Given a point $ P$ inside a triangle $ \triangle ABC$. Let $ D$, $ E$, $ F$ be the orthogonal projections of the point $ P$ on the sides $ BC$, $ CA$, $ AB$, respectively. Let the orthogonal projections of the point $ A$ on the lines $ BP$ and $ CP$ be $ M$ and $ N$, respectively. Prove that the lines $ ME$, $ NF$, $ BC$ are concurrent. Original formulation: Let $ ABC$ be any triangle and $ P$ any point in its interior. Let $ P_1, P_2$ be the feet of the perpendiculars from $ P$ to the two sides $ AC$ and $ BC.$ Draw $ AP$ and $ BP,$ and from $ C$ drop perpendiculars to $ AP$ and $ BP.$ Let $ Q_1$ and $ Q_2$ be the feet of these perpendiculars. Prove that the lines $ Q_1P_2,Q_2P_1,$ and $ AB$ are concurrent.

$ ABC$ is an acute-angled triangle. $ M$ is the midpoint of $ BC$ and $ P$ is the point on $ AM$ such that $ MB = MP$. $ H$ is the foot of the perpendicular from $ P$ to $ BC$. The lines through $ H$ perpendicular to $ PB$, $ PC$ meet $ AB, AC$ respectively at $ Q, R$. Show that $ BC$ is tangent to the circle through $ Q, H, R$ at $ H$. Original Formulation: For an acute triangle $ ABC, M$ is the midpoint of the segment $ BC, P$ is a point on the segment $ AM$ such that $ PM = BM, H$ is the foot of the perpendicular line from $ P$ to $ BC, Q$ is the point of intersection of segment $ AB$ and the line passing through $ H$ that is perpendicular to $ PB,$ and finally, $ R$ is the point of intersection of the segment $ AC$ and the line passing through $ H$ that is perpendicular to $ PC.$ Show that the circumcircle of $ QHR$ is tangent to the side $ BC$ at point $ H.$

Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

In the triangle $ ABC,$ with $ \angle A = 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP = BC.$ Show that $ \angle BFP = \frac{\angle B}{2}.$

$ ABCD$ is a terahedron: $ AD+BD=AC+BC,$ $ BD+CD=BA+CA,$ $ CD+AD=CB+AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA=OB=OC=OD.$ Prove that $ \angle MOP = \angle NOP =\angle NOM.$

$ S$ be a set of $ n$ points in the plane. No three points of $ S$ are collinear. Prove that there exists a set $ P$ containing $ 2n - 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $ S$ lies a point that is an element of $ P.$

In the plane we are given a set $ E$ of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of $ E,$ there exist at least 1593 other points of $ E$ to which it is joined by a path. Show that there exist six points of $ E$ every pair of which are joined by a path. Alternative version: Is it possible to find a set $ E$ of 1991 points in the plane and paths joining certain pairs of the points in $ E$ such that every point of $ E$ is joined with a path to at least 1592 other points of $ E,$ and in every subset of six points of $ E$ there exist at least two points that are not joined?

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $ \,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

Prove that $ \sum_{k = 0}^{995} \frac {( - 1)^k}{1991 - k} {1991 - k \choose k} = \frac {1}{1991}$

Let $ S = \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

Given any integer $ n \geq 2,$ assume that the integers $ a_1, a_2, \ldots, a_n$ are not divisible by $ n$ and, moreover, that $ n$ does not divide $ \sum^n_{i=1} a_i.$ Prove that there exist at least $ n$ different sequences $ (e_1, e_2, \ldots, e_n)$ consisting of zeros or ones such $ \sum^n_{i=1} e_i \cdot a_i$ is divisible by $ n.$

Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) = ax^2 + bx + c$ is a perfect square for $ 2p - 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 - 4ac.$

Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} - a_{1} = a_{3} - a_{2} = \cdots = a_{k} - a_{k - 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x + 4^y = 5^z.$

Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} + 1992^{1991^{1990}}.\]

Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) + 2\cos(2 \pi \alpha) = 0$. Prove that $ \alpha = \frac {2}{3}$.

Let $ \alpha$ be the positive root of the equation $ x^{2} = 1991x + 1$. For natural numbers $ m$ and $ n$ define \[ m*n = mn + \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r = p*(q*r). \]

Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) = f^2(x) - 9.$ Show that the number of distinct integer solutions of $ g(x) = 0$ cannot exceed $ 1995.$

Real constants $ a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $ y = x^3 + ax^2 + bx + c.$ Prove that the square has sides of length $ \sqrt[4]{72}.$

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that (a) $ f(m + f(f(n))) = -f(f(m+ 1) - n$ for all integers $ m$ and $ n;$ (b) $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} - a_{2} + a_{3} - \cdots - a_{n - 1} + a_{n}$, $ a_{2} - a_{3} + a_{3} - \cdots - a_{n} + a_{1}$, $ a_{3} - a_{4} + a_{5} - \cdots - a_{1} + a_{2}$, $ \cdots$, $ a_{n} - a_{1} + a_{2} - \cdots - a_{n - 2} + a_{n - 1}$ are all positive. Determine the set of all `nice' integers.

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n - 1$ the inequality \[ x_i (1 - x_{i+1}) \geq \frac{1}{4} x_1 (1 - x_{n})\]

Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i = 1, \ldots, n,$ and \[ \sum^n_{i=1} a_i = \sum^n_{i=1} b_i.\] Prove the inequality: \[ \sum^n_{i=1} b_i \prod^n_{j = 1, j \neq i} a_j \leq \frac{p}{(n-1)^{n-1}}.\]

Determine the maximum value of the sum \[ \sum_{i < j} x_ix_j (x_i + x_j) \] over all $ n -$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i = 1} x_i = 1.$

An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be bounded if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that \[ \vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1 \] for every pair of distinct nonnegative integers $ i, j$.

We call a set $ S$ on the real line $ \mathbb{R}$ superinvariant if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) = x_0 + a(x - x_0), a > 0$ there exists a translation $ B,$ $ B(x) = x+b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) = B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) = A(u).$ Determine all superinvariant sets.

Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”