For a given positive integer $n$, we wish to construct a circle of six numbers as shown below so that the circle has the following properties: (a) The six numbers are different three-digit numbers, none of whose digits is a 0. (b) Going around the circle clockwise, the first two digits of each number are the last two digits, in the same order, of the previous number. (c) All six numbers are divisible by $n$. The example above shows a successful circle for $n = 2$. For each of $n = 3, 4, 5, 6, 7, 8, 9$, either construct a circle that satisfies these properties, or prove that it is impossible to do so. [asy] pair a = (1,0); defaultpen(linewidth(0.7)); draw(a..-a..a); int[] num = {264,626,662,866,486,648}; for (int i=0;i<6;++i) { dot(a); label(format("$%d$",num[i]),a,a); a=dir(60*i+60); }[/asy]

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$

On a line are given $2k -1$ white segments and $2k -1$ black ones. Assume that each white segment intersects at least $k$ black segments, and each black segment intersects at least $k$ white ones. Prove that there are a black segment intersecting all the white ones, and a white segment intersecting all the black ones.

Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin: $$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$ Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?

Liliana wants to paint a $m\times n$ board. Liliana divides each unit square by one of its diagonals and paint one of the halves of the square with black and the other half with white in such a way that triangles that have a common side haven't the same colour. How many possibilities has Liliana to paint the board?

Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a > 2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

Let $\Gamma$ and $\Gamma'$ be two congruent circles centered at $O$ and $O'$, respectively, and let $A$ be one of their two points of intersection. $B$ is a point on $\Gamma$, $C$ is the second point of intersection of $AB$ and $\Gamma'$, and $D$ is a point on $\Gamma'$ such that $OBDO'$ is a parallelogram. Show that the length of $CD$ does not depend on the position of $B$.

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D$ be a point on the base $BC$ such that $BD:DC = 2: 1$. Note on the segment $AD$ a point $P$ such that $\angle BAC= \angle BPD $. Prove that $\angle BPD = 2 \angle CPD$.

Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2,\ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that $\sum_{1 \leq i \leq k} t_i=4000.$ Find the value of $\sum_{1 \leq i \leq k} a^2_i$. [i]Individual #8[/i]

Consider a sequence of eleven squares that have side lengths $3, 6, 9, 12,\ldots, 33$. Eleven copies of a single square each with area $A$ have the same total area as the total area of the eleven squares of the sequence. Find $A$.

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

In the sequence of numbers 1, 3, 2, ... each term after the first two is equal to the term preceding it minus the term preceding that. The sum of the first one hundred terms of the sequence is $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \minus{}1$

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.

Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$

Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.

The sequence $\{a_n\}_{n\geq 1}$ is defined by $a_{n+2}=7a_{n+1}-a_n$ for positive integers $n$ with initial values $a_1=1$ and $a_2=8$. Another sequence, $\{b_n\}$, is defined by the rule $b_{n+2}=3b_{n+1}-b_n$ for positive integers $n$ together with the values $b_1=1$ and $b_2=2$. Find $\gcd(a_{5000},b_{501})$.

Let $n$ be a positive integer such that $n \geq 3$. Consider a grid with size $n \times n$ where each square can be white or black, in the beginning they are all white. In every step we can change the colors of cells forming a shape like below [img] https://imgtr.ee/images/2023/04/04/k0i9m.png [/img] or any of its rotations. Determine all $n$ such that the whole grid can be black after a finite number of steps.

Let $a, b, c, d$ be reals such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 7$ and $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}= 12$. Find the value of $w =\frac{a}{b}+\frac{c}{d}$ .

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$.

Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?

Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {\sqrt 2}2 \qquad\textbf{(D)}\ \sqrt 3 -1 \qquad\textbf{(E)}\ \text{None of the above} $

Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition: $$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$ for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$

Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$. (a) Prove that $8$ is $100$-discerning. (b) Prove that $9$ is not $100$-discerning. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]