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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 HKQ=90^\circ</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.
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==Problem 5==
 
==Problem 5==
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Let <math>\mathbb{R}</math> be the set of real numbers. Determine all functions <math>f</math>:<math>\mathbb{R}\rightarrow\mathbb{R}</math> satisfying the equation
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Let <math>\mathbb{R}</math> be the set of real numbers. Determine all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> satisfying the equation
  
 
<math>f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)</math>
 
<math>f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)</math>
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[[2015 IMO Problems/Problem 6|Solution]]
 
[[2015 IMO Problems/Problem 6|Solution]]
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==Problem 1==
 
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''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all
 
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''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''
 
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[[2019 IMO Problems/Problem 1|Solution]]
 
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==Problem 2==
 
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In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.
 
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Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.
 
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[[2019 IMO Problems/Problem 2|Solution]]
 
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==Problem 3==
 
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A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:
 
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Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.
 
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Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
 
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[[2019 IMO Problems/Problem 3|Solution]]
 
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==Problem 4==
 
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Find all pairs <math>(k,n)</math> of positive integers such that
 
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<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath>
 
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[[2019 IMO Problems/Problem 4|Solution]]
 
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==Problem 5==
 
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The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:
 
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If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.
 
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(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
 
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(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For
 
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example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math>
 
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possible initial configurations <math>C</math>.
 
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[[2019 IMO Problems/Problem 5|Solution]]
 
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==Problem 6==
 
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Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.
 
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Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.
 
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[[2019 IMO Problems/Problem 6|Solution]]
 
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{{IMO box|year=2015|before=[[2014 IMO Problems]]|after=[[2016 IMO Problems]]}}
 

Latest revision as of 15:00, 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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