Modify Euclid's proof that there are infinitely many primes by assuming the existence of a largest prime and using the integer to arrive at a contradiction.

Let be distinct positive integers. Prove that
Further, determine when equality holds.

Let be a positive integer. Find the smallest positive integer such that for any set of points in the interior of the unit square, there exists a set of rectangles such that the following hold:
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[*]The sides of each rectangle are parallel to the sides of the unit square.
[*]Each point in is not in the interior of any rectangle.
[*]Each point in the interior of the unit square but not in is in the interior of at least one of the rectangles
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(The interior of a polygon does not contain its boundary.)

Holden Mui

Find all functions such that for all and then

Let Prove that
Let Prove that
Let Prove that
Let Prove that

Let Prove that

Let be the altitude of the triangle and be foots of perpendicular lines through on , respectively. Construct the parallelogram and altitudes of the triangle . Prove that lies on

Alice and Bob are playing a game on an () chessboard. Initially, Alice’s queen is placed at the bottom-left corner, and Bob’s queen is at the bottom-right corner. All the other squares on the board are covered by neutral pieces.

Alice starts first, and the two players take turns. In each turn, a player must move their queen to capture a piece. A queen can capture a piece if and only if the piece lies in the same row, column, or diagonal as the queen, and there are no other pieces between them. A player loses if their queen is captured or if there are no remaining pieces they
can capture. For which values of does Alice have a winning strategy?

Determine all natural numbers such that there exist a positive even number such that

is a perfect square.

Find all the integer pairs such that