Let $n$ and $k$ be positive integers. A function $f : \{1, 2, 3, 4, \dots , kn - 1, kn\} \to \{1, \cdots , 5\}$ is [i]good[/i] if $f(j + k) - f(j)$ is multiple of $k$ for every $j = 1, 2. \cdots , kn - k$. [b](a)[/b] Prove that, if $k = 2$, then the number of good functions is a perfect square for every positive integer $n$. [b](b)[/b] Prove that, if $k = 3$, then the number of good functions is a perfect cube for every positive integer $n$.

In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$. [asy] unitsize(1.2 cm); pair A, B, C, D, P, Q, R, S; A = (0,0); D = (3,0); B = (0.8,1.5); C = (3.2,1.5); S = extension(A, incenter(A,B,D), D, incenter(A,C,D)); Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A); P = extension(A, S, B, Q); R = extension(D, S, C, Q); draw(A--D--C--B--cycle); draw(B--Q--C); draw(A--S--D); dot("$A$", A, SW); dot("$B$", B, NW); dot("$C$", C, NE); dot("$D$", D, SE); dot("$P$", P, dir(90)); dot("$Q$", Q, dir(270)); dot("$R$", R, dir(90)); dot("$S$", S, dir(90)); [/asy] Attention: the figure is not drawn to scale.

In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$) or is divisible by $4$ but not by $100$ (such as $2012$). The $200\text{th}$ anniversary of the birth of novelist Charles Dickens was celebrated on February $7$, $2012$, a Tuesday. On what day of the week was Dickens born? $ \textbf{(A)}\ \text{Friday} \qquad\textbf{(B)}\ \text{Saturday} \qquad\textbf{(C)}\ \text{Sunday} \qquad\textbf{(D)}\ \text{Monday} \qquad\textbf{(E)}\ \text{Tuesday} $

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

We say that a set $S$ of natural numbers is [i]synchronous [/i] provided that the digits of $a^2$ are the same (in occurence and numbers, if differently ordered) for all numbers $a$ in $S$. For example, $\{13, 14, 31\}$ is synchronous, since we find $\{13^2, 14^2, 31^2\} = \{169, 196, 961\}$. But $\{119, 121\}$ is not synchronous, for even though $119^2 = 14161$ and $121^2 = 14641$ have the same digits, they occur in different numbers. Show that there exists a synchronous set containing $2021$ different natural numbers.

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi

One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write?

Let $ a$, $ b$, $ c$ be real numbers such that for every two of the equations \[ x^2\plus{}ax\plus{}b\equal{}0, \quad x^2\plus{}bx\plus{}c\equal{}0, \quad x^2\plus{}cx\plus{}a\equal{}0\] there is exactly one real number satisfying both of them. Determine all possible values of $ a^2\plus{}b^2\plus{}c^2$.

If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, what is the sum of $a$ and $b$? $\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 35 \qquad \textbf{(E)}\ 37$

Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$

A prime number $p$ is mundane if there exist positive integers $a$ and $b$ less than $\tfrac{p}{2}$ such that $\tfrac{ab-1}{p}$ is a positive integer. Find, with proof, all prime numbers that are not mundane.

Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8. Prove that the sum \[\sum_{i = 1}^8 P_{i - 1} P_i^2\] is minimal if and only if $Q$ is the centre of the circle.

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

Let $ a$, $ b$ and $ m$ be positive integers. Prove that there exists a positive integer $ n$ such that $ (a^n \minus{} 1)b$ is divisible by $ m$ if and only if $ \gcd (ab, m) \equal{} \gcd (b, m)$.

A tree with $n\geq 2$ vertices is given. (A tree is a connected graph without cycles.) The vertices of the tree have real numbers $x_1,x_2,\dots,x_n$ associated with them. Each edge is associated with the product of the two numbers corresponding to the vertices it connects. Let $S$ be a sum of number across all edges. Prove that \[\sqrt{n-1}\left(x_1^2+x_2^2+\dots+x_n^2\right)\geq 2S.\] (Author: V. Dolnikov)

Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc. Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal. Who made it?

Find the sum (in base $10$) of the three greatest numbers less than $1000_{10}$ that are palindromes in both base $10$ and base $5$.

Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. [i]N. Agakhanov[/i]

Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there?

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective. [i]Note: [/i] A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

How many different positive integers divide $10!$ ?

Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$