ISL 1991

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.

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?

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?

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

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) = $g(f(n))$ $\forall n \in \mathbb{Z}.$

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 $\sum^n_{i = 1} x_i = 1.$

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."