Determine the number of positive integers $n$ satisfying: [list] [*] $n<10^6$ [*] $n$ is divisible by 7 [*] $n$ does not contain any of the digits 2,3,4,5,6,7,8. [/list]

Consider a polyhedron having $100$ edges. (a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex. (b) Prove that for a non-convex polyhedron this number i. can be as great as $96$, ii. cannot be as great as $100$. (A Andjans, Riga

Let $ P (x) $ and $ Q (x) $ be polynomials with real coefficients such that $ P (0)> 0 $ and all coefficients of the polynomial $ S (x) = P (x) \cdot Q (x) $ are integers. Prove that for any positive $ x $ the inequality holds: $$S ({{x} ^ {2}}) - {{S} ^ {2}} (x) \le \frac {1} {4} ({{P} ^ {2}} ({{ x} ^ {3}}) + Q ({{x} ^ {3}})). $$

Liquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm thick. A rectangular box measuring $6$ cm by $3$ cm by $12$ cm is filled with liquid X. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film? $ \textbf{(A)}\ \frac{\sqrt{216}}{\pi}\qquad\textbf{(B)}\ \sqrt{\frac{216}{\pi}}\qquad\textbf{(C)}\ \sqrt{\frac{2160}{\pi}}\qquad\textbf{(D)}\ \frac{216}{\pi}\qquad\textbf{(E)}\ \frac{2160}{\pi} $

The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 $

Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.

If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$? (Note: The roots of the polynomial do not have to be different from each other.)

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 $AB = BC = CA.$