# Difference between revisions of "2016 APMO Problems"

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− | == | + | ==Problem 1== |

+ | We say that a triangle <math>ABC</math> is great if the following holds: for any point <math>D</math> on the side <math>BC</math>, if <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>D</math> to the lines <math>AB</math> and <math>AC</math>, respectively, then the reflection of <math>D</math> in the line <math>PQ</math> lies on the circumcircle of the triangle <math>ABC</math>. Prove that triangle <math>ABC</math> is great if and only if <math>\angle A = 90^{\circ}</math> and <math>AB = AC</math>. | ||

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+ | [[2016 APMO Problems/Problem 1|Solution]] | ||

+ | ==Problem 2== | ||

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+ | A positive integer is called fancy if it can be expressed in the form<cmath>2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},</cmath>where <math>a_1,a_2, \cdots, a_{100}</math> are non-negative integers that are not necessarily distinct. Find the smallest positive integer <math>n</math> such that no multiple of <math>n</math> is a fancy number. | ||

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+ | [[2016 APMO Problems/Problem 2|Solution]] | ||

+ | ==Problem 3== | ||

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+ | Let <math>AB</math> and <math>AC</math> be two distinct rays not lying on the same line, and let <math>\omega</math> be a circle with center <math>O</math> that is tangent to ray <math>AC</math> at <math>E</math> and ray <math>AB</math> at <math>F</math>. Let <math>R</math> be a point on segment <math>EF</math>. The line through <math>O</math> parallel to <math>EF</math> intersects line <math>AB</math> at <math>P</math>. Let <math>N</math> be the intersection of lines <math>PR</math> and <math>AC</math>, and let <math>M</math> be the intersection of line <math>AB</math> and the line through <math>R</math> parallel to <math>AC</math>. Prove that line <math>MN</math> is tangent to <math>\omega</math>. | ||

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+ | [[2016 APMO Problems/Problem 3|Solution]] | ||

+ | ==Problem 4== | ||

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+ | The country Dreamland consists of <math>2016</math> cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer <math>k</math> such that no matter how Starways establishes its flights, the cities can always be partitioned into <math>k</math> groups so that from any city it is not possible to reach another city in the same group by using at most <math>28</math> flights. | ||

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+ | [[2016 APMO Problems/Problem 4|Solution]] | ||

+ | ==Problem 5== | ||

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+ | Find all functions <math>f: \mathbb{R}^+ \to \mathbb{R}^+</math> such that | ||

+ | <cmath>(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),</cmath>for all positive real numbers <math>x, y, z</math>. | ||

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+ | [[2016 APMO Problems/Problem 5|Solution]] | ||

== See Also == | == See Also == | ||

* [[Asian Pacific Mathematics Olympiad]] | * [[Asian Pacific Mathematics Olympiad]] | ||

* [[APMO Problems and Solutions]] | * [[APMO Problems and Solutions]] | ||

* [[Mathematics competition resources]] | * [[Mathematics competition resources]] |

## Latest revision as of 22:09, 11 July 2021

## Problem 1

We say that a triangle is great if the following holds: for any point on the side , if and are the feet of the perpendiculars from to the lines and , respectively, then the reflection of in the line lies on the circumcircle of the triangle . Prove that triangle is great if and only if and .

## Problem 2

A positive integer is called fancy if it can be expressed in the formwhere are non-negative integers that are not necessarily distinct. Find the smallest positive integer such that no multiple of is a fancy number.

## Problem 3

Let and be two distinct rays not lying on the same line, and let be a circle with center that is tangent to ray at and ray at . Let be a point on segment . The line through parallel to intersects line at . Let be the intersection of lines and , and let be the intersection of line and the line through parallel to . Prove that line is tangent to .

## Problem 4

The country Dreamland consists of cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer such that no matter how Starways establishes its flights, the cities can always be partitioned into groups so that from any city it is not possible to reach another city in the same group by using at most flights.

## Problem 5

Find all functions such that for all positive real numbers .