# [Shortlists] International Mathematical Olympiad 2007

### Algebra

1. Real numbers $a_{1}$, $a_{2}$, $\ldots$, $a_{n}$ are given. For each $i$, $(1 \leq i \leq n )$, define $d_{i} = \max \{ a_{j}\mid 1 \leq j \leq i \} - \min \{ a_{j}\mid i \leq j \leq n \}$ and let $d = \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, $\max \{ |x_{i} - a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)$  (b) Show that there are real numbers $x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
2. Consider those functions $f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition $f(m + n) \geq f(m) + f(f(n)) - 1$ for all $m,n \in \mathbb{N}.$ Find all possible values of $f(2007).$
3. Let $n$ be a positive integer, and let $x$ and $y$ be a positive real number such that $x^n + y^n = 1.$ Prove that $\left(\sum^n_{k = 1} \frac {1 + x^{2k}}{1 + x^{4k}} \right) \cdot \left( \sum^n_{k = 1} \frac {1 + y^{2k}}{1 + y^{4k}} \right) < \frac {1}{(1 - x) \cdot (1 - y)}.$
4. Find all functions $f: \mathbb{R}^{ + }\to\mathbb{R}^{ + }$ satisfying $$f\left(x + f\left(y\right)\right) = f\left(x + y\right) + f\left(y\right)$$ for all pairs of positive reals $x$ and $y$. Here, $\mathbb{R}^{ + }$ denotes the set of all positive reals.
5. Let $c > 2,$ and let $a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that $a(m + n) \leq 2 \cdot a(m) + 2 \cdot a(n) \text{ for all } m,n \geq 1,$ and $a\left(2^k \right) \leq \frac {1}{(k + 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $a(n)$ is bounded.
6. Let $a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $$a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$$ Prove that $a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.$
7. Let $n$ be a positive integer. Consider $S = \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x + y + z > 0 \right \}$ as a set of $(n + 1)^{3} - 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $S$ but does not include $(0,0,0)$.

### Combinatorics

1. Let $n > 1$ be an integer. Find all sequences $a_1, a_2, \ldots a_{n^2 + n}$ satisfying the following conditions
• $a_i \in \left\{0,1\right\}$ for all $1 \leq i \leq n^2 + n$;
• $a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n}$ for all $0 \leq i \leq n^2 - n$.
2. A rectangle $D$ is partitioned in several ($\ge2$) rectangles with sides parallel to those of $D$. Given that any line parallel to one of the sides of $D$, and having common points with the interior of $D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $D$'s boundary.
3. Find all positive integers $n$ for which the numbers in the set $S = \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $S \times S \times S$ contains exactly $2007$ ordered triples $\left(x, y, z\right)$ such that
• the numbers $x$, $y$, $z$ are of the same color, and
• the number $x + y + z$ is divisible by $n$.
4. Let $A_0 = (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $k\geq 0$, from the sequence $A_k = (x_1,\dots,x_k)$ we construct a new sequence $A_{k + 1}$ in the following way.
• We choose a partition $\{1,\dots,n\} = I\cup J$, where $I$ and $J$ are two disjoint sets, such that the expression $\left|\sum_{i\in I}x_i - \sum_{j\in J}x_j\right|$ attains the smallest value. (We allow $I$ or $J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily.
• We set $A_{k + 1} = (y_1,\dots,y_n)$ where $y_i = x_i + 1$ if $i\in I$, and $y_i = x_i - 1$ if $i\in J$.
Prove that for some $k$, the sequence $A_k$ contains an element $x$ such that $|x|\geq\frac n2$.
5. In the Cartesian coordinate plane define the strips $$S_n = \{(x,y)|n\le x < n + 1\},\quad n\in\mathbb{Z}$$ and color each strip black or white. Prove that any rectangle which is not a square can be placed in the plane so that its vertices have the same color.
6. In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.
7. Let $\alpha < \frac {3 - \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $n$ and $p > \alpha \cdot 2^n$ for which one can select $2 \cdot p$ pairwise distinct subsets $S_1, \ldots, S_p$, $T_1, \ldots, T_p$ of the set $\{1,2, \ldots, n\}$ such that $S_i \cap T_j \neq \emptyset$ for all $1 \leq i,j \leq p$.
8. Given is a convex polygon $P$ with $n$ vertices. Triangle whose vertices lie on vertices of $P$ is called good if all its sides are equal in length. Prove that there are at most $\frac {2n}{3}$ good triangles.

