Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $. If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences? [i]Proposed by Isabella Grabski[/i]

Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

Given a pentagon $ABCDE$ with all its interior angles less than $180^\circ$. Prove that if $\angle ABC = \angle ADE$ and $\angle ADB = \angle AEC$, then $\angle BAC = \angle DAE$.

Postman Pete has a pedometer to count his steps. The pedometer records up to $ 99999$ steps, then flips over to $ 00000$ on the next step. Pete plans to determine his mileage for a year. On January $ 1$ Pete sets the pedometer to $ 00000$. During the year, the pedometer flips from $ 99999$ to $ 00000$ forty-four times. On December $ 31$ the pedometer reads $ 50000$. Pete takes $ 1800$ steps per mile. Which of the following is closest to the number of miles Pete walked during the year? $ \textbf{(A)}\ 2500 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3500 \qquad \textbf{(D)}\ 4000 \qquad \textbf{(E)}\ 4500$

Compute the number of triangles of different sizes which contain the gray triangle in the figure below. [asy] size(5cm); real n = 4; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray); [/asy] [i]Proposed by Nathan Xiong[/i]

A positive integer $n$ has the property that, for any $2$ integers $a$ and $b$, if $ab + 1$ is divisible by $n$, then $a + b$ is also divisible by $n$. What is the largest possible value of $n$?

(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.$$ Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the [i]nonhomogeneous[/i] differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1.$ Find integers $a,b,c$ such that $(f(-2)-a)(f(2)-b)=c.$

$a,b,c \in (0,1)$ and $x,y,z \in ( 0, \infty)$ reals satisfies the condition $a^x=bc,b^y=ca,c^z=ab$. Prove that \[ \dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4} \] \\

A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones. (L.Steingarts)

Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$ [i]The Problem Selection Committee[/i]

There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

Let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{\left(n-1\right)a_{n-1}+\left(n-2\right)a_{n-2}+...+2a_{2}+1a_{1}}{\left(n-1\right)+\left(n-2\right)+...+2+1}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. [i]Generalization.[/i] Let $\left(b_{1},\ b_{2},\ b_{3},\ ...\right)$ be a monotonically increasing sequence of positive reals, and let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{b_{n-1}a_{n-1}+b_{n-2}a_{n-2}+...+b_{2}a_{2}+b_{1}a_{1}}{b_{n-1}+b_{n-2}+...+b_{2}+b_{1}}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. darij

One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

Determine all pairs $(a, b)$ of integers which satisfy the equality $\frac{a + 2}{b + 1} +\frac{a + 1}{b + 2} = 1 +\frac{6}{a + b + 1}$

Let pairwise different positive integers $a,b, c$ with gcd$(a,b,c) = 1$ are such that $a | (b - c)^2, b | (c- a)^2, c | (a - b)^2$. Prove, that there is no non-degenerate triangle with side lengths $a, b$ and $c$.

Let $a, b$ and $n$ be positive integers with $a>b$ such that all of the following hold: i. $a^{2021}$ divides $n$, ii. $b^{2021}$ divides $n$, iii. 2022 divides $a-b$. Prove that there is a subset $T$ of the set of positive divisors of the number $n$ such that the sum of the elements of $T$ is divisible by 2022 but not divisible by $2022^2$. [i]Proposed by Silouanos Brazitikos, Greece[/i]

Let $14$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $91$. Prove that not all numbers on the paper are different.

A trawler is about to fish in territorial waters of a neighboring country, for what he has no licence. Whenever he throws the net, the coast-guard may stop him with the probability $1/k$, where $k$ is a fixed positive integer. Each throw brings him a fish landing of a fixed weight. However, if the coast-guard stops him, they will confiscate his entire fish landing and demand him to leave the country. The trawler plans to throw the net $n$ times before he returns to territorial waters in his country. Find $n$ for which his expected profit is maximal.

How many ways can you mark $8$ squares of an $8\times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

Evaluate $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n\cdot 2^{n-1}}$.

A square with side $1$ contains a non-self-intersecting polyline of length at least $200$. Prove that there is a straight line parallel to the side of the square that has at least $101$ points in common with this polyline.

Prove that for all $n \ge 3$ there are an infinite number of $n$-sided polygonal numbers which are also the sum of two other (not necessarily different) $n$-sided polygonal numbers! The first $n$-sided polygonal number is $1$. The kth n-sided polygonal number for $k \ge 2$ is the number of different points in a figure that consists of all of the regular $n$-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are $\ell$ cm long where $1 \le \ell \le k - 1$ cm and $\ell$ is an integer. [i]In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly $1$ cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure.[/i] [img]https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png[/img]

Suppose a convex polygon has a perimeter of $1$. Prove that it can be covered with a circle of radius $1/4$.

Joshua's physics teacher, Dr. Lisi, lives next door to the Kubiks and is a long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing and raising chickens as he does trying to map out a $\textit{Theory of Everything}$. Dr. Lisi often poses problems to the Kubik children to challenge them to think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776,-1765,-1754,\ldots.$ Joshua then computes the (positive) difference between the $1980^\text{th}$ term in the sequence, and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?