$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

Given trapezoid $ABCD$ with bases $AB \parallel CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$ , i.e. $CE \perp BD$ and $DE \perp AC$ . By analogy, $AF \perp BD$ and $BF \perp AC$ . Are three points $E , O, F$ located on the same line?

Solve for $x$: \begin{eqnarray*} v - w + x - y + z & = & 79 \\ v + w + x + y + z & = & -1 \\ v + 2w + 4x + 8y + 16z & = & -2 \\ v + 3w + 9x + 27y + 81z & = & -1 \\ v + 5w + 25x + 125y + 625z & = & 79. \end{eqnarray*}

Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

Let $k$ be a positive integer, and set $n=2^k$, $N=\{1, 2, \cdots, n\}$. For any bijective function $f:N\rightarrow N$, if a set $A\subset N$ contains an element $a\in A$ such that $\{a, f(a), f(f(a)), \cdots\} = A$, then we call $A$ as a cycle of $f$. Prove that: among all bijective functions $f:N\rightarrow N$, at least $\frac{n!}{2}$ of them have number of cycles less than or equal to $2k-1$. [i]Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto.[/i] [i]Proposed by CSJL[/i]

Let $n$ and $k$ be positive integers such that $k\leq 2^n$. Banana and Corona are playing the following variant of the guessing game. First, Banana secretly picks an integer $x$ such that $1\leq x\leq n$. Corona will attempt to determine $x$ by asking some questions, which are described as follows. In each turn, Corona chooses $k$ distinct subsets of $\{1, 2, \ldots, n\}$ and, for each chosen set $S$, asks the question "Is $x$ in the set $S$?''. Banana picks one of these $k$ questions and tells both the question and its answer to Corona, who can then start another turn. Find all pairs $(n,k)$ such that, regardless of Banana's actions, Corona could determine $x$ in finitely many turns with absolute certainty. [i]Pitchayut Saengrungkongka, Thailand[/i]

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?

Let $x, y, z$ be real numbers from interval $(0, 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}$$

Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that (a) $|LH| = |KM|$ (b) the line through $B$ perpendicular to $DE$ bisects $HK$.

We have 7 buckets labelled 0-6. Initially bucket 0 is empty, while bucket $n$ (for each $1 \leq n \leq 6$) contains the list $[1,2, \ldots, n]$. Consider the following program: choose a subset $S$ of $[1,2,\ldots,6]$ uniformly at random, and replace the contents of bucket $|S|$ with $S$. Let $\tfrac{p}{q}$ be the probability that bucket 5 still contains $[1,2, \ldots, 5]$ after two executions of this program, where $p,q$ are positive coprime integers. Find $p$.

Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.

Let \( ABC \) be an acute scalene triangle with orthocenter \( H \), and consider \( M \) to be the midpoint of side \( BC \). Define \( P \neq A \) as the intersection point of the circle with diameter \( AH \) and the circumcircle of triangle \( ABC \), and let \( Q \) be the intersection of \( AP \) with \( BC \). Let \( G \neq M \) be the intersection of the circumcircle of triangle \( MPQ \) with the circumcircle of triangle \( AHM \). Show that \( G \) lies on the circle that passes through the feet of the altitudes of triangle \( ABC \).

Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.

Let p>3 be a prime and k>0 an integer. Find the multiplicity of X-1 in the factorization of $ f(X)= X^{p^k-1}+X^{p^k-2}+\cdots+X+1$ modulo p; in other words, find the unique non-negative integer r such that $ (X - 1)^r $ divides f(X) \modulo p, but$ (X - 1)^{r+1} $does not divide f(X) \modulo p.

Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.

The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes? $ \text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190$

In a tennis tournament of $10$ players, everyone played against everyone once. In this tournament, if player $i$ won the match against player $j$, then the total number of matches $i$ lost plus the total number of matches $j$ won is greater than or equal to $8$. We will say that three players $i$, $j$, $k$ form an [i]atypical tri[/i]o if $i$ beat $j$, $j$ beat $k$ and $k$ beat $i$. Prove that in the tournament there were exactly $40$ atypical trios.

In a trapezoid $ABCD$ with base $AB$ let $E$ be the midpoint of side $AD$. Suppose further that $2CD=EC=BC=b$. Let $\angle ECB=120^{\circ}$. Construct the trapezoid and determine its area based on $b$.

The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions: $a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$ $a_{1} + a_{2} = 20$ $a_{3} + a_{4} + ... + a_{20} \le 20$ . What is maximum value of the expression: $a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ? For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?

Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$

Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$ [i]Proposed by William Chao[/i]

Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$ where $x$ is a real number and $p$ is a nonzero integer. Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.