1999 Akdeniz University MO

High Schools-1,2

1

Prove that, we find infinite numbers such that, this number writeable $1999k+1$ for $k \in {\mathbb N}$ and all digits are equal.

2

Prove that, we can't find positive numbers $m$ and $n$ such that, $$m^2+(m+1)^2=n^4+(n+1)^4$$

3

Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

4

In a sequence ,first term is $2$ and after $2.$ term all terms is equal to sum of the previous number's digits' $5.$ power. (Like this $2.$term is $2^5=32$ , $3.$term is $3^5+2^5=243+32=275\dotsm$) Prove that, this infinite sequence has at least $2$ two numbers are equal.

5

A circle centered with $O$. $C$ is a stable point in circle. A chord $[AB]$, parallel to $OC$.Prove that, $$[AC]^2+[BC]^2$$is stable.

High Schools-3

1

Let $n$'s positive divisors sum is $T(n)$. For all $n \geq 3$'s prove that, $$(T(n))^3<n^4$$

2

Find all $(x,y)$ real numbers pairs such that, $$x^7+y^7=x^4+y^4$$

3

For all $x> \sqrt 2$, $y> \sqrt 2$ numbers, prove that $$x^4-x^3y+x^2y^2-xy^3+y^4>x^2+y^2$$

4

Placing $n \in {\mathbb N}$ circles with radius $1$ $unit$ inside a square with side $100$ $unit$ such that, whichever line segment with lenght $10$ $unit$ intersect at least one circle. Prove that $$n \geq 416$$

5

Let $C$ is at a circle. $[AB]$ is a diameter this circle. $D$ is a point at $[AB]$. Perpendicular from $C$ to $[AB]$'s foot on the $[AB]$ is $E$, perpendicular from $A$ to $[CD]$'s foot on the $[CD]$ is $F$. Prove that, $$[DC][FC]=[BD][EA]$$