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Difference between revisions of "2024 USAMO Problems"
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1. Find all integers <math>n \geq 3</math> such that the following property holds: if we list the divisors of <math>n !</math> in increasing order as <math>1=d_1<d_2<\cdots<d_k=n!</math>, then we have
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==Day 1==
 +
===Problem 1===
 +
Find all integers <math>n \geq 3</math> such that the following property holds: if we list the divisors of <math>n !</math> in increasing order as <math>1=d_1<d_2<\cdots<d_k=n!</math>, then we have
  
 
<cmath>d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1}.</cmath>
 
<cmath>d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1}.</cmath>
  
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2. Let <math>S_1, S_2, \ldots, S_{100}</math> be finite sets of integers whose intersection is not empty. For each non-empty <math>T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}</math>, the size of the intersection of the sets in <math>T</math> is a multiple of the number of sets in <math>T</math>. What is the least possible number of elements that are in at least 50 sets?
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===Problem 2===
 +
Let <math>S_1, S_2, \ldots, S_{100}</math> be finite sets of integers whose intersection is not empty. For each non-empty <math>T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}</math>, the size of the intersection of the sets in <math>T</math> is a multiple of the number of sets in <math>T</math>. What is the least possible number of elements that are in at least 50 sets?
  
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3. Let <math>m</math> be a positive integer. A triangulation of a polygon is <math>m</math>-balanced if its triangles can be colored with <math>m</math> colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the <math>m</math> colors. Find all positive integers <math>n</math> for which there exists an <math>m</math>-balanced triangulation of a regular <math>n</math>-gon.
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===Problem 3===
 +
Let <math>m</math> be a positive integer. A triangulation of a polygon is <math>m</math>-balanced if its triangles can be colored with <math>m</math> colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the <math>m</math> colors. Find all positive integers <math>n</math> for which there exists an <math>m</math>-balanced triangulation of a regular <math>n</math>-gon.
  
 
Note: A triangulation of a convex polygon <math>\mathcal{P}</math> with <math>n \geq 3</math> sides is any partitioning of <math>\mathcal{P}</math> into <math>n-2</math> triangles by <math>n-3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the polygon's interior.
 
Note: A triangulation of a convex polygon <math>\mathcal{P}</math> with <math>n \geq 3</math> sides is any partitioning of <math>\mathcal{P}</math> into <math>n-2</math> triangles by <math>n-3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the polygon's interior.
  
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4. Let <math>m</math> and <math>n</math> be positive integers. A circular necklace contains <math>m n</math> beads, each either red or blue. It turned out that no matter how the necklace was cut into <math>m</math> blocks of <math>n</math> consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair <math>(m, n)</math>.
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==Day 2==
 +
===Problem 4===
 +
Let <math>m</math> and <math>n</math> be positive integers. A circular necklace contains <math>m n</math> beads, each either red or blue. It turned out that no matter how the necklace was cut into <math>m</math> blocks of <math>n</math> consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair <math>(m, n)</math>.
  
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5. Point <math>D</math> is selected inside acute triangle <math>A B C</math> so that <math>\angle D A C=</math> <math>\angle A C B</math> and <math>\angle B D C=90^{\circ}+\angle B A C</math>. Point <math>E</math> is chosen on ray <math>B D</math> so that <math>A E=E C</math>. Let <math>M</math> be the midpoint of <math>B C</math>.
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===Problem 5===
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Point <math>D</math> is selected inside acute triangle <math>A B C</math> so that <math>\angle D A C=</math> <math>\angle A C B</math> and <math>\angle B D C=90^{\circ}+\angle B A C</math>. Point <math>E</math> is chosen on ray <math>B D</math> so that <math>A E=E C</math>. Let <math>M</math> be the midpoint of <math>B C</math>.
 
Show that line <math>A B</math> is tangent to the circumcircle of triangle <math>B E M</math>.
 
Show that line <math>A B</math> is tangent to the circumcircle of triangle <math>B E M</math>.
  
