Difference between revisions of "2016 APMO Problems"

Revision as of 21:06, 11 July 2021

Problem 1

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

Problem 2

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

Problem 3

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

Problem 4

The country Dreamland consists of $2016$ 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 $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights.

Problem 5

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$ for all positive real numbers $x, y, z$ .

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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>.
+==Problem 2==
+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.
+==Problem 3==
+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>.
+==Problem 4==
+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.
+==Problem 5==
+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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