Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that: (i) Lines $ BC$ and $ AA'$ are orthogonal. (ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid

For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality: $ x^{2} + y^{2} + z^{2} - xy - yz - zx \geq \max \{\frac {3(x - y)^{2}}{4} , \frac {3(y - z)^{2}}{4} , \frac {3(y - z)^{2}}{4} \}$

Let $ b$ be an even positive integer. Assume that there exist integer $ n > 1$ such that $ \frac {b^{n} - 1}{b - 1}$ is perfect square. Prove that $ b$ is divisible by 8.

Click for solution Let $ \frac{b^{n}-1}{b-1}=k^{2}$, so that $ 1+b+\cdots +b^{n-1}=k^{2}$. The left side of this equation is odd, so $ k$ must be odd also. Since squares are 0, 1, or 4 mod 8, we have $ k^{2}\equiv 1\pmod{8}$, so $ b+b^{2}+\cdots +b^{n-1}\equiv 0\pmod{8}$ $ b(1+b+\cdots +b^{n-2})\equiv 0\pmod{8}$ Since $ 1+b+\cdots +b^{n-2}$ is odd, $ 8|b$.

Given are two disjoint sets $ A$ and $ B$ such that their union is $ \mathbb N$. Prove that for all positive integers $ n$ there exist different numbers $ a$ and $ b$, both greater than $ n$, such that either $ \{ a,b,a + b \}$ is contained in $ A$ or $ \{ a,b,a + b \}$ is contained in $ B$.

Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively. (i) Prove that $ AD=\frac{1}{2}(b+c)$ (ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D=OM+OL$

IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}+b^{2}+c^{2}=1$ prove the inequality: \[ \frac{a^{5}+b^{5}}{ab(a+b)}+ \frac {b^{5}+c^{5}}{bc(b+c)}+\frac {c^{5}+a^{5}}{ca(a+b)}\geq 3(ab+bc+ca)-2.\]

Prove that equation $ p^{4}+q^{4}=r^{4}$ does not have solution in set of prime numbers.

$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$

Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$ at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$. Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$.

If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1+\frac{4a}{b+c}\right)\left(1+\frac{4b}{a+c}\right)\left(1+\frac{4c}{a+b}\right) > 25.\]

Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}+a^{3}+1}{a^{2}b^{2}+ab^{2}+1}$ is an integer.

Click for solution $ \frac {a^{4} + a^{3} + 1}{a^{2}b^{2} + ab^{2} + 1}=k\ge 1$ gives $ (a^2+a)(a^2-kb^2)=k-1\ge 0$ if $ k=1$ then $ a=b$ if $ k>1$ then $ k> a^2>a^2>kb^2$ gives $ b^2=0$ then $ a=b$ or $ b=0$

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?

Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$. Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$ and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$. If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear.

Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}+a^{3}+1}{a^{2}b^{2}+ab^{2}+1}$ is an integer.

Click for solution $ \frac {a^{4} + a^{3} + 1}{a^{2}b^{2} + ab^{2} + 1}=k\ge 1$ gives $ (a^2+a)(a^2-kb^2)=k-1\ge 0$ if $ k=1$ then $ a=b$ if $ k>1$ then $ k> a^2>a^2>kb^2$ gives $ b^2=0$ then $ a=b$ or $ b=0$

A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?

Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.