Difference between revisions of "2014 USAJMO Problems"
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===Problem 1=== | ===Problem 1=== | ||
+ | Let <math>a</math>, <math>b</math>, <math>c</math> be real numbers greater than or equal to <math>1</math>. Prove that <cmath>\min{\left (\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> | ||
[[2014 USAJMO Problems/Problem 1|Solution]] | [[2014 USAJMO Problems/Problem 1|Solution]] | ||
+ | |||
===Problem 2=== | ===Problem 2=== | ||
+ | Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circumcenter and orthocenter of <math>\triangle{ABC}</math>, respectively. | ||
+ | |||
+ | (a) Prove that line <math>OH</math> intersects both segments <math>AB</math> and <math>AC</math>. | ||
+ | |||
+ | (b) Line <math>OH</math> intersects segments <math>AB</math> and <math>AC</math> at <math>P</math> and <math>Q</math>, respectively. Denote by <math>s</math> and <math>t</math> the respective areas of triangle <math>APQ</math> and quadrilateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>. | ||
+ | |||
[[2014 USAJMO Problems/Problem 2|Solution]] | [[2014 USAJMO Problems/Problem 2|Solution]] | ||
+ | |||
===Problem 3=== | ===Problem 3=== | ||
+ | Let <math>\mathbb{Z}</math> be the set of integers. Find all functions <math>f : \mathbb{Z} \rightarrow \mathbb{Z}</math> such that <cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>. | ||
+ | |||
[[2014 USAJMO Problems/Problem 3|Solution]] | [[2014 USAJMO Problems/Problem 3|Solution]] | ||
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===Problem 4=== | ===Problem 4=== | ||
+ | Let <math>b\geq 2</math> be an integer, and let <math>s_b(n)</math> denote the sum of the digits of <math>n</math> when it is written in base <math>b</math>. Show that there are infinitely many positive integers that cannot be represented in the form <math>n+s_b(n)</math>, where <math>n</math> is a positive integer. | ||
+ | |||
[[2014 USAJMO Problems/Problem 4|Solution]] | [[2014 USAJMO Problems/Problem 4|Solution]] | ||
+ | |||
===Problem 5=== | ===Problem 5=== | ||
+ | Let <math>k</math> be a positive integer. Two players <math>A</math> and <math>B</math> play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with <math>A</math> moving first. In his move, <math>A</math> may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, <math>B</math> may choose any counter on the board and remove it. If at any time there are <math>k</math> consecutive grid cells in a line all of which contain a counter, <math>A</math> wins. Find the minimum value of <math>k</math> for which <math>A</math> cannot win in a finite number of moves, or prove that no such minimum value exists. | ||
+ | |||
[[2014 USAJMO Problems/Problem 5|Solution]] | [[2014 USAJMO Problems/Problem 5|Solution]] | ||
+ | |||
===Problem 6=== | ===Problem 6=== | ||
+ | Let <math>ABC</math> be a triangle with incenter <math>I</math>, incircle <math>\gamma</math> and circumcircle <math>\Gamma</math>. Let <math>M,N,P</math> be the midpoints of sides <math>\overline{BC}</math>, <math>\overline{CA}</math>, <math>\overline{AB}</math> and let <math>E,F</math> be the tangency points of <math>\gamma</math> with <math>\overline{CA}</math> and <math>\overline{AB}</math>, respectively. Let <math>U,V</math> be the intersections of line <math>EF</math> with line <math>MN</math> and line <math>MP</math>, respectively, and let <math>X</math> be the midpoint of arc <math>BAC</math> of <math>\Gamma</math>. | ||
+ | |||
+ | (a) Prove that <math>I</math> lies on ray <math>CV</math>. | ||
+ | |||
+ | (b) Prove that line <math>XI</math> bisects <math>\overline{UV}</math>. | ||
+ | |||
[[2014 USAJMO Problems/Problem 6|Solution]] | [[2014 USAJMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | == See Also == | ||
+ | *[[USAJMO Problems and Solutions]] | ||
+ | |||
+ | {{USAJMO box|year=2014|before=[[2013 USAJMO Problems]]|after=[[2015 USAJMO Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 16:42, 5 August 2023
Contents
Day 1
Problem 1
Let ,
,
be real numbers greater than or equal to
. Prove that
Solution
Problem 2
Let be a non-equilateral, acute triangle with
, and let
and
denote the circumcenter and orthocenter of
, respectively.
(a) Prove that line intersects both segments
and
.
(b) Line intersects segments
and
at
and
, respectively. Denote by
and
the respective areas of triangle
and quadrilateral
. Determine the range of possible values for
.
Problem 3
Let be the set of integers. Find all functions
such that
for all
with
.
Day 2
Problem 4
Let be an integer, and let
denote the sum of the digits of
when it is written in base
. Show that there are infinitely many positive integers that cannot be represented in the form
, where
is a positive integer.
Problem 5
Let be a positive integer. Two players
and
play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with
moving first. In his move,
may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move,
may choose any counter on the board and remove it. If at any time there are
consecutive grid cells in a line all of which contain a counter,
wins. Find the minimum value of
for which
cannot win in a finite number of moves, or prove that no such minimum value exists.
Problem 6
Let be a triangle with incenter
, incircle
and circumcircle
. Let
be the midpoints of sides
,
,
and let
be the tangency points of
with
and
, respectively. Let
be the intersections of line
with line
and line
, respectively, and let
be the midpoint of arc
of
.
(a) Prove that lies on ray
.
(b) Prove that line bisects
.
See Also
2014 USAJMO (Problems • Resources) | ||
Preceded by 2013 USAJMO Problems |
Followed by 2015 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.