IMO 1995 — Shortlisted problems

A1 (RUS), IMO2. Let $a$, $b$, and $c$ be positive real numbers such that $abc=1$. Prove that
$$\frac1{a^3(b+c)}+\frac1{b^3(a+c)}+\frac1{c^3(a+b)}\ge\frac32.$$

A2 (SWE). Let $a$ and $b$ be nonnegative integers such that $ab\ge c^2$, where $c$ is an integer. Prove that there is a number $n$ and integers $x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ such that
$$\sum_{i=1}^n x_i^2=a,\qquad\sum_{i=1}^n y_i^2=b,\qquad\sum_{i=1}^n x_iy_i=c.$$

A3 (UKR). Let $n$ be an integer, $n\ge3$. Let $a_1,a_2,\ldots,a_n$ be real numbers such that $2\le a_i\le3$ for $i=1,2,\ldots,n$. If $s=a_1+a_2+\cdots+a_n$, prove that
$$\frac{a_1^2+a_2^2-a_3^2}{a_1+a_2-a_3}+\frac{a_2^2+a_3^2-a_4^2}{a_2+a_3-a_4}+\cdots+\frac{a_n^2+a_1^2-a_2^2}{a_n+a_1-a_2}\le2s-2n.$$

A4 (USA). Let $a$, $b$, and $c$ be given positive real numbers. Determine all positive real numbers $x$, $y$, and $z$ such that
$$x+y+z=a+b+c$$
and
$$4xyz-(a^2x+b^2y+c^2z)=abc.$$

A5 (UKR). Let $\mathbb R$ be the set of real numbers. Does there exist a function $f:\mathbb R\to\mathbb R$ that simultaneously satisfies the following three conditions?
(a) There is a positive number $M$ such that $-M\le f(x)\le M$ for all $x$.
(b) $f(1)=1$.
(c) If $x\ne0$, then
$$f\left(x+\frac1{x^2}\right)=f(x)+\left[f\left(\frac1x\right)\right]^2.$$

A6 (JAP). Let $n$ be an integer, $n\ge3$. Let $x_1,x_2,\ldots,x_n$ be real numbers such that $x_i\lt x_{i+1}$ for $1\le i\le n-1$. Prove that
$$\frac{n(n-1)}2\sum_{i\lt j}x_ix_j\gt \left(\sum_{i=1}^{n-1}(n-i)x_i\right)\left(\sum_{j=2}^n(j-1)x_j\right).$$

G1 (BUL), IMO1. Let $A$, $B$, $C$, and $D$ be distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. $O$ is an arbitrary point on the line $XY$ but not on $AD$. $CO$ intersects the circle with diameter $AC$ again at $M$, and $BO$ intersects the other circle again at $N$. Prove that the lines $AM$, $DN$, and $XY$ are concurrent.

G2 (GER). Let $A$, $B$, and $C$ be noncollinear points. Prove that there is a unique point $X$ in the plane of $ABC$ such that $XA^2+XB^2+AB^2=XB^2+XC^2+BC^2=XC^2+XA^2+CA^2$.

G3 (TUR). The incircle of $ABC$ touches $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. $X$ is a point inside $ABC$ such that the incircle of $XBC$ touches $BC$ at $D$ also, and touches $CX$ and $XB$ at $Y$ and $Z$, respectively. Prove that $EFZY$ is a cyclic quadrilateral.

G4 (UKR). An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$) such that
$$\angle AA_1A_2=\angle AA_2A_1=\angle BB_1B_2=\angle BB_2B_1=\angle CC_1C_2=\angle CC_2C_1.$$
The lines $AA_1$, $BB_1$, and $CC_1$ form a triangle, and the lines $AA_2$, $BB_2$, and $CC_2$ form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

G5 (NZL), IMO5. Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$, $DE=EF=FA$, and $\angle BCD=\angle EFA=\pi/3$ (that is, $60^\circ$). Let $G$ and $H$ be two points interior to the hexagon such that angles $AGB$ and $DHE$ are both $2\pi/3$ (that is, $120^\circ$). Prove that $AG+GB+GH+DH+HE\ge CF$.

G6 (USA). Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A'_1$, $A'_2$, $A'_3$, and $A'_4$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1$, $GA_2$, $GA_3$, and $GA_4$, respectively. Prove that
$$GA_1\cdot GA_2\cdot GA_3\cdot GA_4\le GA'_1\cdot GA'_2\cdot GA'_3\cdot GA'_4$$
and
$$\frac1{GA'_1}+\frac1{GA'_2}+\frac1{GA'_3}+\frac1{GA'_4}\le\frac1{GA_1}+\frac1{GA_2}+\frac1{GA_3}+\frac1{GA_4}.$$

G7 (LAT). $O$ is a point inside a convex quadrilateral $ABCD$ of area $S$. $K$, $L$, $M$, and $N$ are interior points of the sides $AB$, $BC$, $CD$, and $DA$ respectively. If $OKBL$ and $OMDN$ are parallelograms, prove that $\sqrt S\ge\sqrt{S_1}+\sqrt{S_2}$, where $S_1$ and $S_2$ are the areas of $ONAK$ and $OLCM$ respectively.

G8 (COL). Let $ABC$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ again at $C'$ and $B'$, respectively. Prove that $BB'$, $CC'$, and $HH'$ are concurrent, where $H$ and $H'$ are the orthocenters of triangles $ABC$ and $AB'C'$ respectively.

N1 (ROM). Let $k$ be a positive integer. Prove that there are infinitely many perfect squares of the form $n2^k-7$, where $n$ is a positive integer.

N2 (RUS). Let $\mathbb Z$ denote the set of all integers. Prove that for any integers $A$ and $B$, one can find an integer $C$ for which $M_1=\{x^2+Ax+B:x\in\mathbb Z\}$ and $M_2=\{2x^2+2x+C:x\in\mathbb Z\}$ do not intersect.

N3 (CZE), IMO3. Determine all integers $n\gt 3$ such that there are $n$ points $A_1,A_2,\ldots,A_n$ in the plane that satisfy the following two conditions simultaneously:
(a) No three lie on the same line.
(b) There exist real numbers $p_1,p_2,\ldots,p_n$ such that the area of $\triangle A_iA_jA_k$ is equal to $p_i+p_j+p_k$, for $1\le i\lt j\lt k\le n$.

N4 (BUL). Find all positive integers $x$ and $y$ such that $x+y^2+z^3=xyz$, where $z$ is the greatest common divisor of $x$ and $y$.

N5 (IRE). At a meeting of $12k$ people, each person exchanges greetings with exactly $3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

N6 (POL), IMO6. Let $p$ be an odd prime. Find the number of $p$-element subsets $A$ of $\{1,2,\ldots,2p\}$ such that the sum of all elements of $A$ is divisible by $p$.