A1 (SLO) Let $a,b,$ and $c$ be positive real numbers such that $abc=1$. Prove that
$$\frac{ab}{a^5+b^5+ab}+\frac{bc}{b^5+c^5+bc}+\frac{ca}{c^5+a^5+ca}\le1.$$
When does equality hold?
A2 (IRE) Let $a_1\ge a_2\ge\cdots\ge a_n$ be real numbers such that
$$a_1^k+a_2^k+\cdots+a_n^k\ge0$$
for all integers $k\gt 0$. Let $p=\max\{|a_1|,\ldots,|a_n|\}$. Prove that $p=a_1$ and that
$$(x-a_1)(x-a_2)\cdots(x-a_n)\le x^n-a_1^n$$
for all $x\gt a_1$.
A3 (GRE) Let $a\gt 2$ be given, and define recursively
$$a_0=1,\qquad a_1=a,\qquad a_{n+1}=\left(\frac{a_n^2}{a_{n-1}^2}-2\right)a_n.$$
Show that for all $k\in\mathbb N$, we have
$$\frac1{a_0}+\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_k}\lt \frac12\left(2+a-\sqrt{a^2-4}\right).$$
A4 (KOR) Let $a_1,a_2,\ldots,a_n$ be nonnegative real numbers, not all zero.
(a) Prove that $x^n-a_1x^{n-1}-\cdots-a_{n-1}x-a_n=0$ has precisely one positive real root.
(b) Let $A=\sum_{j=1}^na_j$, $B=\sum_{j=1}^nja_j$, and let $R$ be the positive real root of the equation in part (a). Prove that
$$A^A\le R^B.$$
A5 (ROM) Let $P(x)$ be the real polynomial function $P(x)=ax^3+bx^2+cx+d$. Prove that if $|P(x)|\le1$ for all $x$ such that $|x|\le1$, then
$$|a|+|b|+|c|+|d|\le7.$$
A6 (IRE) Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
for some polynomials $f(x),g(x)$ having integer coefficients. If $k_0$ denotes the least such $k$, determine $k_0$ as a function of $n$.
A6′ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that
$$k=f(x)(x+1)^n+g(x)(x^n+1)$$
for some polynomials $f(x),g(x)$ having integer coefficients. If $k_0$ denotes the least such $k$, show that $k_0=2^q$, where $q$ is the odd integer determined by $n=q2^r$, $r\in\mathbb N$.
A6″ Prove that for each positive integer $n$, there exist polynomials $f(x),g(x)$ having integer coefficients such that
$$f(x)(x+1)^{2^n}+g(x)(x^{2^n}+1)=2.$$
A7 (ARM) Let $f$ be a function from the set of real numbers $\mathbb R$ into itself such that for all $x\in\mathbb R$, we have $|f(x)|\le1$ and
$$f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac16\right)+f\left(x+\frac17\right).$$
Prove that $f$ is a periodic function (that is, there exists a nonzero real number $c$ such that $f(x+c)=f(x)$ for all $x\in\mathbb R$).
A8 (ROM) IMO3 Let $\mathbb N_0$ denote the set of nonnegative integers. Find all functions $f$ from $\mathbb N_0$ into itself such that
$$f(m+f(n))=f(f(m))+f(n),\qquad\forall m,n\in\mathbb N_0.$$
A9 (POL) Let the sequence $a(n)$, $n=1,2,3,\ldots$, be generated as follows: $a(1)=0$, and for $n\gt 1$,
$$a(n)=a([n/2])+(-1)^{\frac{n(n+1)}2}.$$
(Here $[t]$ = the greatest integer $\le t$.)
(a) Determine the maximum and minimum value of $a(n)$ over $n\le1996$ and find all $n\le1996$ for which these extreme values are attained.
(b) How many terms $a(n)$, $n\le1996$, are equal to $0$?
G1 (GBR) Let triangle $ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A,B,C$. Let $E$ be the foot of the altitude $BH$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet $HR$ in $X$. Prove that $EX$ is parallel to $AP$.
G2 (CAN) IMO2 Let $P$ be a point inside $\triangle ABC$ such that
$$\angle APB-\angle C=\angle APC-\angle B.$$
Let $D,E$ be the incenters of $\triangle APB,\triangle APC$ respectively. Show that $AP,BD$ and $CE$ meet in a point.
G3 (GBR) Let $ABC$ be an acute-angled triangle with $BC\gt CA$. Let $O$ be the circumcenter, $H$ its orthocenter, and $F$ the foot of its altitude $CH$. Let the perpendicular to $OF$ at $F$ meet the side $CA$ at $P$. Prove that $\angle FHP=\angle BAC$.
Possible second part: What happens if $|BC|\le|CA|$ (the triangle still being acute-angled)?
G4 (USA) Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP,BP,CP$ meet the sides $BC,CA,AB$ in the points $A_1,B_1,C_1$ respectively. Prove that
$$A_1B_1\cdot B_1C_1\cdot C_1A_1\ge A_1B\cdot B_1C\cdot C_1A.$$
G5 (ARM) IMO5 Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $AF$. Let $R_A,R_C,R_E$ be the circumradii of triangles $FAB,BCD,DEF$ respectively, and let $P$ denote the perimeter of the hexagon. Prove that
$$R_A+R_C+R_E\ge\frac P2.$$
G6 (ARM) Let the sides of two rectangles be $\{a,b\}$ and $\{c,d\}$ with $a\lt c\le d\lt b$ and $ab\lt cd$. Prove that the first rectangle can be placed within the second one if and only if
$$(b^2-a^2)^2\le(bd-ac)^2+(bc-ad)^2.$$
G7 (GBR) Let $ABC$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $AO$ meet the circle $BOC$ again in $A'$, let $BO$ meet the circle $COA$ again in $B'$, and let $CO$ meet the circle $AOB$ again in $C'$. Prove that
$$OA'\cdot OB'\cdot OC'\ge8R^3.$$
When does equality hold?
G8 (RUS) Let $ABCD$ be a convex quadrilateral, and let $R_A,R_B,R_C,$ and $R_D$ denote the circumradii of the triangles $DAB,ABC,BCD,$ and $CDA$ respectively. Prove that $R_A+R_C\gt R_B+R_D$ if and only if
$$\angle A+\angle C\gt \angle B+\angle D.$$
G9 (UKR) In the plane are given a point $O$ and a polygon $\mathcal F$ (not necessarily convex). Let $P$ denote the perimeter of $\mathcal F$, $D$ the sum of the distances from $O$ to the vertices of $\mathcal F$, and $H$ the sum of the distances from $O$ to the lines containing the sides of $\mathcal F$. Prove that
$$D^2-H^2\ge\frac{P^2}4.$$
N1 (UKR) Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this number and the next number on the circle, in a given direction (that is, the numbers $a,b,c,d$ are replaced by $a-b,b-c,c-d,d-a$). Is it possible after $1996$ such steps to have numbers $a,b,c,d$ such that the numbers $|bc-ad|,|ac-bd|,|ab-cd|$ are primes?
N2 (RUS) IMO4 The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?