The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.

Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that $$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$ [b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $ [b]b)[/b] Show that if $ a+b=1, $ then $ f $ is constant.

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

If $f(1) = 1$ and $f(n+1) = \frac{2f(n) + 1}{2}$, then find $f(237)$. $\text{(A) }117\qquad\text{(B) }118\qquad\text{(C) }119\qquad\text{(D) }120\qquad\text{(E) }121$

Prove that all solutions of the equation $0.001x^3 + x^2 - 1 = 0$ are irrational numbers. (A number $x$ is said to be [i]irrational[/i], if one cannot write $x = m/n$, with $m$ and $n$ integer numbers.)

Let $x_1,x_2,\cdots,x_n$ be positive real numbers such that $x_1+x_2+\cdots+x_n=1$ $(n\ge 2)$. Prove that\[\sum_{i=1}^n\frac{x_i}{x_{i+1}-x^3_{i+1}}\ge \frac{n^3}{n^2-1}.\]here $x_{n+1}=x_1.$

Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.

The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$.

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties: (i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$; (ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite. Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic. [i]Proposed by Ang Jie Jun, Singapore[/i]

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

Given any positive integer $n$, let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$. Prove that (1) $a_{n+1}>a_n$; (2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$.

Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area

If $p$ is a prime number greater than $2$ and $a, b, c$ integers not divisible by $p$, prove that the equation \[ax^2 + by^2 = pz + c\] has an integer solution.

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

Find the sum $\sum_{i=1}^{\infty} \frac{n}{2^n}.$

The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is $ \displaystyle \textbf{(A)}\ \sqrt {\frac {a}{b}} \qquad \textbf{(B)}\ \ \frac {a}{b} \qquad \textbf{(C)}\ \ \frac {a^2}{b^2} \qquad \textbf{(D)}\ \ \frac {a \plus{} b}{2b} \qquad \textbf{(E)}\ \ \frac {a^2 \plus{} 2ab}{b^2 \plus{} 2ab}$ [asy]unitsize(2cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); fill(Arc((1/3,0),2/3,0,180)--reverse(Arc((-2/3,0),1/3,180,360))--reverse(Arc((0,0),1,0,180))--cycle,mediumgray); draw(unitcircle); draw(Arc((-2/3,0),1/3,360,180)); draw(Arc((1/3,0),2/3,0,180)); label("$a$",(-2/3,0)); label("$b$",(1/3,0)); draw((-2/3+1/15,0)--(-1/3,0),EndArrow(4)); draw((-2/3-1/15,0)--(-1,0),EndArrow(4)); draw((1/3+1/15,0)--(1,0),EndArrow(4)); draw((1/3-1/15,0)--(-1/3,0),EndArrow(4));[/asy]

Given the larger of two circles with center $ P$ and radius $ p$ and the smaller with center $ Q$ and radius $ q$. Draw $ PQ$. Which of the following statements is false? $ \textbf{(A)}\ p\minus{}q\text{ can be equal to }\overline{PQ}\\ \textbf{(B)}\ p\plus{}q\text{ can be equal to }\overline{PQ}\\ \textbf{(C)}\ p\plus{}q\text{ can be less than }\overline{PQ}\\ \textbf{(D)}\ p\minus{}q\text{ can be less than }\overline{PQ}\\ \textbf{(E)}\ \text{none of these}$

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.

$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ . Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

How many positive integers $ b$ have the property that $ \log_b729$ is a positive integer? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$