# [Shortlists] International Mathematical Olympiad 1995

### Algebra

1. Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$
2. Let $a$ and $b$ be non-negative integers such that $ab \geq c^2,$ 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 $\sum^n_{i=1} x^2_i = a, \sum^n_{i=1} y^2_i = b, \text{ and } \sum^n_{i=1} x_iy_i = c.$
3. Let $n$ be an integer, $n \geq 3.$ Let $a_1, a_2, \ldots, a_n$ be real numbers such that $2 \leq a_i \leq 3$ for $i = 1, 2, \ldots, n.$ If $s = a_1 + a_2 + \ldots + a_n,$ prove that $\frac{a^2_1 + a^2_2 - a^2_3}{a_1 + a_2 - a_3} + \frac{a^2_2 + a^2_3 - a^2_4}{a_2 + a_3 - a_4} + \ldots + \frac{a^2_n + a^2_1 - a^2_2}{a_n + a_1 - a_2} \leq 2s - 2n.$
4. Find all of the positive real numbers like $x,y,z,$ such that
i) $x + y + z = a + b + c$
ii) $4xyz = a^2x + b^2y + c^2z + abc$
5. Let $\mathbb{R}$ be the set of real numbers. Does there exist a function $f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions?
a) There is a positive number $M$ such that $\forall x:$ $- M \leq f(x) \leq M.$
b) The value of $f(1)$ is $1$.
c) If $x \neq 0,$ then $f \left(x + \frac {1}{x^2} \right) = f(x) + \left[ f \left(\frac {1}{x} \right) \right]^2$
6. Let $n$ be an integer,$n \geq 3.$ Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_i < x_{i+1}$ for $1 \leq i \leq n - 1$. Prove that $\frac{n(n-1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n-1}_{i=1} (n-i)\cdot x_i \right) \cdot \left(\sum^{n}_{j=2} (j-1) \cdot x_j \right)$

### Geometry

1. Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.
2. Let $A, B$ and $C$ be non-collinear points. Prove that there is a unique point $X$ in the plane of $ABC$ such that $XA^2 + XB^2 + AB^2 = XB^2 + XC^2 + BC^2 = XC^2 + XA^2 + CA^2.$
3. The incircle of triangle $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ at $D, E, F$ respectively. $X$ is a point inside triangle of $\triangle ABC$ such that the incircle of triangle $\triangle XBC$ touches $BC$ at $D$, and touches $CX$ and $XB$ at $Y$ and $Z$ respectively. Show that $E, F, Z, Y$ are concyclic.
4. 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 $B$) so 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 $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
5. Let $ABCDEF$ be a convex hexagon with $AB = BC = CD$ and $DE = EF = FA$, such that $\angle BCD = \angle EFA = \frac {\pi}{3}$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB = \angle DHE = \frac {2\pi}{3}$. Prove that $$AG + GB + GH + DH + HE \geq CF.$$
6. 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 $GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4$ and $\frac{1}{GA'_1} + \frac{1}{GA'_2} + \frac{1}{GA'_3} + \frac{1}{GA'_4} \leq \frac{1}{GA_1} + \frac{1}{GA_2} + \frac{1}{GA_3} + \frac{1}{GA_4}.$
7. Let ABCD be a convex quadrilateral and O a point inside it. Let the parallels to the lines BC, AB, DA, CD through the point O meet the sides AB, BC, CD, DA of the quadrilateral ABCD at the points E, F, G, H, respectively. Then, prove that $\sqrt {\left|AHOE\right|} + \sqrt {\left|CFOG\right|}\leq\sqrt {\left|ABCD\right|}$, where $\left|P_1P_2...P_n\right|$ is an abbreviation for the non-directed area of an arbitrary polygon $P_1P_2...P_n$.
8. Suppose that $ABCD$ is a cyclic quadrilateral. Let $E = AC\cap BD$ and $F = AB\cap CD$. Denote by $H_{1}$ and $H_{2}$ the orthocenters of triangles $EAD$ and $EBC$, respectively. Prove that the points $F$, $H_{1}$, $H_{2}$ are collinear.

### Number Theory, Combinatorics

1. Let $k$ be a positive integer. Show that there are infinitely many perfect squares of the form $n \cdot 2^k - 7$ where $n$ is a positive integer.
2. 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 $$M_1 = \{x^2 + Ax + B : x \in \mathbb{Z}\} \\ M_2 = {2x^2 + 2x + C : x \in \mathbb{Z}}$$ do not intersect.
3. Determine all integers $n > 3$ for which there exist $n$ points $A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $r_{1},\cdots ,r_{n}$ such that for $1\leq i < j < k\leq n$, the area of $\triangle A_{i}A_{j}A_{k}$ is $r_{i} + r_{j} + r_{k}$.
4. Find all $x,y$ and $z$ in positive integer: $z + y^{2} + x^{3} = xyz$ and $x = \gcd(y,z)$.
5. 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?
6. Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $p$?
7. Does there exist an integer $n > 1$ which satisfies the following condition? The set of positive integers can be partitioned into $n$ nonempty subsets, such that an arbitrary sum of $n - 1$ integers, one taken from each of any $n - 1$ of the subsets, lies in the remaining subset.
8. Let $p$ be an odd prime. Determine positive integers $x$ and $y$ for which $x \leq y$ and $\sqrt{2p} - \sqrt{x} - \sqrt{y}$ is non-negative and as small as possible.

### Sequences

1. Does there exist a sequence $F(1), F(2), F(3), \ldots$ of non-negative integers that simultaneously satisfies the following three conditions?
a) Each of the integers $0, 1, 2, \ldots$ occurs in the sequence.
b) Each positive integer occurs in the sequence infinitely often.
c) For any $n \geq 2,$ $F(F(n^{163})) = F(F(n)) + F(F(361)).$
2. Find the maximum value of $x_{0}$ for which there exists a sequence $x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $x_{0} = x_{1995}$, such that $x_{i - 1} + \frac {2}{x_{i - 1}} = 2x_{i} + \frac {1}{x_{i}},$ for all $i = 1,\cdots ,1995$.
3. For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and $x_{n+1} = \frac{x_n p(x_n)}{q(x_n)}$ for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
4. Suppose that $x_1, x_2, x_3, \ldots$ are positive real numbers for which $x^n_n = \sum^{n-1}_{j=0} x^j_n$ for $n = 1, 2, 3, \ldots$ Prove that $\forall n,$ $2 - \frac{1}{2^{n-1}} \leq x_n < 2 - \frac{1}{2^n}.$
5. For positive integers $n,$ the numbers $f(n)$ are defined inductively as follows: $f(1) = 1,$ and for every positive integer $n,$ $f(n+1)$ is the greatest integer $m$ such that there is an arithmetic progression of positive integers $a_1 < a_2 < \ldots < a_m = n$ for which $f(a_1) = f(a_2) = \ldots = f(a_m).$ Prove that there are positive integers $a$ and $b$ such that $f(an+b) = n+2$ for every positive integer $n.$
6. Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $f: \mathbb{N} \mapsto \mathbb{N}$ satisfying $f(m + f(n)) = n + f(m + 95)$ for all $m$ and $n$ in $\mathbb{N}.$ What is the value of $\sum^{19}_{k = 1} f(k)?$
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