Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$. Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.

Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]

A certain value of $x$ satisfies $1 + x + 5 - 1 = 7$. What is this value of $x$? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{impossible to determine}$

Compute the second smallest positive whole number that has exactly $6$ positive whole number divisors (including itself).

Prove that the sum of digits of the number $1972^n$ is not bounded from above when $n$ tends to infinity.

Each vertex of a regular $1997$-gon is labeled with an integer, so that the sum of the integers is $1$. We write down the sums of the first $k$ integers read counterclockwise, starting from some vertex $(k=1,2,\ldots,1997)$. Can we always choose the starting vertex so that all these sums are positive? If yes, how many possible choices are there?

A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$. $ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad \textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad \textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad \textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad \textbf{(E)}\ 8 \plus{} \sqrt{2}$

We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles are there?

In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? [asy] defaultpen(linewidth(0.7)); size(120); pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC; draw(A--B--C--cycle); for(int i = 1; i < 4; ++i) { AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4); draw(AB[i-1] -- AC[i-1]); } filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]

For each $n\in\mathbb N$, determine the sign of $n^6+5n^5\sin n+1$. For which $n\in\mathbb N$ does it hold that $\frac{n^2+5n\cos n+1}{n^6+5n^5\sin n+1}\ge10^{-4}$.

Let $ a$, $ b$, $ c$, and $ d$ be real numbers with $ |a\minus{}b|\equal{}2$, $ |b\minus{}c|\equal{}3$, and $ |c\minus{}d|\equal{}4$. What is the sum of all possible values of $ |a\minus{}d|$? $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Kan Krao Park is a circular park that has $21$ entrances and a straight line walkway joining each pair of two entrances. No three walkways meet at a single point. Some walkways are paved with bricks, while others are paved with asphalt. At each intersection of two walkways, excluding the entrances, is planted lotus if the two walkways are paved with the same material, and is planted waterlily if the two walkways are paved with different materials. Each walkway is decorated with lights if and only if the same type of plant is placed at least $45$ different points along that walkway. Prove that there are at least $11$ walkways decorated with lights and paved with the same material.

Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$ Prove that: $\deg(P)$ is an even number and find $P(x).$

In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.

Let $x_1, x_2,... , x_{84}$ be the roots of the equation $x^{84} + 7x - 6 = 0$. Compute $\sum_{k=1}^{84} \frac{x_k}{x_k-1}$.

No matter how two copies of a convex polygon are placed inside a square, they always have a common point. Prove that no matter how three copies of the same polygon are placed inside this square, they also have a common point.

In Mexico City, in order to limit traffic flow, for each private car is given one day a week on which it cannot go on the city streets. A wealthy family of 10 bribed the police, and for each car they are given 2 days, one of which the police chooses as a ''no travel'' day. What is the smallest number of cars a family needs to buy so that each family member can drive independently every day, if the approval of “no travel” days for cars occurs sequentially? [hide=original wording]В городе Мехико в целях ограничения транспортного потока для каждой частной автомашины устанавливаются один деньв неделю, в который она не может выезжать на улицы города. Состоятельная семья из 10 человек подкупила полицию, и для каждой машины они называют 2 дня, один из которых полиция выбирает в качестве ''невыездного'' дня. Какое наименьшее количество машин нужно купить семье, чтобы каждый день каждый член семьи мог самостоятельно ездить, если утверждение ''невыездных'' дней для автомобилей идет последовательно?[/hide]

Find all functions $f:Q^+ \to Q^+$ such that for any $x,y \in Q^+$ : $$y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]$$

A set $M$ of positive integers is called [i]connected[/i] if for any element $x\in M$ at least one of the numbers $x-1,x+1$ is in $M$. Let $U_n$ be the number of the connected subsets of $\{1,2,\ldots,n\}$. a) Compute $U_7$; b) Find the smallest number $n$ such that $U_n \geq 2006$.

Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$. Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$. Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$. Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$. Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$. Prove: $MF$ and $AO$ are orthogonal to each other.

The fourth-degree polynomial $P(x)$ is such that the equation $P(x)=x$ has $4$ roots, and any equation of the form $P(x)=c$ has no more two roots. Prove that the equation $P(x)=-x$ too has no more than two roots.

A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ in point $P$, $DM$ meets $AC$ in point $Q$. Prove that the area of quadrilateral $APQD$ is equal to the half of the area of the square.

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

$1001$ marbles are drawn at random and without replacement from a jar of $2020$ red marbles and $n$ blue marbles. Find the smallest positive integer $n$ such that the probability that there are more blue marbles chosen than red marbles is strictly greater than $\frac{1}{2}$. [i]Proposed by Taiki Aiba[/i]

Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$