Difference between revisions of "2007 Indonesia MO Problems"

(2007 Indonesia MO problems are up!)
 
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Let <math> ABC</math> be a triangle with <math> \angle ABC=\angle ACB=70^{\circ}</math>. Let point <math> D</math> on side <math> BC</math> such that <math> AD</math> is the altitude, point <math> E</math> on side <math> AB</math> such that <math> \angle ACE=10^{\circ}</math>, and point <math> F</math> is the intersection of <math> AD</math> and <math> CE</math>. Prove that <math> CF=BC</math>.
 
Let <math> ABC</math> be a triangle with <math> \angle ABC=\angle ACB=70^{\circ}</math>. Let point <math> D</math> on side <math> BC</math> such that <math> AD</math> is the altitude, point <math> E</math> on side <math> AB</math> such that <math> \angle ACE=10^{\circ}</math>, and point <math> F</math> is the intersection of <math> AD</math> and <math> CE</math>. Prove that <math> CF=BC</math>.
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[[2007 Indonesia MO Problems/Problem 1|Solution]]
  
 
===Problem 2===
 
===Problem 2===
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(b) Prove that there are finitely many positive integers <math> n</math> which satisfy <math> p(n)=k^2-k+1</math>.
 
(b) Prove that there are finitely many positive integers <math> n</math> which satisfy <math> p(n)=k^2-k+1</math>.
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[[2007 Indonesia MO Problems/Problem 2|Solution]]
  
 
===Problem 3===
 
===Problem 3===
  
 
Let <math> a,b,c</math> be positive real numbers which satisfy <math> 5(a^2+b^2+c^2)<6(ab+bc+ca)</math>. Prove that these three inequalities hold: <math> a+b>c</math>, <math> b+c>a</math>, <math> c+a>b</math>.
 
Let <math> a,b,c</math> be positive real numbers which satisfy <math> 5(a^2+b^2+c^2)<6(ab+bc+ca)</math>. Prove that these three inequalities hold: <math> a+b>c</math>, <math> b+c>a</math>, <math> c+a>b</math>.
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[[2007 Indonesia MO Problems/Problem 3|Solution]]
  
 
===Problem 4===
 
===Problem 4===
  
 
A 10-digit arrangement <math> 0,1,2,3,4,5,6,7,8,9</math> is called beautiful if (i) when read left to right, <math> 0,1,2,3,4</math> form an increasing sequence, and <math> 5,6,7,8,9</math> form a decreasing sequence, and (ii) <math> 0</math> is not the leftmost digit. For example, <math> 9807123654</math> is a beautiful arrangement. Determine the number of beautiful arrangements.
 
A 10-digit arrangement <math> 0,1,2,3,4,5,6,7,8,9</math> is called beautiful if (i) when read left to right, <math> 0,1,2,3,4</math> form an increasing sequence, and <math> 5,6,7,8,9</math> form a decreasing sequence, and (ii) <math> 0</math> is not the leftmost digit. For example, <math> 9807123654</math> is a beautiful arrangement. Determine the number of beautiful arrangements.
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[[2007 Indonesia MO Problems/Problem 4|Solution]]
  
 
==Day 2==
 
==Day 2==
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[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
 
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
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[[2007 Indonesia MO Problems/Problem 5|Solution]]
  
 
===Problem 6===
 
===Problem 6===
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<cmath> z = x^3 + x - 8.</cmath>
 
<cmath> z = x^3 + x - 8.</cmath>
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[[2007 Indonesia MO Problems/Problem 6|Solution]]
  
 
===Problem 7===
 
===Problem 7===
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(b) Prove that <math> QR</math> is perpendicular to line <math> AB</math>.
 
(b) Prove that <math> QR</math> is perpendicular to line <math> AB</math>.
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[[2007 Indonesia MO Problems/Problem 7|Solution]]
  
 
===Problem 8===
 
===Problem 8===
  
 
Let <math> m</math> and <math> n</math> be two positive integers. If there are infinitely many integers <math> k</math> such that <math> k^2+2kn+m^2</math> is a perfect square, prove that <math> m=n</math>.
 
Let <math> m</math> and <math> n</math> be two positive integers. If there are infinitely many integers <math> k</math> such that <math> k^2+2kn+m^2</math> is a perfect square, prove that <math> m=n</math>.
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[[2007 Indonesia MO Problems/Problem 8|Solution]]
  
 
==See Also==
 
==See Also==
 
{{Indonesia MO box|year=2007|before=[[2006 Indonesia MO]]|after=[[2008 Indonesia MO]]}}
 
{{Indonesia MO box|year=2007|before=[[2006 Indonesia MO]]|after=[[2008 Indonesia MO]]}}

Latest revision as of 17:18, 12 February 2020

Day 1

Problem 1

Let $ABC$ be a triangle with $\angle ABC=\angle ACB=70^{\circ}$. Let point $D$ on side $BC$ such that $AD$ is the altitude, point $E$ on side $AB$ such that $\angle ACE=10^{\circ}$, and point $F$ is the intersection of $AD$ and $CE$. Prove that $CF=BC$.

Solution

Problem 2

For every positive integer $n$, $b(n)$ denote the number of positive divisors of $n$ and $p(n)$ denote the sum of all positive divisors of $n$. For example, $b(14)=4$ and $p(14)=24$. Let $k$ be a positive integer greater than $1$.

(a) Prove that there are infinitely many positive integers $n$ which satisfy $b(n)=k^2-k+1$.

(b) Prove that there are finitely many positive integers $n$ which satisfy $p(n)=k^2-k+1$.

Solution

Problem 3

Let $a,b,c$ be positive real numbers which satisfy $5(a^2+b^2+c^2)<6(ab+bc+ca)$. Prove that these three inequalities hold: $a+b>c$, $b+c>a$, $c+a>b$.

Solution

Problem 4

A 10-digit arrangement $0,1,2,3,4,5,6,7,8,9$ is called beautiful if (i) when read left to right, $0,1,2,3,4$ form an increasing sequence, and $5,6,7,8,9$ form a decreasing sequence, and (ii) $0$ is not the leftmost digit. For example, $9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.

Solution

Day 2

Problem 5

Let $r$, $s$ be two positive integers and $P$ a 'chessboard' with $r$ rows and $s$ columns. Let $M$ denote the maximum value of rooks placed on $P$ such that no two of them attack each other.

(a) Determine $M$.

(b) How many ways to place $M$ rooks on $P$ such that no two of them attack each other?

[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]

Solution

Problem 6

Find all triples $(x,y,z)$ of real numbers which satisfy the simultaneous equations

\[x = y^3 + y - 8\]

\[y = z^3 + z - 8\]

\[z = x^3 + x - 8.\]

Solution

Problem 7

Points $A,B,C,D$ are on circle $S$, such that $AB$ is the diameter of $S$, but $CD$ is not the diameter. Given also that $C$ and $D$ are on different sides of $AB$. The tangents of $S$ at $C$ and $D$ intersect at $P$. Points $Q$ and $R$ are the intersections of line $AC$ with line $BD$ and line $AD$ with line $BC$, respectively.

(a) Prove that $P$, $Q$, and $R$ are collinear.

(b) Prove that $QR$ is perpendicular to line $AB$.

Solution

Problem 8

Let $m$ and $n$ be two positive integers. If there are infinitely many integers $k$ such that $k^2+2kn+m^2$ is a perfect square, prove that $m=n$.

Solution

See Also

2007 Indonesia MO (Problems)
Preceded by
2006 Indonesia MO
1 2 3 4 5 6 7 8 Followed by
2008 Indonesia MO
All Indonesia MO Problems and Solutions
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