Prove that there are no $2003$ odd positive integers whose product equals their sum. Is the previous proposition true for $2001$ odd positive integers?

The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle. [asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy] The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.

Let $x_1 \dots, x_{42}$, be real numbers such that $5x_{i+1}-x_i-3x_ix_{i+1}=1$ for each $1 \le i \le 42$, with $x_1=x_{43}$. Find all the product of all possible values for $x_1 + x_2 + \dots + x_{42}$. [i] Proposed by Michael Ma [/i]

Find all the solutions of the system $$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$

There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal $k$ such that the following holds: There are $k$ customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.

Find a $7$-digit integer divisible by $128$, whose decimal representation contains only the digits $2$ and $3$.

If we divide number $1^{1990}+2^{1990}+3^{1990}+...+1990^{1990}$ with $10$, what remainder will we find?

Let $ ABC$ be a triangle with $ \angle ABC\equal{}\angle ACB\equal{}70^{\circ}$. Let point $ D$ on side $ BC$ such that $ AD$ is the altitude, point $ E$ on side $ AB$ such that $ \angle ACE\equal{}10^{\circ}$, and point $ F$ is the intersection of $ AD$ and $ CE$. Prove that $ CF\equal{}BC$.

Find the largest positive integer $n$ such that $n^3 + 4n^2 - 15n - 18$ is the cube of an integer.

How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of above} $

Alice and Bob take turns taking balloons from a box containing infinitely many balloons. In the first turn, Alice takes $k_1$ amount of balloons, where $\gcd(30;k_1)\neq1$. Then, on his first turn, Bob takes $k_2$ amount of ballons where $k_1<k_2<2k_1$. After first turn, Alice and Bob alternately takes as many balloons as his/her partner has. Is it possible for Bob to take $k_2$ amount of balloons at first, such that after a finite amount of turns, one of them have a number of balloons that is a multiple of $2025^{2025}$?

For each $ k\in\mathbb{N} ,k\le 2000, $ Let $ r_k $ be the remainder of the division of $ k $ by $ 4, $ and $ r'_k $ be the remainder of the division of $ k $ by $ 3. $ Prove that there is an unique $ m\in\mathbb{N} ,m\le 1999 $ such that $$ r_1+r_2+\cdots +r_m=r'_{m+1} +r'_{m+2} +\cdots r'_{2000} . $$ [i]Mircea Fianu[/i]

Find the maximum and minimum areas of the region enclosed by the curve $y=|x|e^{|x|}$ and the line $y=a\ (0\leq a\leq e)$ at $[-1,\ 1]$.

How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

Three natural numbers greater than or equal to $2$ are written, not necessarily different, and from them a sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the penultimate one is $b$ and the last one is $c$, it is written $x$ such that $$x\cdot c=a+b+186.$$Determine all the possible values of the three numbers initially written so that when the process continues indefinitely all the written numbers are natural numbers greater than or equal to $2$.

In January Kolya and Vasya have been assessed at school $20$ times and each has been given $20$ marks (each being an integer no greater than $5$ , with both Kolya and Vasya receiving at least twos on each occasion). Kolya has been given as many fives as Vasya fours, as many fours as Vasya threes, as many threes as Vasya twos and as many twos as Vasya fives. If each has the same average mark , determine how many twos were given to Kolya. (S . Fomin, Leningrad)

Let $\{b_n\}_{n\geq 1}^{\infty}$ be a sequence of positive integers. The sequence $\{a_n\}_{n\geq 1}^{\infty}$ is defined as follows: $a_1$ is a fixed positive integer and \[a_{n+1}=a_n^{b_n}+1 ,\qquad \forall n\geq 1.\] Find all positive integers $m\geq 3$ with the following property: If the sequence $\{a_n\mod m\}_{n\geq 1 }^{\infty}$ is eventually periodic, then there exist positive integers $q,u,v$ with $2\leq q\leq m-1$, such that the sequence $\{b_{v+ut}\mod q\}_{t\geq 1}^{\infty}$ is purely periodic.

Natural $a,b,c$ are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every $a \times a$, every $b \times b$, every $c \times c$ squares is even. Is it true, that every number in table must be even?

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

Two coloured points are marked on a line, with the blue one at the left and the red one at the right. You may add to the line two neighbouring points of the same color (both red or both blue) or delete two such points (neighbouring means that there is no coloured point between these two). Prove that after several such transformation you cannot again get only two points on the line in which the red one is at the left and the blue one is at the right. (A Belov)

For a convex quadrilateral $ABCD$, call a point in the interior of $ABCD$ [b]balanced[/b], if (1) $P$ is not on $AC,BD$ (2) Let $AP,BP,CP,DP$ intersect the boundaries of $ABCD$ at $A', B', C', D'$, respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximum possible number of balanced points.