Prove that exist infinite integers $n$ so that $n^2$ divides $2^n+3^n$. Thanks

[color=darkred]Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions \[f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}\] such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$[/color]

There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?

[u]Round 1[/u] [b]1.1.[/b] A person asks for help every $3$ seconds. Over a time period of $5$ minutes, how many times will they ask for help? [b]1.2.[/b] In a big bag, there are $14$ red marbles, $15$ blue marbles, and$ 16$ white marbles. If Anuj takes a marble out of the bag each time without replacement, how many marbles does Anuj need to remove to be sure that he will have at least $3$ red marbles? [b]1.3.[/b] If Josh has $5$ distinct candies, how many ways can he pick $3$ of them to eat? [u]Round 2[/u] [b]2.1.[/b] Annie has a circular pizza. She makes $4$ straight cuts. What is the minimum number of slices of pizza that she can make? [b]2.2.[/b] What is the sum of the first $4$ prime numbers that can be written as the sum of two perfect squares? [b]2.3.[/b] Consider a regular octagon $ABCDEFGH$ inscribed in a circle of area $64\pi$. If the length of arc $ABC$ is $n\pi$, what is $n$? [u]Round 3[/u] [b]3.1.[/b] Let $ABCDEF$ be an equiangular hexagon with consecutive sides of length $6, 5, 3, 8$, and $3$. Find the length of the sixth side. [b]3.2.[/b] Jack writes all of the integers from $ 1$ to $ n$ on a blackboard except the even primes. He selects one of the numbers and erases all of its digits except the leftmost one. He adds up the new list of numbers and finds that the sum is $2020$. What was the number he chose? [b]3.3.[/b] Our original competition date was scheduled for April $11$, $2020$ which is a Saturday. The numbers $4116$ and $2020$ have the same remainder when divided by $x$. If $x$ is a prime number, find the sum of all possible $x$. [u]Round 4[/u] [b]4.1.[/b] The polynomials $5p^2 + 13pq + cq^2$ and $5p^2 + 13pq - cq^2$ where $c$ is a positive integer can both be factored into linear binomials with integer coefficients. Find $c$. [b]4.2.[/b] In a Cartesian coordinate plane, how many ways are there to get from $(0, 0)$ to $(2, 3)$ in $7$ moves, if each move consists of a moving one unit either up, down, left, or right? [b]4.3.[/b] Bob the Builder is building houses. On Monday he finds an empty field. Each day starting on Monday, he finishes building a house at noon. On the $n$th day, there is a $\frac{n}{8}$ chance that a storm will appear at $3:14$ PM and destroy all the houses on the field. At any given moment, Bob feels sad if and only if there is exactly $1$ house left on the field that is not destroyed. The probability that he will not be sad on Friday at $6$ PM can be expressed as $p/q$ in simplest form. Find $p + q$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784570p24468605]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid? b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen on the line $FX$ so that the quadrilateral $AFEM$ is cyclic. Let lines $AM$ and $DE$ intersect at point $L$ and let $Q$ be the midpoint of segment $AE$. Point $T$ is given on the line $LQ$ such that the quadrilateral $ALDT$ is cyclic. Let $S$ be a point such that the quadrilateral $TFSA$ is a parallelogram, and let $N$ be the second point of intersection of the circumcircle of triangle $ASX$ and the line $TS$. Prove that the circumcircles of triangles $TAN$ and $LSA$ are tangent to each other.

If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.

There is a unit circle that starts out painted white. Every second, you choose uniformly at random an arc of arclength $1$ of the circle and paint it a new color. You use a new color each time, and new paint covers up old paint. Let $c_n$ be the expected number of colors visible after $n$ seconds. Compute $\lim_{n\to \infty} c_n$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

[b]2.[/b] Let $A(x)$ denote the number of positive integers $n$ not greater than $x$ and having at least one prime divisor greater than $\sqrt[3]{n}$. Prove that $\lim_{x\to \infty} \frac {A(x)}{x}$ exists. [b](N. 15)[/b]

Compute the largest prime factor of $111^2+11^3+1^1$. [i]2020 CCA Math Bonanza Lightning Round #3.3[/i]

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]

The number of diagonals that can be drawn in a polygon of 100 sides is: $\textbf{(A)}\ 4850 \qquad \textbf{(B)}\ 4950\qquad \textbf{(C)}\ 9900 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 8800$

The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles. Determine the smallest number of circles needed to do this. [asy] unitsize(1 cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((0,1)); dot((2,1)); dot((0,2)); dot((1,2)); dot((2,2)); [/asy]

Let $P$ is a regular $(2n+1)$-gon in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $\overline{ES}$ contains no other points that lie on the sides of $P$ except $S$ . We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$ , at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$ , we consider them colorless). Find the largest positive integer $n$ for which such a coloring is possible.

In an acute-angled triangle $ABC$, in which $AB<AC$, the point $M$ is the midpoint of the side $BC, K$ is the midpoint of the broken line segment $BAC$ . Prove that $\sqrt2 KM > AB$. (George Naumenko)

Given positive integers $a,b,c$ which are pairwise relatively prime, show that \[2abc-ab-bc-ac \] is the biggest number that can't be expressed in the form $xbc+yca+zab$ with $x,y,z$ being natural numbers.

A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at: $\textbf{(A)}\ 3\text{:}45\text{ p.m.} \qquad \textbf{(B)}\ 3\text{:}30\text{ p.m.} \qquad \textbf{(C)}\ 5\text{:}00\text{ p.m.} \qquad \textbf{(D)}\ 2\text{:}45\text{ p.m.} \qquad \textbf{(E)}\ 5\text{:}30\text{ p.m.}$

A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

Find all pairs $(x, y)$ of real numbers such that \[y^2 - [x]^2 = 19.99 \text{ and } x^2 + [y]^2 = 1999\] where $f(x)=[x]$ is the floor function.

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous functions such that $\lim_{x\rightarrow \infty} f(x) =\infty$ and $\forall x,y\in \mathbb{R}, |x-y|>\varphi, \exists n<\varphi^{2023}, n\in \mathbb{N}$ such that $$f^n(x)+f^n(y)=x+y$$

Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers; [b]a)[/b] Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$. [b]b)[/b] If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$. [b]c)[/b] If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.