Show that for any two positive real numbers \(x,y\) it is true that \(x^2+y^2 \ge 2xy\).
Find all pairs of integers \((x,y)\) so that the following equation is true \(xy = y+x\).
Calculate the following squares in the shortest possible way (without
a calculator or any other device):
a) \(1001^2\) b) \(9998^2\) c) \(20003^2\) d) \(497^2\)
Real numbers \(x,y\) are such that \(x^2 +x \le y\). Show that \(y^2 +y \ge x\).
The perimeter of the triangle \(\triangle ABC\) is \(10\). Let \(D,E,F\) be the midpoints of the segments \(AB,BC,AC\) respectively. What is the perimeter of \(\triangle DEF\)?
Let \(\triangle ABC\) be a triangle and \(D\) be a point on the edge \(BC\) so that the segment \(AD\) bisects the angle \(\angle BAC\). Show that \(\frac{|AB|}{|BD|}=\frac{|AC|}{|CD|}\).
Proposed by USA for IMO 1993. For positive real numbers \(a,b,c,d\) prove that \[\frac{a}{b+2c+3d} + \frac{b}{c+2d+3a} + \frac{c}{d+2a+3b} + \frac{d}{a+2b+3c} \geq \frac{2}{3}.\]
Prove the \(AM-GM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\frac{a_1+a_2+...+a_n}{n}\geq \sqrt[n]{a_1a_2...a_n}.\]
For non-negative real numbers \(a,b,c\) prove that \[a^3+b^3+c^3 \geq \frac{(a+b+c)(a^2+b^2+c^2)}{3}\geq a^2b+b^2c+c^2a.\]
Prove Nesbitt’s inequality, which states that for positive real numbers \(a,b,c\) we have \[\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq \frac{3}{2}.\]