Difference between revisions of "2005 Indonesia MO Problems"

(Created page with "==Day 1== ===Problem 1=== Let <math> n</math> be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length...")
 
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===Problem 3===
 
===Problem 3===
  
Let <math> k</math> and <math> m</math> be positive integers such that <math> \displaystyle\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)</math> is an integer.
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Let <math> k</math> and <math> m</math> be positive integers such that <math>\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)</math> is an integer.
  
 
(a) Prove that <math> \sqrt{k}</math> is rational.
 
(a) Prove that <math> \sqrt{k}</math> is rational.
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===Problem 5===
 
===Problem 5===
  
For an arbitrary real number <math> x</math>, <math> \lfloor x\rfloor</math> denotes the greatest integer not exceeding <math> x</math>. Prove that there is exactly one integer <math> m</math> which satisfy <math> \displaystyle m-\left\lfloor \frac{m}{2005}\right\rfloor=2005</math>.
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For an arbitrary real number <math> x</math>, <math> \lfloor x\rfloor</math> denotes the greatest integer not exceeding <math> x</math>. Prove that there is exactly one integer <math> m</math> which satisfy <math>m-\left\lfloor \frac{m}{2005}\right\rfloor=2005</math>.
  
 
[[2005 Indonesia MO Problems/Problem 5|Solution]]
 
[[2005 Indonesia MO Problems/Problem 5|Solution]]
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<math> y(z + x) = z^2 + x^2 - 2</math>
 
<math> y(z + x) = z^2 + x^2 - 2</math>
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<math> z(x + y) = x^2 + y^2 - 2</math>.
  
 
[[2005 Indonesia MO Problems/Problem 6|Solution]]
 
[[2005 Indonesia MO Problems/Problem 6|Solution]]
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==See Also==
 
==See Also==
 
{{Indonesia MO box|year=2005|before=[[2004 Indonesia MO]]|after=[[2006 Indonesia MO]]}}
 
{{Indonesia MO box|year=2005|before=[[2004 Indonesia MO]]|after=[[2006 Indonesia MO]]}}
 
<math> z(x + y) = x^2 + y^2 - 2</math>.
 

Latest revision as of 00:31, 5 September 2018

Day 1

Problem 1

Let $n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $n$.

Solution

Problem 2

For an arbitrary positive integer $n$, define $p(n)$ as the product of the digits of $n$ (in decimal). Find all positive integers $n$ such that $11p(n)=n^2-2005$.

Solution

Problem 3

Let $k$ and $m$ be positive integers such that $\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)$ is an integer.

(a) Prove that $\sqrt{k}$ is rational.

(b) Prove that $\sqrt{k}$ is a positive integer.

Solution

Problem 4

Let $M$ be a point in triangle $ABC$ such that $\angle AMC=90^{\circ}$, $\angle AMB=150^{\circ}$, $\angle BMC=120^{\circ}$. The centers of circumcircles of triangles $AMC,AMB,BMC$ are $P,Q,R$, respectively. Prove that the area of $\triangle PQR$ is greater than the area of $\triangle ABC$.

Solution

Day 2

Problem 5

For an arbitrary real number $x$, $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. Prove that there is exactly one integer $m$ which satisfy $m-\left\lfloor \frac{m}{2005}\right\rfloor=2005$.

Solution

Problem 6

Find all triples $(x,y,z)$ of integers which satisfy

$x(y + z) = y^2 + z^2 - 2$

$y(z + x) = z^2 + x^2 - 2$

$z(x + y) = x^2 + y^2 - 2$.

Solution

Problem 7

Let $ABCD$ be a convex quadrilateral. Square $AB_1A_2B$ is constructed such that the two vertices $A_2,B_1$ is located outside $ABCD$. Similarly, we construct squares $BC_1B_2C$, $CD_1C_2D$, $DA_1D_2A$. Let $K$ be the intersection of $AA_2$ and $BB_1$, $L$ be the intersection of $BB_2$ and $CC_1$, $M$ be the intersection of $CC_2$ and $DD_1$, and $N$ be the intersection of $DD_2$ and $AA_1$. Prove that $KM$ is perpendicular to $LN$.

Solution

Problem 8

There are $90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.

Solution

See Also

2005 Indonesia MO (Problems)
Preceded by
2004 Indonesia MO
1 2 3 4 5 6 7 8 Followed by
2006 Indonesia MO
All Indonesia MO Problems and Solutions
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