Difference between revisions of "2005 Indonesia MO Problems"
Rockmanex3 (talk | contribs) (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...") |
Rockmanex3 (talk | contribs) m (2005 Indonesia MO Problems are up!) |
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===Problem 3=== | ===Problem 3=== | ||
− | Let <math> k</math> and <math> m</math> be positive integers such that <math> | + | 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> | + | 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> | ||
+ | |||
+ | <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]]}} | ||
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Latest revision as of 00:31, 5 September 2018
Contents
Day 1
Problem 1
Let be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is .
Problem 2
For an arbitrary positive integer , define as the product of the digits of (in decimal). Find all positive integers such that .
Problem 3
Let and be positive integers such that is an integer.
(a) Prove that is rational.
(b) Prove that is a positive integer.
Problem 4
Let be a point in triangle such that , , . The centers of circumcircles of triangles are , respectively. Prove that the area of is greater than the area of .
Day 2
Problem 5
For an arbitrary real number , denotes the greatest integer not exceeding . Prove that there is exactly one integer which satisfy .
Problem 6
Find all triples of integers which satisfy
.
Problem 7
Let be a convex quadrilateral. Square is constructed such that the two vertices is located outside . Similarly, we construct squares , , . Let be the intersection of and , be the intersection of and , be the intersection of and , and be the intersection of and . Prove that is perpendicular to .
Problem 8
There are contestants in a mathematics competition. Each contestant gets acquainted with at least 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.
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 |