A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

[list] [*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this? [*] Previous problem for the grid of $5\times 5$ lattice.[/list]

Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school. Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?

Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. Then it's again Betül's turn. And the game continues until two stones remain. When two stones remain, Ali confesses the marked stone. At least in how many moves can Betül guarantee to find out the marked stone? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 19 $

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? $\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $

Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

Let $N$ be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when $N$ is divided by 1000.

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? $\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $

Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that [b](a)[/b] The sum of the elements of $A$ is $0,$ [b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$. Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that \[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\] Where $|X|$ denote the numbers of the elements in set $X$.

Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?

Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.

The $n$-element set of real numbers is given, where $n \geq 6$. Prove that there exist at least $n-1$ two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.

Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

In a store, there are 7 cases containing 128 apples altogether. Let $ N$ be the greatest number such that one can be certain to find a case with at least $ N$ apples. Then, the last digit of $ N$ is $ \text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$