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Difference between revisions of "2024 USAJMO Problems"

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== Day 1 ==
 
  
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=== Problem 1 ===
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==Day 1==
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Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=7</math> and <math>CD=8</math>. Points <math>P</math> and <math>Q</math> are selected on line segment <math>AB</math> so that <math>AP=BQ=3</math>. Points <math>R</math> and <math>S</math> are selected on line segment <math>CD</math> so that <math>CR=DS=2</math>. Prove that <math>PQRS</math> is a quadrilateral.
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===Problem 1===
 +
Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB = 7</math> and <math>CD = 8</math>. Points <math>P</math> and <math>Q</math> are selected on segment <math>AB</math> such that <math>AP = BQ = 3</math>. Points <math>R</math> and <math>S</math> are selected on segment <math>CD</math> such that <math>CR = DS = 2</math>. Prove that <math>PQRS</math> is a cyclic quadrilateral.
  
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=== Problem 2 ===
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[[2024 USAJMO Problems/Problem 1|Solution]]
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===Problem 2===
 
Let <math>m</math> and <math>n</math> be positive integers. Let <math>S</math> be the set of integer points <math>(x,y)</math> with <math>1\leq x\leq2m</math> and <math>1\leq y\leq2n</math>. A configuration of <math>mn</math> rectangles is called ''happy'' if each point in <math>S</math> is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
 
Let <math>m</math> and <math>n</math> be positive integers. Let <math>S</math> be the set of integer points <math>(x,y)</math> with <math>1\leq x\leq2m</math> and <math>1\leq y\leq2n</math>. A configuration of <math>mn</math> rectangles is called ''happy'' if each point in <math>S</math> is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
  
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=== Problem 3 ===
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[[2024 USAJMO Problems/Problem 2|Solution]]
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Let <math>a(n)</math> be the sequence defined by <math>a(1)=2</math> and <math>a(n+1)=(a(n))^{n+1}-1~ for each integer </math>n\geq1<math>. Suppose that </math>p>2<math> is prime and </math>k<math> is a positive integer. Prove that some term of the sequence </math>a(n)<math> is divisible by </math>p^k$.
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===Problem 3===
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Let <math>a(n)</math> be the sequence defined by <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>.
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[[2024 USAJMO Problems/Problem 3|Solution]]
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==Day 2==
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===Problem 4===
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Let <math>n \ge 3</math> be an integer. Rowan and Colin play a game on an <math>n \times n</math> grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is <math>orderly</math> if:
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*no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and
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*no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring;
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In terms of <math>n</math>, how many orderly colorings are there?
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[[2024 USAJMO Problems/Problem 4|Solution]]
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===Problem 5===
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Find all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> that satisfy
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<cmath>f(x^2-y)+2yf(x)=f(f(x))+f(y)</cmath>
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for all <math>x,y\in\mathbb{R}</math>.
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[[2024 USAJMO Problems/Problem 5|Solution]]
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===Problem 6===
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Point <math>D</math> is selected inside acute triangle <math>ABC</math> so that <math>\angle DAC=\angle ACB</math> and <math>\angle BDC=90^\circ+\angle BAC</math>. Point <math>E</math> is chosen on ray <math>BD</math> so that <math>AE=EC</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Show that line <math>AB</math> is tangent to the circumcircle of triangle <math>BEM</math>.
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[[2024 USAJMO Problems/Problem 6|Solution]]
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== See also ==
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{{USAJMO box|year=2024|before=[[2023 USAJMO Problems]]|after=[[2025 USAJMO Problems]]}}
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{{MAA Notice}}

Latest revision as of 16:11, 12 June 2024

Day 1

Problem 1

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Points $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral.

Solution

Problem 2

Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq2m$ and $1\leq y\leq2n$. A configuration of $mn$ rectangles is called happy if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.

Solution

Problem 3

Let $a(n)$ be the sequence defined by $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq1$. Suppose that $p>2$ is prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$.

Solution

Day 2

Problem 4

Let $n \ge 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is $orderly$ if:

  • no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and
  • no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring;

In terms of $n$, how many orderly colorings are there?

Solution

Problem 5

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(x^2-y)+2yf(x)=f(f(x))+f(y)\] for all $x,y\in\mathbb{R}$.

Solution

Problem 6

Point $D$ is selected inside acute triangle $ABC$ so that $\angle DAC=\angle ACB$ and $\angle BDC=90^\circ+\angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE=EC$. Let $M$ be the midpoint of $BC$. Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.

Solution

See also

2024 USAJMO (Problems • Resources)
Preceded by
2023 USAJMO Problems 		Followed by
2025 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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