Find $\displaystyle n \in \mathbb N$, $\displaystyle n \geq 2$, and the digits $\displaystyle a_1,a_2,\ldots,a_n$ such that \[ \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n . \]

We define the weight $W$ of a positive integer as follows: $W(1) = 0$, $W(2) = 1$, $W(p) = 1 + W(p + 1)$ for every odd prime $p$, $W(c) = 1 + W(d)$ for every composite $c$, where $d$ is the greatest proper factor of $c$. Compute the greatest possible weight of a positive integer less than 100.

Find all prime numbers $p$ for which $\frac{2^{p-1} -1}{p}$ is a perfect square.

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with \[\omega(n)\ge\omega(P(n)).\] Greece (Minos Margaritis - Iasonas Prodromidis)

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Patrick tosses four four-sided dice, each numbered $1$ through $4$. What's the probability their product is a multiple of four?

For how many integers $1\leq n \leq 2010$, $2010$ divides $1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5 $

Three outlaws entered an inn one evening and ordered baked potatoes. They agreed that the first outlaw would eat half of all the potatoes, the second would eat a third, and the third would eat a sixth. However, being tired, they fell asleep. After the potatoes were served, the first outlaw woke up in the middle of the night, ate half the potatoes, and went back to sleep. Then the second outlaw woke up, ate a third of the remaining potatoes, and also went back to sleep. Finally, near morning, the third outlaw woke up, ate a sixth of the remaining potatoes, and went back to sleep. In the morning, they saw that $10$ potatoes were left on the table. How many potatoes did they originally order?

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

Three doctors, four nurses, and three patients stand in a line in random order. The probability that there is at least one doctor and at least one nurse between each pair of patients is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

For some positive integer $n>m$, it turns out that it is representable as sum of $2021$ non-negative integer powers of $m$, and that it is representable as sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0, 1, \dots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

Ten closed intervals, each of length $1$, are placed in the interval $[0,4]$. Show that there is a point in the larger interval that belongs to at least four of the smaller intervals.

Let $a, b, c$ be positive reals such that $4(a+b+c)\geq\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Define \begin{align*} &A =\sqrt{\frac{3a}{a+2\sqrt{bc}}}+\sqrt{\frac{3b}{b+2\sqrt{ca}}}+\sqrt{\frac{3c}{c+2\sqrt{ab}}} \\ &B =\sqrt{a}+\sqrt{b}+\sqrt{c} \\ &C =\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}. \end{align*} Prove that $$A\leq 2B\leq 4C.$$

In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called [i]ascending [/i] if $a_i < a_j$ and [i]descending[/i] if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ distinct numbers has at least $k$ non-overlapping ascending pairs or at least $k$ non-overlapping descending pairs.

In a perpendicular triangle the perimeter is 60 and the altitude on the hypotenuse is 12. Then, the length of the hypotenuse is $ \text{(A)}\ 24 \qquad \text{(B)}\ 25 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 28$

Let $G$ be the centroid of a triangle $\triangle ABC$ and let $AG, BG, CG$ meet its circumcircle at $P, Q, R$ respectively. Let $AD, BE, CF$ be the altitudes of the triangle. Prove that the radical center of circles $(DQR),(EPR),(FPQ)$ lies on Euler Line of $\triangle ABC$. [i]Proposed by Ivan Chai, Malaysia.[/i]

The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one. If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!! Show that, we can repeat this operation until we have all $1996$ numbers are equal Note: It's not necessary to choose the same quantity of numbers in each operation!!!

Let $d_i(k)$ the number of divisors of $k$ greater than $i$. Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$. Find all $n \in N$ such that $f(n)$ is a perfect square.

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

Let $ S$ be the set of ordered triples $ (x,y,z)$ of real numbers for which \[ \log_{10} (x \plus{} y) \equal{} z\text{ and }\log_{10} (x^2 \plus{} y^2) \equal{} z \plus{} 1. \]There are real numbers $ a$ and $ b$ such that for all ordered triples $ (x,y,z)$ in $ S$ we have $ x^3 \plus{} y^3 \equal{} a \cdot 10^{3z} \plus{} b \cdot 10^{2z}$. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ \frac {15}{2}\qquad \textbf{(B)}\ \frac {29}{2}\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ \frac {39}{2}\qquad \textbf{(E)}\ 24$

Given $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. For how many $x_0$ is it true that $x_0=x_5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{infinitely many} $

Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below: