A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than $1$ in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than $4$ in absolute value.

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that: $$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}} +\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$

Let $ p $ be a natural number and let two partitions $ \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} $ of a finite set $ \mathcal{M} . $ Knowing that, whenever an element of $ \mathcal{A} $ doesn´t have any elements in common with another of $ \mathcal{B} , $ it holds that the number of elements of these two is greater than $ p, $ prove that $ \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . $ Can equality hold?

Let's call several numbers written in a row a 'combination of numbers'. In the country of Robotland, some combinations of numbers have been declared prohibited. It is known that there are a finite number of forbidden combinations and there is an infinite decimal fraction that does not contain forbidden combinations. Prove that there is an infinite periodic decimal fraction that does not contain prohibited combinations.

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

Given positive integers $n, k$ such that $n\ge 4k$, find the minimal value $\lambda=\lambda(n,k)$ such that for any positive reals $a_1,a_2,\ldots,a_n$, we have \[ \sum\limits_{i=1}^{n} {\frac{{a}_{i}}{\sqrt{{a}_{i}^{2}+{a}_{{i}+{1}}^{2}+{\cdots}{{+}}{a}_{{i}{+}{k}}^{2}}}} \le \lambda\] Where $a_{n+i}=a_i,i=1,2,\ldots,k$

Find minimum number $n$ that: 1) $80|n$ 2) we can permute 2 different numbers in $n$ to get $m$ and $80|m$

Let $k$ be a positive integer and $a_1, a_2,$ $\cdot \cdot \cdot$ $, a_k$ digits. Prove that there exists a positive integer $n$ such that the last $2k$ digits of $2^n$ are, in the following order, $a_1, a_2,$ $\cdot \cdot \cdot$ $, a_k , b_1, b_2,$ $\cdot \cdot \cdot$ $, b_k$, for certain digits $b_1, b_2,$ $\cdot \cdot \cdot$ $, b_k$

[asy] size(150); dotfactor=4; draw(circle((0,0),4)); draw(circle((10,-6),3)); pair O,A,P,Q; O = (0,0); A = (10,-6); P = (-.55, -4.12); Q = (10.7, -2.86); dot("$O$", O, NE); dot("$O'$", A, SW); dot("$P$", P, SW); dot("$Q$", Q, NE); draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle); draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle); draw(P--Q--cycle); //Credit to happiface for the diagram[/asy] In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is $\textbf{(A) }\text{the average of the lengths of the internal and external common tangents}\qquad$ $\textbf{(B) }\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}\qquad$ $\textbf{(C) }\text{always equal to the length of an external common tangent}\qquad$ $\textbf{(D) }\text{greater than the length of an external common tangent}\qquad$ $\textbf{(E) }\text{the geometric mean of the lengths of the internal and external common tangents}$

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

For real numbers $ a$ and $ b$, define $ a \diamond b \equal{} \sqrt{a^2 \plus{} b^2}$. What is the value of \[(5\diamond 12)\diamond ((\minus{}12) \diamond (\minus{}5))?\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{17}{2}\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 13\sqrt{2}\qquad \textbf{(E)}\ 26$

One afternoon a bakery finds that it has $300$ cups of flour and $300$ cups of sugar on hand. Annie and Sam decide to use this to make and sell some batches of cookies and some cakes. Each batch of cookies will require $1$ cup of flour and $3$ cups of sugar. Each cake will require $2$ cups of flour and $1$ cup of sugar. Annie thinks that each batch of cookies should sell for $2$ dollars and each cake for $1$ dollar, but Sam thinks that each batch of cookies should sell for $1$ dollar and each cake should sell for $3$ dollars. Find the difference between the maximum dollars of income they can receive if they use Sam's selling plan and the maximum dollars of income they can receive if they use Annie's selling plan.

Given isosceles triangle $ABC$ ($AB = AC$). A straight line $\ell$ is drawn through its vertex $B$ at a right angle with $AB$ . On the straight line $AC$, an arbitrary point $D$ is taken, different from the vertices, and a straight line is drawn through it at a right angle with $AC$, intersecting $\ell$ at the point $F$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the circumscribed circle of triangle $ABD$.

Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]