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Difference between revisions of "2017 USAJMO Problems"
 
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[[2017 USAJMO Problems/Problem 2|Solution]]
 
[[2017 USAJMO Problems/Problem 2|Solution]]
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===Problem 3===
 
===Problem 3===
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>.
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>.
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===Problem 4===
 
===Problem 4===
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Are there any triples <math>(a,b,c)</math> of positive integers such that <math>(a-2)(b-2)(c-2) + 12</math> is prime that properly divides the number <math>a^2 + b^2 + c^2 + abc - 2017</math>?
+
Are there any triples <math>(a,b,c)</math> of positive integers such that <math>(a-2)(b-2)(c-2) + 12</math> is a prime that properly divides the positive number <math>a^2 + b^2 + c^2 + abc - 2017</math>?
  
 
[[2017 USAJMO Problems/Problem 4|Solution]]
 
[[2017 USAJMO Problems/Problem 4|Solution]]
 +
 
===Problem 5===
 
===Problem 5===
 
(<math>*</math>) Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acute triangle <math>ABC</math>. Points <math>M</math> and <math>D</math> lie on side <math>BC</math> such that <math>BM = CM</math> and <math>\angle BAD = \angle CAD</math>. Ray <math>MO</math> intersects the circumcircle of triangle <math>BHC</math> in point <math>N</math>. Prove that <math>\angle ADO = \angle HAN</math>.  
 
(<math>*</math>) Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acute triangle <math>ABC</math>. Points <math>M</math> and <math>D</math> lie on side <math>BC</math> such that <math>BM = CM</math> and <math>\angle BAD = \angle CAD</math>. Ray <math>MO</math> intersects the circumcircle of triangle <math>BHC</math> in point <math>N</math>. Prove that <math>\angle ADO = \angle HAN</math>.  
  
 
[[2017 USAJMO Problems/Problem 5|Solution]]
 
[[2017 USAJMO Problems/Problem 5|Solution]]
 +
 
===Problem 6===
 
===Problem 6===
 
Let <math>P_1, \ldots, P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2 + y^2 = 1</math> other than <math>(1,0)</math>. Each point is colored either red or blue, with exactly <math>n</math> of them red and exactly <math>n</math> of them blue. Let <math>R_1, \ldots, R_n</math> be any ordering of the red points. Let <math>B_1</math> be the nearest blue point to <math>R_1</math> traveling counterclockwise around the circle starting from <math>R_1</math>. Then let <math>B_2</math> be the nearest of the remaining blue points to <math>R_2</math> traveling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all the blue points <math>B_1, \ldots, B_n</math>. Show that the number of counterclockwise arcs of the form <math>R_i \rightarrow B_i</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1, \ldots, R_n</math> of the red points.  
 
Let <math>P_1, \ldots, P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2 + y^2 = 1</math> other than <math>(1,0)</math>. Each point is colored either red or blue, with exactly <math>n</math> of them red and exactly <math>n</math> of them blue. Let <math>R_1, \ldots, R_n</math> be any ordering of the red points. Let <math>B_1</math> be the nearest blue point to <math>R_1</math> traveling counterclockwise around the circle starting from <math>R_1</math>. Then let <math>B_2</math> be the nearest of the remaining blue points to <math>R_2</math> traveling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all the blue points <math>B_1, \ldots, B_n</math>. Show that the number of counterclockwise arcs of the form <math>R_i \rightarrow B_i</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1, \ldots, R_n</math> of the red points.  
  
 
[[2017 USAJMO Problems/Problem 6|Solution]]
 
[[2017 USAJMO Problems/Problem 6|Solution]]
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{{USAJMO box|year=2017|before=[[2016 USAJMO Problems]]|after=[[2018 USAJMO Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}
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{{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}}
 

Latest revision as of 15:44, 5 August 2023

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

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $a^b + b^a$ is divisible by $a + b.$

Solution

Problem 2

Consider the equation \[\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.\]

(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.

(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

Solution

Problem 3

($*$) Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$; let lines $PB$ and $CA$ intersect at $E$; and let lines $PC$ and $AB$ intersect at $F$. Prove that the area of triangle $DEF$ is twice the area of triangle $ABC$.

Solution

Day 2

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 4

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is a prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$?

Solution

Problem 5

($*$) Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.

Solution

Problem 6

Let $P_1, \ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \ldots, R_n$ of the red points.

Solution

2017 USAJMO (Problems • Resources)
Preceded by
2016 USAJMO Problems 		Followed by
2018 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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