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Difference between revisions of "2015 IMO Problems"

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==Problem 3==
 
==Problem 3==
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Let <math>ABC</math> be an acute triangle with <math>AB > AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HQA = 90◦</math> and let <math>K</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ = 90◦</math> . Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math> and <math>Q</math> are all different and lie on <math>\gamma</math> in this order.
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Let <math>ABC</math> be an acute triangle with <math>AB > AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HQA = 90◦</math> and let <math>K</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ = 90◦</math> . Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math> and <math>Q</math> are all different and lie on <math>\Gamma</math> in this order.
  
 
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
 
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.

Latest revision as of 14:00, 1 June 2024

Problem 1

We say that a finite set $\mathcal{S}$ in the plane is balanced if, for any two different points $A$, $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three points $A$, $B$, $C$ in $\mathcal{S}$, there is no point $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

  1. Show that for all integers $n\geq 3$, there exists a balanced set consisting of $n$ points.
  2. Determine all integers $n\geq 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Solution

Problem 2

Determine all triples of positive integers $(a,b,c)$ such that each of the numbers \[ab-c,\; bc-a,\; ca-b\] is a power of 2.

(A power of 2 is an integer of the form $2^n$ where $n$ is a non-negative integer ).

Solution

Problem 3

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90◦$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90◦$ . Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Solution

Problem 4

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Solution

Problem 5

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation

$f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$

for all real numbers $x$ and $y$.

Solution

Problem 6

The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:

(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.

Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.

Solution

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