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Difference between revisions of "2025 USAJMO Problems"

(2025 USAJO)
 
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yikes! somebody is cheating for the answer!
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__TOC__
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==Day 1==
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===Problem 1===
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Let <math>\mathbb Z</math> be the set of integers, and let <math>f\colon \mathbb Z \to \mathbb Z</math> be a function. Prove that there are infinitely many integers <math>c</math> such that the function <math>g\colon \mathbb Z \to \mathbb Z</math> defined by <math>g(x) = f(x) + cx</math> is not bijective.
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Note: A function <math>g\colon \mathbb Z \to \mathbb Z</math> is bijective if for every integer <math>b</math>, there exists exactly one integer <math>a</math> such that <math>g(a) = b</math>.
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[[2025 USAJMO Problems/Problem 1|Solution]]
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===Problem 2===
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Let <math>k</math> and <math>d</math> be positive integers. Prove that there exists a positive integer <math>N</math> such that for every odd integer <math>n>N</math>, the digits in the base-<math>2n</math> representation of <math>n^k</math> are all greater than <math>d</math>.
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[[2025 USAJMO Problems/Problem 2|Solution]]
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===Problem 3===
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Let <math>m</math> and <math>n</math> be positive integers, and let <math>\mathcal R</math> be a <math>2m\times 2n</math> grid of unit squares.
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A domino is a <math>1\times2</math> or <math>2\times1</math> rectangle. A subset <math>S</math> of grid squares in <math>\mathcal R</math> is domino-tileable if dominoes can be placed to cover every square of <math>S</math> exactly once with no domino extending outside of <math>S</math>. Note: The empty set is domino tileable.
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An up-right path is a path from the lower-left corner of <math>\mathcal R</math> to the upper-right corner of <math>\mathcal R</math> formed by exactly <math>2m+2n</math> edges of the grid squares.
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Determine, with proof, in terms of <math>m</math> and <math>n</math>, the number of up-right paths that divide <math>\mathcal R</math> into two domino-tileable subsets.
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[[2025 USAJMO Problems/Problem 3|Solution]]
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==Day 2==
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===Problem 4===
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Let <math>n</math> be a positive integer, and let <math>a_0,\,a_1,\dots,\,a_n</math> be nonnegative integers such that <math>a_0\ge a_1\ge \dots\ge a_n.</math> Prove that
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<cmath>\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.</cmath>
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Note: <math>\binom{k}{2}=\frac{k(k-1)}{2}</math> for all nonnegative integers <math>k</math>.
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[[2025 USAJMO Problems/Problem 4|Solution]]
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===Problem 5===
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Let <math>H</math> be the orthocenter of acute triangle <math>ABC</math>, let <math>F</math> be the foot of the altitude from <math>C</math> to <math>AB</math>, and let <math>P</math> be the reflection of <math>H</math> across <math>BC</math>. Suppose that the circumcircle of triangle <math>AFP</math> intersects line <math>BC</math> at two distinct points <math>X</math> and <math>Y</math>. Prove that <math>C</math> is the midpoint of <math>XY</math>.
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[[2025 USAJMO Problems/Problem 5|Solution]]
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===Problem 6===
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Let <math>S</math> be a set of integers with the following properties:
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<math>\bullet</math> <math>\{ 1, 2, \dots, 2025 \} \subseteq S</math>.
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<math>\bullet</math> If <math>a, b \in S</math> and <math>\gcd(a, b) = 1</math>, then <math>ab \in S</math>.
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<math>\bullet</math> If for some <math>s \in S</math>, <math>s + 1</math> is composite, then all positive divisors of <math>s + 1</math> are in <math>S</math>.
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Prove that <math>S</math> contains all positive integers.
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[[2025 USAJMO Problems/Problem 6|Solution]]
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== See also ==
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{{USAJMO box|year=2025|before=[[2024 USAJMO Problems]]|after=[[2026 USAJMO Problems]]}}
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{{MAA Notice}}

Revision as of 01:22, 22 March 2025

Day 1

Problem 1

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective. Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.

Solution

Problem 2

Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

Solution

Problem 3

Let $m$ and $n$ be positive integers, and let $\mathcal R$ be a $2m\times 2n$ grid of unit squares.

A domino is a $1\times2$ or $2\times1$ rectangle. A subset $S$ of grid squares in $\mathcal R$ is domino-tileable if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside of $S$. Note: The empty set is domino tileable.

An up-right path is a path from the lower-left corner of $\mathcal R$ to the upper-right corner of $\mathcal R$ formed by exactly $2m+2n$ edges of the grid squares.

Determine, with proof, in terms of $m$ and $n$, the number of up-right paths that divide $\mathcal R$ into two domino-tileable subsets.

Solution

Day 2

Problem 4

Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that \[\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.\] Note: $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.

Solution

Problem 5

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

Solution

Problem 6

Let $S$ be a set of integers with the following properties:

$\bullet$ $\{ 1, 2, \dots, 2025 \} \subseteq S$.

$\bullet$ If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.

$\bullet$ If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$.

Prove that $S$ contains all positive integers.

Solution

See also

2025 USAJMO (Problems • Resources)
Preceded by
2024 USAJMO Problems 		Followed by
2026 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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