Difference between revisions of "2021 USAJMO Problems"

Revision as of 08:17, 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.

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

@@ Line 20: / Line 20: @@
 ==Day 2==
 ===Problem 4===
-Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021? (A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.) Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021? (A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.)
+Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021? (A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.) Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021?
+(A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.)
 [[2021 USAJMO Problems/Problem 4|Solution]]
 ===Problem 5===
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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]]
-|-
-| colspan="3" style="text-align:center;" | '''[[USAJMO Problems and Solutions | All USAJMO Problems and Solutions]]'''
-|}<includeonly></includeonly><noinclude>
-{{MAA Notice}}
-{| class="wikitable" style="margin:0.5em auto; font-size:95%; border:1px solid black; width:40%;"
-| style="background:#ccf;text-align:center;" colspan="3" | '''[[2021 USAJMO]]''' ('''[[2021 USAJMO Problems|Problems]]''' • [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=176&year={{{year}}} Resources])
-|-
-| width="50%" align="center" rowspan="{{{rowsp|1}}}" | {{{beforetext|Preceded&nbsp;by<br/>}}}'''{{{before|[[2020 USOJMO]]}}}'''
-| width="50%" align="center" rowspan="{{{rowsf|1}}}" | {{{aftertext|Followed&nbsp;by<br/>}}}'''{{{after|[[2022 USAJMO]]}}}'''
-|-
-| colspan="3" style="text-align:center;" | [[2021 USAJMO Problems/Problem 1|1]] '''•''' [[2021 USAJMO Problems/Problem 2|2]] '''•''' [[2021 USAJMO Problems/Problem 3|3]] '''•''' [[2021 USAJMO Problems/Problem 4|4]] '''•''' [[2021 USAJMO Problems/Problem 5|5]] '''•''' [[2021 USAJMO Problems/Problem 6|6]]
 |-
 | colspan="3" style="text-align:center;" | '''[[USAJMO Problems and Solutions | All USAJMO Problems and Solutions]]'''
 |}<includeonly></includeonly><noinclude>

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