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Difference between revisions of "2023 USAJMO Problems"

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__TOC__
 
==Day 1==
 
==Day 1==
  
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==Problem 1==
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===Problem 1===
 
Find all triples of positive integers <math>(x,y,z)</math> that satisfy the equation
 
Find all triples of positive integers <math>(x,y,z)</math> that satisfy the equation
 
<cmath>
 
<cmath>
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</cmath>
 
</cmath>
  
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==Problem 2==
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[[2023 USAJMO Problems/Problem 1|Solution]]
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===Problem 2===
 
In an acute triangle <math>ABC</math>, let <math>M</math> be the midpoint of <math>\overline{BC}</math>. Let <math>P</math> be the foot of the perpendicular from <math>C</math> to <math>AM</math>. Suppose that the circumcircle of triangle <math>ABP</math> intersects line <math>BC</math> at two distinct points <math>B</math> and <math>Q</math>. Let <math>N</math> be the midpoint of <math>\overline{AQ}</math>. Prove that <math>NB=NC</math>.
 
In an acute triangle <math>ABC</math>, let <math>M</math> be the midpoint of <math>\overline{BC}</math>. Let <math>P</math> be the foot of the perpendicular from <math>C</math> to <math>AM</math>. Suppose that the circumcircle of triangle <math>ABP</math> intersects line <math>BC</math> at two distinct points <math>B</math> and <math>Q</math>. Let <math>N</math> be the midpoint of <math>\overline{AQ}</math>. Prove that <math>NB=NC</math>.
  
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==Problem 3==
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[[2023 USAJMO Problems/Problem 2|Solution]]
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===Problem 3===
 
Consider an <math>n</math>-by-<math>n</math> board of unit squares for some odd positive integer <math>n</math>. We say that a collection <math>C</math> of identical dominoes is a maximal grid-aligned configuration on the board if <math>C</math> consists of <math>(n^2-1)/2</math> dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: <math>C</math> then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let <math>k(C)</math> be the number of distinct maximal grid-aligned configurations obtainable from <math>C</math> by repeatedly sliding dominoes. Find the maximum value of <math>k(C)</math> as a function of <math>n</math>.
 
Consider an <math>n</math>-by-<math>n</math> board of unit squares for some odd positive integer <math>n</math>. We say that a collection <math>C</math> of identical dominoes is a maximal grid-aligned configuration on the board if <math>C</math> consists of <math>(n^2-1)/2</math> dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: <math>C</math> then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let <math>k(C)</math> be the number of distinct maximal grid-aligned configurations obtainable from <math>C</math> by repeatedly sliding dominoes. Find the maximum value of <math>k(C)</math> as a function of <math>n</math>.
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 +
[[2023 USAJMO Problems/Problem 3|Solution]]
  
 
==Day 2==
 
==Day 2==
  
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==Problem 4==
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===Problem 4===
 
Two players, <math>B</math> and <math>R</math>, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with <math>B</math>. On <math>B</math>'s turn, <math>B</math> selects one white unit square and colors it blue. On <math>R</math>'s turn, <math>R</math> selects two white unit squares and colors them red. The players alternate until <math>B</math> decides to end the game. At this point, <math>B</math> gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score <math>B</math> can guarantee?
 
Two players, <math>B</math> and <math>R</math>, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with <math>B</math>. On <math>B</math>'s turn, <math>B</math> selects one white unit square and colors it blue. On <math>R</math>'s turn, <math>R</math> selects two white unit squares and colors them red. The players alternate until <math>B</math> decides to end the game. At this point, <math>B</math> gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score <math>B</math> can guarantee?
  
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(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.
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(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)
 +
 
 +
[[2023 USAJMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
 
A positive integer <math>a</math> is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer <math>n</math> on the board with <math>n+a</math>, and on Bob's turn he must replace some even integer <math>n</math> on the board with <math>n/2</math>. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
 
A positive integer <math>a</math> is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer <math>n</math> on the board with <math>n+a</math>, and on Bob's turn he must replace some even integer <math>n</math> on the board with <math>n/2</math>. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
  
 
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of <math>a</math> and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
 
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of <math>a</math> and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
  
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==Problem 6==
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[[2023 USAJMO Problems/Problem 5|Solution]]
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===Problem 6===
 
Isosceles triangle <math>ABC</math>, with <math>AB=AC</math>, is inscribed in circle <math>\omega</math>. Let <math>D</math> be an arbitrary point inside <math>BC</math> such that <math>BD\neq DC</math>. Ray <math>AD</math> intersects <math>\omega</math> again at <math>E</math> (other than <math>A</math>). Point <math>F</math> (other than <math>E</math>) is chosen on <math>\omega</math> such that <math>\angle DFE = 90^\circ</math>. Line <math>FE</math> intersects rays <math>AB</math> and <math>AC</math> at points <math>X</math> and <math>Y</math>, respectively. Prove that <math>\angle XDE = \angle EDY</math>.
 
Isosceles triangle <math>ABC</math>, with <math>AB=AC</math>, is inscribed in circle <math>\omega</math>. Let <math>D</math> be an arbitrary point inside <math>BC</math> such that <math>BD\neq DC</math>. Ray <math>AD</math> intersects <math>\omega</math> again at <math>E</math> (other than <math>A</math>). Point <math>F</math> (other than <math>E</math>) is chosen on <math>\omega</math> such that <math>\angle DFE = 90^\circ</math>. Line <math>FE</math> intersects rays <math>AB</math> and <math>AC</math> at points <math>X</math> and <math>Y</math>, respectively. Prove that <math>\angle XDE = \angle EDY</math>.
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[[2023 USAJMO Problems/Problem 6|Solution]]
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 +
== See also ==
 +
 +
{{USAJMO box|year=2023|before=[[2022 USAJMO Problems]]|after=[[2024 USAJMO Problems]]}}
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{{MAA Notice}}

Latest revision as of 20:38, 19 March 2024

Day 1

Problem 1

Find all triples of positive integers $(x,y,z)$ that satisfy the equation \begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}

Solution

Problem 2

In an acute triangle $ABC$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the foot of the perpendicular from $C$ to $AM$. Suppose that the circumcircle of triangle $ABP$ intersects line $BC$ at two distinct points $B$ and $Q$. Let $N$ be the midpoint of $\overline{AQ}$. Prove that $NB=NC$.

Solution

Problem 3

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$.

Solution

Day 2

Problem 4

Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$'s turn, $B$ selects one white unit square and colors it blue. On $R$'s turn, $R$ selects two white unit squares and colors them red. The players alternate until $B$ decides to end the game. At this point, $B$ gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score $B$ can guarantee?

(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)

Solution

Problem 5

A positive integer $a$ is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer $n$ on the board with $n+a$, and on Bob's turn he must replace some even integer $n$ on the board with $n/2$. Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.

After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of $a$ and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.

Solution

Problem 6

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$.

Solution

See also

2023 USAJMO (Problems • Resources)
Preceded by
2022 USAJMO Problems 		Followed by
2024 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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