Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

A necklace contains 2024 pearls, each one of them having one of the following colours: black, green and yellow. Each moment, we will switch each one of all pearls simultaneously to a new one following the following rules: i) If its two neighbours are of the same colour, then it'll be switched to that same colour. ii) If its two neighbours are of different colours, then it'll be switched to the third colour. a) Does there exist any necklace that can be transformed into a necklace that consists of only yellow pearls if initially half of the pearls are black and the other half is green? b) Does there exist a necklace that can be transformed into a necklace that consists of only yellow pearls if initially 998 pearls are black and the rest 1026 pearls are green?

$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the festival last?

Let triangle $\triangle{ABC}$ have a right angle at $C$, and let $M$ be the midpoint of the hypotenuse $AB$. Choose a point $D$ on line $BC$ so that angle $\angle{CDM}$ measures $30$ degrees. Prove that the segments $AC$ and $MD$ have equal lengths.

A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$.

[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

Let $\Sigma$ be a finite set. For $x,y \in \Sigma^{\ast}$, define \[x\preceq y\] if $x$ is a sub-string ([b]not necessarily contiguous[/b]) of $y$. For example, $ac \preceq abc$. We call a set $S\subseteq \Sigma^{\ast}$ [b][u]good[/u][/b] if $\forall x,y \in \Sigma^{\ast}$, $$ x\preceq y, \; y \in S \; \; \; \Rightarrow \; x\in S .$$ Prove or disprove: Every good set is regular.

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.

The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer? $\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}$

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Let be a board with $2007 \times 2007$ cells, we colour with black $P$ cells such that: $\bullet$ there are no 3 colored cells that form a L-trinomes in any of its 4 orientations Find the minimum value of $P$, such that when you colour one cell more, this configuration can't keep the condition above.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]

Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.

Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$ [i]Proposed by Radu-Andrei Lecoiu[/i]

Find all real numbers $x$ such that: \[ 2^x + 3^x + 6^x - 4^x - 9^x = 1, \] and prove that there are no others.

Solve the following equation in the set of integers $x^5 + 2 = 3 \cdot 101^y$.

Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)

In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.

Let $\,n>6\,$ be an integer and $\,a_{1},a_{2},\ldots,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.

A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$. Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.

In the set of equations $ z^x \equal{} y^{2x}, 2^z \equal{} 2\cdot4^x, x \plus{} y \plus{} z \equal{} 16$, the integral roots in the order $ x,y,z$ are: $ \textbf{(A)}\ 3,4,9 \qquad\textbf{(B)}\ 9, \minus{} 5 \minus{} ,12 \qquad\textbf{(C)}\ 12, \minus{} 5,9 \qquad\textbf{(D)}\ 4,3,9 \qquad\textbf{(E)}\ 4,9,3$

Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list] [i]Proposed by Espen Slettnes[/i]