Difference between revisions of "2021 USAJMO Problems"

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[[2021 USAJMO Problems/Problem 2|Solution]]
 
[[2021 USAJMO Problems/Problem 2|Solution]]
 
===Problem 3===
 
===Problem 3===
An equilateral triangle <math>\Delta</math> of side length <math>L>0</math> is given. Suppose that <math>n</math> equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside <math>\Delta</math>, such that each unit equilateral triangle has sides parallel to <math>\Delta</math>, but with opposite orientation. (An example with <math>n=2</math> is drawn below.)[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]Prove that<cmath>n \leq \frac{2}{3} L^{2}.</cmath>
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An equilateral triangle <math>\Delta</math> of side length <math>L>0</math> is given. Suppose that <math>n</math> equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside <math>\Delta</math>, such that each unit equilateral triangle has sides parallel to <math>\Delta</math>, but with opposite orientation. (An example with <math>n=2</math> is drawn below.)
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<asy>
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draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5));
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filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5));
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filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5));
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</asy>
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Prove that<cmath>n \leq \frac{2}{3} L^{2}.</cmath>
 +
 
 
[[2021 USAJMO Problems/Problem 3|Solution]]
 
[[2021 USAJMO Problems/Problem 3|Solution]]
 
==Day 2==
 
==Day 2==

Revision as of 00:43, 16 April 2021

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

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\] Solution

Problem 2

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 3

An equilateral triangle $\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\Delta$, such that each unit equilateral triangle has sides parallel to $\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.) [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy] Prove that\[n \leq \frac{2}{3} L^{2}.\]

Solution

Day 2

Problem 4

Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021? (A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.) Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021? (A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.) Solution

Problem 5

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 6

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 \\ 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

2020 USOJMO (ProblemsResources)
Preceded by
2019 USAJMO
Followed by
2021 USAJMO
1 2 3 4 5 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

2021 USAJMO (ProblemsResources)
Preceded by
2020 USOJMO
Followed by
2022 USAJMO
1 2 3 4 5 6
All USAJMO Problems and Solutions