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6. Let <math>n>2</math> be an integer and let <math>\ell \in\{1,2, \ldots, n\}</math>. A collection <math>A_1, \ldots, A_k</math> of (not necessarily distinct) subsets of <math>\{1,2, \ldots, n\}</math> is called <math>\ell</math>-large if <math>\left|A_i\right| \geq \ell</math> for all <math>1 \leq i \leq k</math>. Find, in terms of <math>n</math> and <math>\ell</math>, the largest real number <math>c</math> such that the inequality
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===Problem 6===
 +
Let <math>n>2</math> be an integer and let <math>\ell \in\{1,2, \ldots, n\}</math>. A collection <math>A_1, \ldots, A_k</math> of (not necessarily distinct) subsets of <math>\{1,2, \ldots, n\}</math> is called <math>\ell</math>-large if <math>\left|A_i\right| \geq \ell</math> for all <math>1 \leq i \leq k</math>. Find, in terms of <math>n</math> and <math>\ell</math>, the largest real number <math>c</math> such that the inequality
 
<cmath>\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2</cmath>
 
<cmath>\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2</cmath>
 
holds for all positive integers <math>k</math>, all nonnegative real numbers <math>x_1, \ldots, x_k</math>, and all <math>\ell</math>-large collections <math>A_1, \ldots, A_k</math> of subsets of <math>\{1,2, \ldots, n\}</math>.
 
holds for all positive integers <math>k</math>, all nonnegative real numbers <math>x_1, \ldots, x_k</math>, and all <math>\ell</math>-large collections <math>A_1, \ldots, A_k</math> of subsets of <math>\{1,2, \ldots, n\}</math>.
  
 
Note: For a finite set <math>S,|S|</math> denotes the number of elements in <math>S</math>.
 
Note: For a finite set <math>S,|S|</math> denotes the number of elements in <math>S</math>.

Revision as of 19:22, 23 March 2024

Day 1

Problem 1

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n !$ in increasing order as $1=d_1<d_2<\cdots<d_k=n!$, then we have

\[d_2-d_1 \leq d_3-d_2 \leq \cdots \leq d_k-d_{k-1}.\]

Problem 2

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}$, the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$. What is the least possible number of elements that are in at least 50 sets?

Problem 3

Let $m$ be a positive integer. A triangulation of a polygon is $m$-balanced if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon.

Note: A triangulation of a convex polygon $\mathcal{P}$ with $n \geq 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior.

Day 2

Problem 4

Let $m$ and $n$ be positive integers. A circular necklace contains $m n$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$.

Problem 5

Point $D$ is selected inside acute triangle $A B C$ so that $\angle D A C=$ $\angle A C B$ and $\angle B D C=90^{\circ}+\angle B A C$. Point $E$ is chosen on ray $B D$ so that $A E=E C$. Let $M$ be the midpoint of $B C$. Show that line $A B$ is tangent to the circumcircle of triangle $B E M$.

Problem 6

Let $n>2$ be an integer and let $\ell \in\{1,2, \ldots, n\}$. A collection $A_1, \ldots, A_k$ of (not necessarily distinct) subsets of $\{1,2, \ldots, n\}$ is called $\ell$-large if $\left|A_i\right| \geq \ell$ for all $1 \leq i \leq k$. Find, in terms of $n$ and $\ell$, the largest real number $c$ such that the inequality \[\sum_{i=1}^k \sum_{j=1}^k x_i x_j \frac{\left|A_i \cap A_j\right|^2}{\left|A_i\right| \cdot\left|A_j\right|} \geq c\left(\sum_{i=1}^k x_i\right)^2\] holds for all positive integers $k$, all nonnegative real numbers $x_1, \ldots, x_k$, and all $\ell$-large collections $A_1, \ldots, A_k$ of subsets of $\{1,2, \ldots, n\}$.

Note: For a finite set $S,|S|$ denotes the number of elements in $S$.

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