Difference between revisions of "2013 Indonesia MO Problems"
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===Problem 4=== | ===Problem 4=== | ||
− | + | Suppose <math>p > 3</math> is a prime number and | |
<cmath>S = \sum_{2 \le i < j < k \le p-1} ijk</cmath> | <cmath>S = \sum_{2 \le i < j < k \le p-1} ijk</cmath> | ||
Prove that <math>S+1</math> is divisible by <math>p</math>. | Prove that <math>S+1</math> is divisible by <math>p</math>. | ||
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a. Prove that <math>2013</math> is strong. | a. Prove that <math>2013</math> is strong. | ||
+ | |||
b. If <math>m</math> is strong, determine the smallest <math>y</math> (in terms of <math>m</math>) such that <math>y^{my} + 1</math> is divisible by <math>2^m</math>. | b. If <math>m</math> is strong, determine the smallest <math>y</math> (in terms of <math>m</math>) such that <math>y^{my} + 1</math> is divisible by <math>2^m</math>. | ||
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a. Find a 9-element balanced set. | a. Find a 9-element balanced set. | ||
+ | |||
b. Prove that no set of <math>2013</math> elements can be balanced. | b. Prove that no set of <math>2013</math> elements can be balanced. | ||
Latest revision as of 00:29, 13 August 2018
Contents
Day 1
Problem 1
In a grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
Problem 2
Let be an acute triangle and be its circumcircle. The bisector of intersects at [another point] . Let be a point on and inside . Lines passing that are parallel to and intersects on respectively. Lines intersects at points respectively. Prove that are concurrent.
Problem 3
Determine all positive real such that for any positive reals , at least one of is greater than or equal to .
Problem 4
Suppose is a prime number and Prove that is divisible by .
Day 2
Problem 5
Let be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics such that:
- have positive leading coefficients and zero discriminants (and hence each has a double root)
- The roots of are different
Problem 6
A positive integer is called "strong" if there exists a positive integer such that is divisible by .
a. Prove that is strong.
b. If is strong, determine the smallest (in terms of ) such that is divisible by .
Problem 7
Let be a parallelogram. Construct squares on the outer side of the parallelogram. Construct a square having as one of its sides and it is on the outer side of and call its center . Similarly do it for to obtain . Prove that .
Problem 8
Let be a set of positive integers. is called "balanced" if [and only if] the number of 3-element subsets of whose elements add up to a multiple of is equal to the number of 3-element subsets of whose elements add up to not a multiple of .
a. Find a 9-element balanced set.
b. Prove that no set of elements can be balanced.
See Also
2013 Indonesia MO (Problems) | ||
Preceded by 2012 Indonesia MO |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by 2014 Indonesia MO |
All Indonesia MO Problems and Solutions |