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Difference between revisions of "2018 USAMO Problems"

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Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), 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.
 
Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), 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.
  
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==Problem 1==
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===Problem 1===
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For each positive integer <math>n</math>, find the number of <math>n</math>-digit positive integers that satisfy both of the following conditions:
 
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<math>\bullet</math> no two consecutive digits are equal, and
 
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<math>\bullet</math> the last digit is a prime.
 
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Solution
 
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==Problem 1==
 
 
Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>
 
Let <math>a,b,c</math> be positive real numbers such that <math>a+b+c=4\sqrt[3]{abc}</math>. Prove that <cmath>2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.</cmath>
  
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Solution
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[[2018 USAMO Problems/Problem 1|Solution]]
  
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===Problem 2===
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==Problem 2==
 
 
Find all functions <math>f:(0,\infty) \rightarrow (0,\infty)</math> such that
 
Find all functions <math>f:(0,\infty) \rightarrow (0,\infty)</math> such that
  
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for all <math>x,y,z >0</math> with <math>xyz =1.</math>
 
for all <math>x,y,z >0</math> with <math>xyz =1.</math>
  
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Solution
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[[2018 USAMO Problems/Problem 2|Solution]]
  
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===Problem 3===
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==Problem 3==
 
 
For a given integer <math>n\ge 2,</math> let <math>\{a_1,a_2,…,a_m\}</math> be the set of positive integers less than <math>n</math> that are relatively prime to <math>n.</math> Prove that if every prime that divides <math>m</math> also divides <math>n,</math> then <math>a_1^k+a_2^k + \dots + a_m^k</math> is divisible by <math>m</math> for every positive integer <math>k.</math>
 
For a given integer <math>n\ge 2,</math> let <math>\{a_1,a_2,…,a_m\}</math> be the set of positive integers less than <math>n</math> that are relatively prime to <math>n.</math> Prove that if every prime that divides <math>m</math> also divides <math>n,</math> then <math>a_1^k+a_2^k + \dots + a_m^k</math> is divisible by <math>m</math> for every positive integer <math>k.</math>
  
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Solution
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[[2018 USAMO Problems/Problem 3|Solution]]
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==Day 2==
 
==Day 2==
 
Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), 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.
 
Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), 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.
  
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==Problem 4==
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===Problem 4===
 
Let <math>p</math> be a prime, and let <math>a_1, \dots, a_p</math> be integers. Show that there exists an integer <math>k</math> such that the numbers <cmath>a_1 + k, a_2 + 2k, \dots, a_p + pk</cmath>produce at least <math>\tfrac{1}{2} p</math> distinct remainders upon division by <math>p</math>.
 
Let <math>p</math> be a prime, and let <math>a_1, \dots, a_p</math> be integers. Show that there exists an integer <math>k</math> such that the numbers <cmath>a_1 + k, a_2 + 2k, \dots, a_p + pk</cmath>produce at least <math>\tfrac{1}{2} p</math> distinct remainders upon division by <math>p</math>.
  
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Solution
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[[2018 USAMO Problems/Problem 4|Solution]]
  
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===Problem 5===
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==Problem 5==
 
 
In convex cyclic quadrilateral <math>ABCD,</math> we know that lines <math>AC</math> and <math>BD</math> intersect at <math>E,</math> lines <math>AB</math> and <math>CD</math> intersect at <math>F,</math> and lines <math>BC</math> and <math>DA</math> intersect at <math>G.</math> Suppose that the circumcircle of <math>\triangle ABE</math> intersects line <math>CB</math> at <math>B</math> and <math>P</math>, and the circumcircle of <math>\triangle ADE</math> intersects line <math>CD</math> at <math>D</math> and <math>Q</math>, where <math>C,B,P,G</math> and <math>C,Q,D,F</math> are collinear in that order. Prove that if lines <math>FP</math> and <math>GQ</math> intersect at <math>M</math>, then <math>\angle MAC = 90^{\circ}.</math>
 
In convex cyclic quadrilateral <math>ABCD,</math> we know that lines <math>AC</math> and <math>BD</math> intersect at <math>E,</math> lines <math>AB</math> and <math>CD</math> intersect at <math>F,</math> and lines <math>BC</math> and <math>DA</math> intersect at <math>G.</math> Suppose that the circumcircle of <math>\triangle ABE</math> intersects line <math>CB</math> at <math>B</math> and <math>P</math>, and the circumcircle of <math>\triangle ADE</math> intersects line <math>CD</math> at <math>D</math> and <math>Q</math>, where <math>C,B,P,G</math> and <math>C,Q,D,F</math> are collinear in that order. Prove that if lines <math>FP</math> and <math>GQ</math> intersect at <math>M</math>, then <math>\angle MAC = 90^{\circ}.</math>
  
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Solution
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[[2018 USAMO Problems/Problem 5|Solution]]
  
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===Problem 6===
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Let <math>a_n</math> be the number of permutations <math>(x_1, x_2, \dots, x_n)</math> of the numbers <math>(1,2,\dots, n)</math> such that the <math>n</math> ratios <math>\frac{x_k}{k}</math> for <math>1\le k\le n</math> are all distinct. Prove that <math>a_n</math> is odd for all <math>n\ge 1.</math>
  
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==Problem 6==
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[[2018 USAMO Problems/Problem 6|Solution]]
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Let <math>a_n</math> be the number of permutations <math>(x_1, x_2, \dots, x_n)</math> of the numbers <math>(1,2,\dots, n)</math> such that the <math>n</math> ratios <math>\frac{x_k}{k}</math> for <math>1\le k\le n</math> are all distinct. Prove that <math>a_n</math> is odd for all <math>n\ge 1.</math>
 
  
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Solution
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{{USAMO newbox|year=2018|before=[[2017 USAMO Problems]]|after=[[2019 USAMO Problems]]}}

Latest revision as of 13:48, 22 November 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

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$. Prove that \[2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.\]

Solution

Problem 2

Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that

\[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1.$

Solution

Problem 3

For a given integer $n\ge 2,$ let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n.$ Prove that if every prime that divides $m$ also divides $n,$ then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k.$

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

Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \[a_1 + k, a_2 + 2k, \dots, a_p + pk\]produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.

Solution

Problem 5

In convex cyclic quadrilateral $ABCD,$ we know that lines $AC$ and $BD$ intersect at $E,$ lines $AB$ and $CD$ intersect at $F,$ and lines $BC$ and $DA$ intersect at $G.$ Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^{\circ}.$

Solution

Problem 6

Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1.$

Solution

2018 USAMO (Problems • Resources)
Preceded by
2017 USAMO Problems 		Followed by
2019 USAMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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