### Geometry

1. In triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.
2. Denote by $M$ midpoint of side $BC$ in an isosceles triangle $\triangle ABC$ with $AC = AB$. Take a point $X$ on a smaller arc $\angle MA$ of circumcircle of triangle $\triangle ABM$. Denote by $T$ point inside of angle $BMA$ such that $\angle TMX = 90$ and $TX = BX$. Prove that $\angle MTB - \angle CTM$ does not depend on choice of $X$.
3. The diagonals of a trapezoid $ABCD$ intersect at point $P$. Point $Q$ lies between the parallel lines $BC$ and $AD$ such that $\angle AQD = \angle CQB$, and line $CD$ separates points $P$ and $Q$. Prove that $\angle BQP = \angle DAQ$.
4. Consider five points $A$, $B$, $C$, $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.
5. Let $ABC$ be a fixed triangle, and let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $CA$, $AB$, respectively. Let $P$ be a variable point on the circumcircle. Let lines $PA_1$, $PB_1$, $PC_1$ meet the circumcircle again at $A'$, $B'$, $C'$, respectively. Assume that the points $A$, $B$, $C$, $A'$, $B'$, $C'$ are distinct, and lines $AA'$, $BB'$, $CC'$ form a triangle. Prove that the area of this triangle does not depend on $P$.
6. Determine the smallest positive real number $k$ with the following property. Let $ABCD$ be a convex quadrilateral, and let points $A_1$, $B_1$, $C_1$, and $D_1$ lie on sides $AB$, $BC$, $CD$, and $DA$, respectively. Consider the areas of triangles $AA_1D_1$, $BB_1A_1$, $CC_1B_1$ and $DD_1C_1$; let $S$ be the sum of the two smallest ones, and let $S_1$ be the area of quadrilateral $A_1B_1C_1D_1$. Then we always have $kS_1\ge S$.
7. Given an acute triangle $ABC$ with $\angle B > \angle C$. Point $I$ is the incenter, and $R$ the circumradius. Point $D$ is the foot of the altitude from vertex $A$. Point $K$ lies on line $AD$ such that $AK = 2R$, and $D$ separates $A$ and $K$. Lines $DI$ and $KI$ meet sides $AC$ and $BC$ at $E,F$ respectively. Let $IE = IF$. Prove that $\angle B\leq 3\angle C$.
8. Point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the incircle of triangle $CPD$, and let $I$ be its incenter. Suppose that $\omega$ is tangent to the incircles of triangles $APD$ and $BPC$ at points $K$ and $L$, respectively. Let lines $AC$ and $BD$ meet at $E$, and let lines $AK$ and $BL$ meet at $F$. Prove that points $E$, $I$, and $F$ are collinear.

### Number Theory

1. Find all pairs of natural numbers $(a, b)$ such that $7^a - 3^b$ divides $a^4 + b^2$.
2. Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.
3. Let $X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $Y$ of $X$ such that $a - b + c - d + e$ is not divisible by 47 for any $a,b,c,d,e \in Y.$
4. For every integer $k \geq 2,$ prove that $2^{3k}$ divides the number $\binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}$ but $2^{3k + 1}$ does not.
5. Find all surjective functions $f: \mathbb{N} \to \mathbb{N}$ such that for every $m,n \in \mathbb{N}$ and every prime $p,$ the number $f(m + n)$ is divisible by $p$ if and only if $f(m) + f(n)$ is divisible by $p$.
6. Let $k$ be a positive integer. Prove that the number $(4 \cdot k^2 - 1)^2$ has a positive divisor of the form $8kn - 1$ if and only if $k$ is even
7. For a prime $p$ and a given integer $n$ let $\nu_p(n)$ denote the exponent of $p$ in the prime factorisation of $n!$. Given $d \in \mathbb{N}$ and $\{p_1,p_2,\ldots,p_k\}$ a set of $k$ primes, show that there are infinitely many positive integers $n$ such that $d\mid \nu_{p_i}(n)$ for all $1 \leq i \leq k$.
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