Difference between revisions of "2016 APMO Problems"
Line 1: | Line 1: | ||
==Problem 1== | ==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>. | ||
== See Also == | == See Also == |
Revision as of 21:06, 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
.