such that $abc=1$ prove the inequality: $4(\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}) \leq 3(2+a+b+c+\frac{1}{a}+\frac{1}{b}+ \frac{1}{c})^{\frac{2}{3}}$.

Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals. [i]Vasile Pop[/i]