Difference between revisions of "2014 EGMO Problems"
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Determine all positive integers <math>n\geq 2</math> for which there exist integers <math>x_1,x_2,\ldots ,x_{n-1}</math> satisfying the condition that if <math>0<i<n,0<j<n, i\neq j</math> and <math>n</math> divides <math>2i+j</math>, then <math>x_i<x_j</math>. | Determine all positive integers <math>n\geq 2</math> for which there exist integers <math>x_1,x_2,\ldots ,x_{n-1}</math> satisfying the condition that if <math>0<i<n,0<j<n, i\neq j</math> and <math>n</math> divides <math>2i+j</math>, then <math>x_i<x_j</math>. | ||
− | [[ | + | [[2014 EGMO Problems/Problem 4|Solution]] |
===Problem 5=== | ===Problem 5=== | ||
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Prove that <math>\angle ACP = \angle QCB</math>. | Prove that <math>\angle ACP = \angle QCB</math>. | ||
− | [[ | + | [[2014 EGMO Problems/Problem 5|Solution]] |
===Problem 6=== | ===Problem 6=== | ||
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for all real numbers <math>x</math> and <math>y</math>. | for all real numbers <math>x</math> and <math>y</math>. | ||
− | [[ | + | [[2014 EGMO Problems/Problem 6|Solution]] |
Revision as of 12:47, 24 December 2022
Contents
[hide]Day 1
Problem 1
Determine all real constants such that whenever , and are the lengths of sides of a triangle, then so are , , .
Problem 2
Let and be points in the interiors of sides and , respectively, of a triangle , such that . Let the lines and meet at . Prove that the incentre of triangle , the orthocentre of triangle and the midpoint of the arc of the circumcircle of triangle are collinear.
Problem 3
We denote the number of positive divisors of a positive integer by and the number of distinct prime divisors of by . Let be a positive integer. Prove that there exist infinitely many positive integers such that and does not divide for any positive integers satisfying .
Day 2
Problem 4
Determine all positive integers for which there exist integers satisfying the condition that if and divides , then .
Problem 5
Let be a positive integer. We have boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
Prove that .
Problem 6
Determine all functions satisfying the condition for all real numbers and .