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Difference between revisions of "2016 USAJMO Problems"

(Created page with "==Day 1== ===Problem 1=== The isosceles triangle <math>\triangle ABC</math>, with <math>AB=AC</math>, is inscribed in the circle <math>\omega</math>. Let <math>P</math> be a v...")
 
 
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Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>{1, 2,...,N}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <math>N</math>.
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Find, with proof, the least integer <math>N</math> such that if any <math>2016</math> elements are removed from the set <math>\{1, 2,\dots,N\}</math>, one can still find <math>2016</math> distinct numbers among the remaining elements with sum <math>N</math>.
  
 
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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

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