Difference between revisions of "2023 USAMO Problems"

Revision as of 13:14, 24 April 2023

Day 1

Problem 1

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 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 2

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that, for all $x, y \in \mathbb{R}^{+}$ , $f(xy + f(x)) = xf(y) + 2$

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

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 5

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$ , $2$ , $\dots$ , $n^2$ in a $n \times n$ table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

Solution

Problem 6

Solution

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 18: / Line 18: @@
 ==Day 2==
 ===Problem 4===
+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.
 [[2023 USAMO Problems/Problem 4|Solution]]

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Difference between revisions of "2023 USAMO Problems"

Revision as of 13:14, 24 April 2023

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6