Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2016 USAJMO Problems"
 
Line 36: Line 36:
  
 
[[2016 USAJMO Problems/Problem 6|Solution]]
 
[[2016 USAJMO Problems/Problem 6|Solution]]
−
 
−
== See Also ==
 
−
*[[USAJMO Problems and Solutions]]
 
  
 
{{USAJMO box|year=2016|before=[[2015 USAJMO Problems]]|after=[[2017 USAJMO Problems]]}}
 
{{USAJMO box|year=2016|before=[[2015 USAJMO Problems]]|after=[[2017 USAJMO Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:43, 5 August 2023

Day 1

Problem 1

The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively.

Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.

Solution

Problem 2

Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.

Solution

Problem 3

Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$.

Solution

Day 2

Problem 4

Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\{1, 2,\dots,N\}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.

Solution

Problem 5

Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, respectively.

Given that \[AH^2=2\cdot AO^2,\]prove that the points $O,P,$ and $Q$ are collinear.

Solution

Problem 6

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, \[(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.\]

Solution

2016 USAJMO (Problems • Resources)
Preceded by
2015 USAJMO Problems 		Followed by
2017 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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2016_USAJMO_Problems&oldid=196452"