MathDB

1

Problem 6 Midpoint of segment

ABC

be a triangle with

H

its orthocenter. The circle with diameter

[AC]

cuts the circumcircle of triangle

ABH

at

K

. Prove that the point of intersection of the lines

CK

and

BH

is the midpoint of the segment

[BH]

Problem 5

1

Problem 5 winning strategy

The numbers from

1

to

2017

are written on a board. Deka and Farid play the following game : each of them, on his turn, erases one of the numbers. Anyone who erases a multiple of

2, 3

or

5

loses and the game is over. Is there a winning strategy for Deka ?

Problem 4

1

Problem 4 Equation with integer part

Find all the real numbers

x

such that

\frac{1}{[x]}+\frac{1}{[2x]}=\{x\}+\frac{1}{3}

where

[x]

denotes the integer part of

x

and

\{x\}=x-[x]

. For example,

[2.5]=2, \{2.5\} = 0.5

and

[-1.7]= -2, \{-1.7\} = 0.3

Problem 3

1

Problem 3 number of solutions for integer equation

Let

n

be a positive integer. - Find, in terms of

n

, the number of pairs

(x,y)

of positive integers that are solutions of the equation :

x^2-y^2=10^2.30^{2n}

- Prove further that this number is never a square

Problem 2

1

Inequality x^2y+y^2z+z^2x \geq 2(x+y+z)-3 [PAMO 2017, P2]

Let

x,y

, and

z

be positive real numbers such that

xy+yz+zx=3xyz

. Prove that

x^2y+y^2z+z^2x \geq 2(x+y+z)-3.

In which cases do we have equality?

Problem 1

1

Problem 1 sequence of integer is square

We consider the real sequence

(x_n)

defined by

x_0=0, x_1=1

and

x_{n+2}=3x_{n+1}-2x_n

for

n=0,1,...

We define the sequence

(y_n)

by

y_n=x_n^2+2^{n+2}

for every non negative integer

n

. Prove that for every

n>0

,

y_n

is the square of an odd integer

2017 Pan African

Subcontests

Problem 6 Midpoint of segment

Problem 5 winning strategy

Problem 4 Equation with integer part

Problem 3 number of solutions for integer equation

Inequality x^2y+y^2z+z^2x \geq 2(x+y+z)-3 [PAMO 2017, P2]

Problem 1 sequence of integer is square