Difference between revisions of "2021 USAJMO Problems"

Revision as of 17:11, 15 April 2021

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 $n \leq \frac{2}{3} L^{2}.$

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*}

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+==Problem 1==
+Let <math>\mathbb{N}</math> denote the set of positive integers. Find all functions <math>f : \mathbb{N} \rightarrow \mathbb{N}</math> such that for positive integers <math>a</math> and <math>b,</math><cmath>f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.</cmath>
+==Problem 2==
+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.
+==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>
+==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.)
+==Problem 5==
+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.) Given this information, find all possible values for the number of elements of <math>S</math>.
+==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*}

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