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Difference between revisions of "2021 USAMO Problems"

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==Day 1==
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<math>\textbf{Note:}</math> For any geometry problem whose statement begins with an asterisk <math>(*)</math>, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
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===Problem 1===
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<math>(*)</math> Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent.
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[[2021 USAMO Problems/Problem 1|Solution]]
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===Problem 2===
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The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions.
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A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?
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[[2021 USAMO Problems/Problem 2|Solution]]
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===Problem 3===
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Let <math>n \geq 2</math> be an integer. An <math>n \times n</math> board is initially empty. Each minute, you may perform one of three moves:
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If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.
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If all cells in a column have a stone, you may remove all stones from that column.
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If all cells in a row have a stone, you may remove all stones from that row.
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<asy>
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unitsize(20);
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draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0));
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fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey);
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draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2));
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draw((0,2)--(4,2));
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draw((2,4)--(2,0));
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</asy>
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For which <math>n</math> is it possible that, after some non-zero number of moves, the board has no stones?
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[[2021 USAMO Problems/Problem 3|Solution]]
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==Day 2==
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===Problem 4===
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A finite set <math>S</math> of positive integers has the property that, for each <math>s \in S,</math> and each positive integer divisor <math>d</math> of <math>s</math>, there exists a unique element <math>t \in S</math> satisfying <math>\text{gcd}(s, t) = d</math>. (The elements <math>s</math> and <math>t</math> could be equal.)
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Given this information, find all possible values for the number of elements of <math>S</math>.
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[[2021 USAMO Problems/Problem 4|Solution]]
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===Problem 5===
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Let <math>n \geq 4</math> be an integer. Find all positive real solutions to the following system of <math>2n</math> equations:
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<cmath>\begin{align*}
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a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\
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a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\
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a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\
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&\vdots & &\vdots \\
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a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1}
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\end{align*}</cmath>
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[[2021 USAMO Problems/Problem 5|Solution]]
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===Problem 6===
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<math>(*)</math> Let <math>ABCDEF</math> be a convex hexagon satisfying <math>\overline{AB} \parallel \overline{DE}</math>, <math>\overline{BC} \parallel \overline{EF}</math>, <math>\overline{CD} \parallel \overline{FA}</math>, and
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<cmath> AB \cdot DE = BC \cdot EF = CD \cdot FA. </cmath>
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Let <math>X</math>, <math>Y</math>, and <math>Z</math> be the midpoints of <math>\overline{AD}</math>, <math>\overline{BE}</math>, and <math>\overline{CF}</math>. Prove that the circumcenter of <math>\triangle ACE</math>, the circumcenter of <math>\triangle BDF</math>, and the orthocenter of <math>\triangle XYZ</math> are collinear.
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[[2021 USAMO Problems/Problem 6 |Solution]]
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==See Also==
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{{USAMO newbox|year=2021|before=[[2020 USAMO Problems]]|after=[[2022 USAMO Problems]]}}
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{{MAA Notice}}

Latest revision as of 12:48, 22 November 2023

Day 1

$\textbf{Note:}$ For any geometry problem whose statement begins with an asterisk $(*)$, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

$(*)$ Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that\[\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.\]Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.

Solution

Problem 2

The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions.

A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?

Solution

Problem 3

Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves: If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. If all cells in a column have a stone, you may remove all stones from that column. If all cells in a row have a stone, you may remove all stones from that row. [asy] unitsize(20); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); draw((0,2)--(4,2)); draw((2,4)--(2,0)); [/asy] For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?

Solution

Day 2

Problem 4

A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)

Given this information, find all possible values for the number of elements of $S$.

Solution

Problem 5

Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:

\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

Solution

Problem 6

$(*)$ Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and \[AB \cdot DE = BC \cdot EF = CD \cdot FA.\] Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.

Solution

See Also

2021 USAMO (Problems • Resources)
Preceded by
2020 USAMO Problems 		Followed by
2022 USAMO Problems
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. AMC logo.png

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