Consider the right-angled triangle with right and , . On the ray we take two points and so that and . Consider the squares and containing inside and , respectively. If and , and , prove that and that , and are collinear.

From a point inside a square we raise a segment perpendicular to the plane of the square. Show that the projections of on the planes , , and are coplanar if and only if .

Let . Proof that
have no positive integer solution .

Let a,b,c be real numbers such that a^2(b+c)+b^2(c+a)+c^2(a+b)=3(a+b+c-1) and a+b+c differnet by 0.Prove that ab+bc+ca=3 if and only if abc=1

Let a,b,c be positive real numbers such that a+b+c=3abc.Prove that
a^2+b^2+c^2+3>=2(ab+bc+ca).

Show that there does not exist a function
satisfying the condition that for all ,


~Korea 2017 P7

Ana, Bárbara, and Cecilia play a game with the following rules:
[list]
[*] In each round, exactly one player wins.
[*] The two losing players each give half of their current money to the winner.
[/list]
The game proceeds as follows:

[list=1]
[*] Ana wins the first round.
[*] Bárbara wins the second round.
[*] Cecilia wins the third round.
[/list]
At the end of the game, the players have the following amounts:
[list]
[*] Ana: C$35
[*] Bárbara: C$75
[*] Cecilia: C$150
[/list]
How much money did each of them have at the beginning?

Find all integer pairs such that:

Juan and Pedro play the following game:
[list]
[*] There are initially 65 tokens.
[*] The players alternate turns, starting with Juan.
[*] On each turn, a player may remove between 1 and 7 tokens.
[*] The player who removes the last token wins.
[/list]
Describe and justify a strategy that guarantees Juan a win.

Two circles of radius 2 are tangent to each other and to a straight line. A third circle is placed so that it is tangent to both of the other circles and also tangent to the same straight line.

What is the radius of the third circle?

IMAGE

Let be a positive integer, and let denote the integers modulo . Determine the number of functions satisfying for all .