IR-MO 2004

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$ ,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$ , such that for every line $XX'$ passing through $O$ , where $X$ and $X'$ are boundary points of $A$ , then $$\frac12 \leq \frac {OX}{OX'} \leq 2.$$

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$ .We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$

We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$ . Find the smallest k that: $S(F) \leq k.P(F)^2$

$\mathbb{P}$ is a n-gon with sides $l_1 ,...,l_n$ and vertices on a circle. Prove that no n-gon with this sides has area more than $\mathbb{P}$

Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$ , respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$ , respectively. Prove that $\measuredangle ROS=\measuredangle BAC$ .

$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$ . Prove for $A$ in plane: $$S(A) \geq S(f(A))$$ where $S(A)$ is area of $A$

assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle at A" in the point $X_a$ ; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.

$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$ (Notice that $\overline {...000} \in \mathbb{N}_{10}$ ) Also we easily have $+,*$ in $\mathbb{N}_{10}$ . first $k$ number of $a*b$ = first $k$ nubmer of (first $k$ number of a * first $k$ number of b) first $k$ number of $a+b$ = first $k$ nubmer of (first $k$ number of a + first $k$ number of b) Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$ Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.

Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$ : $$a_i | a_{i+1}$$

We define $f: \mathbb{N} \rightarrow \mathbb{N}$ , $f(n) = \sum_{k = 1}^{n}(k,n)$ .

a) Show that if $\gcd(m,n)=1$ then we have $f(mn)=f(m)\cdot f(n)$ ;

b) Show that $\sum_{d|n}f(d) = nd(n)$ .

This problem is easy but nobody solved it. point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$ .We know $v \geq v'$ .If we know the point of beginning of path of $A$ , then $B$ must be where at first that $B$ can catch $A$ .

Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .

Let $p=4k+1$ be a prime. Prove that $p$ has at least $\frac{\phi(p-1)}2$ primitive roots.

Prove that for any $n$ , there is a subset $\{a_1,\dots,a_n\}$ of $\mathbb N$ such that for each subset $S$ of $\{1,\dots,n\}$ , $\sum_{i\in S}a_i$ has the same set of prime divisors.

Find all integer solutions of $p^3=p^2+q^2+r^2$ where $p,q,r$ are primes.

$p(x)$ is a polynomial in $\mathbb{Z}[x]$ such that for each $m,n\in \mathbb{N}$ there is an integer $a$ such that $n\mid p(a^m)$ . Prove that $0$ or $1$ is a root of $p(x)$ .

$a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $p$ such that for some $k$ $$p\mid a_1^k + \dots + a_n^k.$$

Suppose that $\mathcal F$ is a family of subsets of $X$ . $A,B$ are two subsets of $X$ s.t. each element of $\mathcal{F}$ has non-empty intersection with $A, B$ . We know that no subset of $X$ with $n - 1$ elements has this property. Prove that there is a representation $A,B$ in the form $A = \{a_1,\dots,a_n\}$ and $B = \{b_1,\dots,b_n\}$ , such that for each $1\leq i\leq n$ , there is an element of $\mathcal F$ containing both $a_i, b_i$ .

$\mathcal F$ is a family of 3-subsets of set $X$ . Every two distinct elements of $X$ are exactly in $k$ elements of $\mathcal F$ . It is known that there is a partition of $\mathcal F$ to sets $X_1,X_2$ such that each element of $\mathcal F$ has non-empty intersection with both $X_1,X_2$ . Prove that $|X|\leq4$ .

In triangle $ABC$ , points $M,N$ lie on line $AC$ such that $MA=AB$ and $NB=NC$ . Also $K,L$ lie on line $BC$ such that $KA=KB$ and $LA=LC$ . It is know that $KL=\frac12{BC}$ and $MN=AC$ . Find angles of triangle $ABC$ .

Finitely many convex subsets of $\mathbb R^3$ are given, such that every three have non-empty intersection. Prove that there exists a line in $\mathbb R^3$ that intersects all of these subsets.

Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.

$\Delta_1,\ldots,\Delta_n$ are $n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.

Find all prime numbers $p$ such that $p = m^2 + n^2$ and $p\mid m^3+n^3-4$ .

Find all polynomials $p\in\mathbb Z[x]$ such that $(m,n)=1\Rightarrow (p(m),p(n))=1$