Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.

Let $f:\mathbb{N}\to\mathbb{N}_0$ be a function that satisfies \[ f(mn) = mf(n) + nf(m)\] for all positive integers $m,n$ and $f(2024)=10120$. Prove that there are two integers $m,n$ with $m\ne n$ such that $f(m)=f(n)$.

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

We say that three different integers are [i]friendly[/i] if one of them divides the product of the other two. Let $n$ be a positive integer. a) Show that, between $n^2$ and $n^2+n$, exclusive, does not exist any triplet of friendly numbers. b) Determine if for each $n$ exists a triplet of friendly numbers between $n^2$ and $n^2+n+3\sqrt{n}$ , exclusive.

Let $B,C,D,E$ be four pairwise distinct collinear points and let $A$ be a point not on line $BC$. Now let the circumcircle of $ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$ Show that $DEFG$ is cyclic if and only if $AB=AC$

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Prove that the area of the circle inscribed in a regular hexagon is greater than $90\%$ of the area of the hexagon.

A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?

Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?

Prove that there exists a positive non-prime integer such that if any three of its neighboring digits are replaced with any given triple of the digits, the number remains non-prime. Does there exist a $1997$-digit such number?

Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

Find all functions $f : R \to R$ such that $$2f(x)f(x + y) -f(x^2) =\frac{x}{2}(f(2x) + 4f(f(y)))$$ for all $x, y \in R$.

Suppose that $f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\}$ satisfies $f(1)=1$ and \[f^{2005}(x)=x\quad\text{for}\ x=1,2,\ldots ,1600. \] $(a)$ Prove that $f$ has a fixed point different from $1$. $(b)$ Find all $n>1600$ such that any $f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\}$ satisfying the above condition has at least two fixed points.

Let $n\geq 1$ be an integer. Find all rings $(A,+,\cdot)$ such that all $x\in A\setminus\{0\}$ satisfy $x^{2^{n}+1}=1$.

The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice. In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it. (a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council? (b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

Show that in a convex region in the plane whose boundary contains at most a finite number of straight line segments and whose area is greater than $\frac{\pi}{4}$ there is at least one pair of points a unit distance apart.

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

In Ace Attorney, Phoenix Wright is rolling a standard fair $20$-sided die. He can roll this die up to three times. After each roll, Phoenix can yell "Objection!'' to roll again, or "Hold It!'' to stop and keep his current number. If Phoenix plays optimally to maximize his final number, find the expected value of this number.

The sum of the numbers $a,b$ and $c$ is zero, and their product is negative. Prove that the number $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$ is positive.

$n$ parallel segments of lengths $a_1 \le a_2 \le a_3 \le ... \le a_n$ were painted to mark an airport atrium. However, the architect decided that the $n$ segments should have equal length. If the cost per meter of extending the lines is equal to the cost of reducing them, how long should the lines be in order to minimize costs?

Prove that two congruent regular hexagons can be cut up into (altogether) $6$ parts such that these $6$ parts can be composed to form an equilateral triangle (without gaps or overlaps).