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Difference between revisions of "2021 USAJMO Problems"
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==Problem 6==
 
==Problem 6==
 
Let <math>n \geq 4</math> be an integer. Find all positive real solutions to the following system of <math>2n</math> 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*}
 
Let <math>n \geq 4</math> be an integer. Find all positive real solutions to the following system of <math>2n</math> 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*}
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{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:40%;"
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| style="background:#ccf;text-align:center;" colspan="3" | '''[[2020 USOJMO]]''' ('''[[2020 USOJMO Problems|Problems]]''' • [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176&year={{{year}}} Resources])
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| width="50%" align="center" rowspan="{{{rowsp|1}}}" | {{{beforetext|Preceded&nbsp;by<br/>}}}'''{{{before|[[2019 USAJMO]]}}}'''
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| width="50%" align="center" rowspan="{{{rowsf|1}}}" | {{{aftertext|Followed&nbsp;by<br/>}}}'''{{{after|[[2021 USAJMO]]}}}'''
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|-
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| colspan="3" style="text-align:center;" | [[2020 USOJMO Problems/Problem 1|1]] '''•''' [[2020 USOJMO Problems/Problem 2|2]] '''•''' [[2020 USOJMO Problems/Problem 3|3]] '''•''' [[2020 USOJMO Problems/Problem 4|4]] '''•''' [[2020 USOJMO Problems/Problem 5|5]] '''•''' [[2020 USOJMO Problems/Problem 6|6]]
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| colspan="3" style="text-align:center;" | '''[[USAJMO Problems and Solutions | All USAJMO Problems and Solutions]]'''
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Revision as of 17:16, 15 April 2021

Day 1

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 $b,$\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]

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.

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\[n \leq \frac{2}{3} L^{2}.\]

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.)

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 $S$.

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

2020 USOJMO (Problems • Resources)
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2019 USAJMO 		Followed by
2021 USAJMO
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All USAJMO Problems and Solutions
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