Difference between revisions of "2007 Indonesia MO Problems"
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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
Contents
Day 1
Problem 1
Let be a triangle with . Let point on side such that is the altitude, point on side such that , and point is the intersection of and . Prove that .
Problem 2
For every positive integer , denote the number of positive divisors of and denote the sum of all positive divisors of . For example, and . Let be a positive integer greater than .
(a) Prove that there are infinitely many positive integers which satisfy .
(b) Prove that there are finitely many positive integers which satisfy .
Problem 3
Let be positive real numbers which satisfy . Prove that these three inequalities hold: , , .
Problem 4
A 10-digit arrangement is called beautiful if (i) when read left to right, form an increasing sequence, and form a decreasing sequence, and (ii) is not the leftmost digit. For example, is a beautiful arrangement. Determine the number of beautiful arrangements.
Day 2
Problem 5
Let , be two positive integers and a 'chessboard' with rows and columns. Let denote the maximum value of rooks placed on such that no two of them attack each other.
(a) Determine .
(b) How many ways to place rooks on such that no two of them attack each other?
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
Problem 6
Find all triples of real numbers which satisfy the simultaneous equations
Problem 7
Points are on circle , such that is the diameter of , but is not the diameter. Given also that and are on different sides of . The tangents of at and intersect at . Points and are the intersections of line with line and line with line , respectively.
(a) Prove that , , and are collinear.
(b) Prove that is perpendicular to line .
Problem 8
Let and be two positive integers. If there are infinitely many integers such that is a perfect square, prove that .
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 |