2014 USAJMO Problems
Let , , be real numbers greater than or equal to . Prove that Solution
Let be a non-equilateral, acute triangle with , and let and denote the circumcenter and orthocenter of , respectively.
(a) Prove that line intersects both segments and .
(b) Line intersects segments and at and , respectively. Denote by and the respective areas of triangle and quadrilateral . Determine the range of possible values for .
Let be the set of integers. Find all functions such that for all with .
Let be an integer, and let denote the sum of the digits of when it is written in base . Show that there are infinitely many positive integers that cannot be represented in the form , where is a positive integer.
Let be a positive integer. Two players and play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with moving first. In his move, may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, may choose any counter on the board and remove it. If at any time there are consecutive grid cells in a line all of which contain a counter, wins. Find the minimum value of for which cannot win in a finite number of moves, or prove that no such minimum value exists.
Let be a triangle with incenter , incircle and circumcircle . Let be the midpoints of sides , , and let be the tangency points of with and , respectively. Let be the intersections of line with line and line , respectively, and let be the midpoint of arc of .
(a) Prove that lies on ray .
(b) Prove that line bisects .
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