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Difference between revisions of "2023 USAMO Problems"

(Created page with "==Day 1== ===Problem 1=== 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 perpen...")
 
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===Problem 5===
 
===Problem 5===
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Let <math>n\geq3</math> be an integer. We say that an arrangement of the numbers <math>1</math>, <math>2</math>, <math>\dots</math>, <math>n^2</math> in a <math>n \times n</math> 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 <math>n</math> is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
  
 
[[2023 USAMO Problems/Problem 5|Solution]]
 
[[2023 USAMO Problems/Problem 5|Solution]]

Revision as of 13:13, 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

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

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