a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.

In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.

A museum has the shape of a (not necessarily convex) 3$n$-gon. Prove that $n$ custodians can be positioned so as to control all of the museum’s space.

Three congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$. There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$. [asy] size(250); filldraw(ellipse((2.2,0),2,1),grey); filldraw(ellipse((0,-2),4,2),white); filldraw(ellipse((0,+2),4,2),white); filldraw(ellipse((6.94,0),4,2),white);[/asy]

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive integer $n$ such that $\frac{T(n^2)}{T(n)}=R$?

Given is a cyclic quadrilateral $ABCD$. Points $K, L, M, N$ lying on sides $AB, BC, CD, DA$, respectively, satisfy $\angle ADK=\angle BCK$, $\angle BAL=\angle CDL$, $\angle CBM =\angle DAM$, $\angle DCN =\angle ABN$. Prove that lines $KM$ and $LN$ are perpendicular.

Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. What is the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$? $ \textbf{(A)}\ \frac{\sqrt 2}2 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \frac32 \qquad\textbf{(E)}\ 2 $