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2007 Hello IMO

  1. Find the least natural number not divisible by $11$ satisfying the condition: by replaceing an arbitrary digit a of it by the digit $b$ ($b$ equals to $a+1$ or $a-1),$ the obtaining number is divisible by $11$.
  2. Consider an acute, scalene triangle $A B C$. Let $H$, $I$, $O$ be respectively its orthocenter, incenter and circumcenter. Prove that there is no vertex or there are exactly two vertices of triangle $A B C$ lying on the circle passing through $H$, $I$, $O$.
  3. Prove that $$x y z+2\left(x^{2}+y^{2}+z^{2}\right)+8 \geq 5(x+y+z)$$ for arbitrary positive real numbers $x, y, z$.
  4. Consider a board of size $5 \times 5 .$ Can we color $16$ small squares of this board so that in each subboard of size $2 \times 2$ there are at most two small squares which are colored?
  5. On the plane, let be given some points colored red and some points colored blue; points with distinct colors are joined so that
    • each red point is joined with at least one blue point,
    • each blue point is joined with one or two red points.
      Prove that one can erase less than one haff of the given points so that for the remainning points, each blue point is joined with exactly one red point.
    1. Let $A B C$ be a right-angled triangle, with right angle at $A$. Find all sets of the six distinct points $(M, N, P, Q, R, S)$ satisfying simultaneously the following conditions
      • $N$, $P$ lie on the side $A B$; $Q$, $R$ lie on the side $B C$; $S$, $M$ lie on the side $A C$.
      • $M N=P Q=R S$.
      • the hexagon $MNPQRS$ is a convex hexagon,  inscribable in a circle, with concurrent principal diagonals $M Q$, $N R$, $P S$.
    2. Let be given a real number $k$ in the interval $(-1 ; 2)$ and three pairwise distinct real numbers $a$, $b$, $c$. Prove that $$\left(a^{2}+b^{2}+c^{2}+k(a b+b c+c a)\right) \left(\frac{1}{(a-b)^{2}}+\frac{1}{(b-c)^{2}}+\frac{1}{(c-a)^{2}}\right) \geq \frac{9(2-k)}{4}.$$ When does equality occur?
    3. Does there exist a positive integer a such that in the sequence of numbers $\left(a_{n}\right)$ defined by $a_{n}=n^{3}+a^{3}$ for all $n=1,2,3, \ldots$ every two consecutive terms are coprime integers?
    4. Does there exist a positive integer $n$ such that one can assign to each vertex $A_{1}, A_{2}, \ldots, A_{n}$ of a convex $n$ -polygon an integer (these $n$ integers are not necessarily distinct) so that
      • the sum of these $n$ integers is equal to $2007$, and
      • for every $i=1,2, \ldots, n,$ the number assigned to $A_{1}$ is equal to the absolute value of the difference of the numbers assigned to $A_{t=1}$ and $A_{1}+2\left(\right.$ with the convention $A_{n}+1 \equiv A_{1}$ and $\left.A_{n}+2 \equiv A_{2}\right) ?$
    5. Suppose that in the coordinate plane every integral point (i. e. point with integral coordinates) was colored in one of two given colors. Prove that there exists a infinite set of integral points of the same color. forming. a figure admitting a center of symmetry.
    6. Let $A^{\prime}, B^{\prime}, C^{\prime}$ be the feet of the altitudes of nonright triangle $A B C$ issued from $A$, $B$, $C$ respectively. Let $D$, $E$, $F$ be the incenters of triangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$, $C A^{\prime} B^{\prime}$ respectively. Calculate the circumradius of triangle $D E F$ in terms of $B C=a$, $C A=b$, $A B=c$.
    7. Let be given a natural number $n>6$. Consider all natural number belonging to the interval $\left(n(n-1) ; n^{2}\right)$ which are coprime with $n(n-1) .$ Prove that the greates common divisor of these natural numbers is $1$.
    8. Find all functions $f$ defined on the set of all real number $\mathbb{R},$ with values in $\mathbb{R},$ satisfying the condition $$f(x-y)+f(x y)=f(x)-f(y)+f(x) \cdot f(y)$$ for all real numbers $x$, $y$.
    9. A group of students consists of $15$ boys and $15$ girls. On The Day of Women, some boys congratulated some girls by telephone. Suppose that we can only partition the group into 15 pairs such that each pair consists of a boy and a girl which had been congratulated by this boy. How many congratulations there were at most?
    10. Describe all ways to color all natural numbers in blue, red and yellow so that
      • each natural number is colored in one color and each color is used to color an infinite set of natural numbers,
      • the number 2 is not colored in blue,
      • if the natural number $a$ is colored in red and the natural number $b$ is colored in yellow then the number $a+b$ is colored in blue.

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    2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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    Mathematics & Youth: 2007 Hello IMO
    2007 Hello IMO
    Mathematics & Youth
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