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Difference between revisions of "2013 USAJMO Problems"

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Please redirect to http://www.artofproblemsolving.com/Wiki/index.php?title=2013_USAJMO.
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==Day 1==
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===Problem 1===
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Are there integers <math>a</math> and <math>b</math> such that <math>a^5b+3</math> and <math>ab^5+3</math> are both perfect cubes of integers?
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[[2013 USAJMO Problems/Problem 1|Solution]]
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===Problem 2===
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Each cell of an <math>m\times n</math> board is filled with some nonnegative integer.  Two numbers in the filling are said to be ''adjacent'' if their cells share a common side.  (Note that two numbers in cells that share only a corner are not adjacent).  The filling is called a ''garden'' if it satisfies the following two conditions:
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(i) The difference between any two adjacent numbers is either <math>0</math> or <math>1</math>.
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(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to <math>0</math>.
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Determine the number of distinct gardens in terms of <math>m</math> and <math>n</math>.
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[[2013 USAJMO Problems/Problem 2|Solution]]
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===Problem 3===
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In triangle <math>ABC</math>, points <math>P,Q,R</math> lie on sides <math>BC,CA,AB</math> respectively.  Let <math>\omega_A</math>, <math>\omega_B</math>, <math>\omega_C</math> denote the circumcircles of triangles <math>AQR</math>, <math>BRP</math>, <math>CPQ</math>, respectively.  Given the fact that segment <math>AP</math> intersects <math>\omega_A</math>, <math>\omega_B</math>, <math>\omega_C</math> again at <math>X,Y,Z</math> respectively, prove that <math>YX/XZ=BP/PC</math>.
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[[2013 USAMO Problems/Problem 1|Solution]]
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==Day 2==
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===Problem 4===
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Let <math>f(n)</math> be the number of ways to write <math>n</math> as a sum of powers of <math>2</math>, where we keep track of the order of the summation. For example, <math>f(4)=6</math> because <math>4</math> can be written as <math>4</math>, <math>2+2</math>, <math>2+1+1</math>, <math>1+2+1</math>, <math>1+1+2</math>, and <math>1+1+1+1</math>. Find the smallest <math>n</math> greater than <math>2013</math> for which <math>f(n)</math> is odd.
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[[2013 USAJMO Problems/Problem 4|Solution]]
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===Problem 5===
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Quadrilateral <math>XABY</math> is inscribed in the semicircle <math>\omega</math> with diameter <math>XY</math>.  Segments <math>AY</math> and <math>BX</math> meet at <math>P</math>.  Point <math>Z</math> is the foot of the perpendicular from <math>P</math> to line <math>XY</math>.  Point <math>C</math> lies on <math>\omega</math> such that line <math>XC</math> is perpendicular to line <math>AZ</math>.  Let <math>Q</math> be the intersection of segments <math>AY</math> and <math>XC</math>.  Prove that <cmath>\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.</cmath>
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[[2013 USAJMO Problems/Problem 5|Solution]]
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===Problem 6===
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Find all real numbers <math>x,y,z\geq 1</math> satisfying <cmath>\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.</cmath>
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[[2013 USAMO Problems/Problem 4|Solution]]
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== See Also ==
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*[[USAJMO Problems and Solutions]]
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{{USAJMO box|year=2013|before=[[2012 USAJMO Problems]]|after=[[2014 USAJMO Problems]]}}
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{{MAA Notice}}

Latest revision as of 15:41, 5 August 2023

Day 1

Problem 1

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Solution

Problem 2

Each cell of an $m\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:

(i) The difference between any two adjacent numbers is either $0$ or $1$.

(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$.

Determine the number of distinct gardens in terms of $m$ and $n$.

Solution

Problem 3

In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.

Solution

Day 2

Problem 4

Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.

Solution

Problem 5

Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]

Solution

Problem 6

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

Solution

See Also

2013 USAJMO (Problems • Resources)
Preceded by
2012 USAJMO Problems 		Followed by
2014 USAJMO Problems
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

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