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Explore the Back-StoryGiven a triangle $A B C$ with $\angle A C B=120^{\circ}$. The point $L$ is marked on the side $A B$ so that $C L$ is the bisector of $\angle A C B$. The points $N$ and $K$ are marked on the sides $A C$ and $B C$, respectively, so that $C N+C K=C L$. Prove that the triangle $K L N$ is equilateral.

Given a prime number $p$ such that the number $2 p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36 .

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers $a, b, c$ we have

$$

f(a)=b c ; f(b)=c a ; f(c)=a b .

$$

Determine $f(a+b+c)$ in terms of $a, b, c$.

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition:*for any two different digits from 1, 2, 3, 4, 5, 6, 7, 8 there exists a number in $X$ which contains both of them.*

Determine the smallest possible value of $N$.

The side-lengths $a, b, c$ of a triangle $A B C$ are positive integers. Let

$$

T_n=(a+b+c)^{2 n}-(a-b+c)^{2 n}-(a+b-c)^{2 n}+(a-b-c)^{2 n}

$$

for any positive integer $n$. If $\frac{T_2}{2 T_1}=2023$ and $a>b>c$, determine all possible perimeters of the triangle $A B C$.

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at $P$. The point $Q$ is chosen on the segment $B C$ so that $P Q$ is perpendicular to $A C$. Prove that the line joining the centres of the circumcircles of triangles $A P D$ and $B Q D$ is parallel to $A D$.

Let $\mathbb{N}$ be the set of all positive integers and $S=\left{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right}$. Find the largest positive integer $m$ such that $m$ divides $a b c d$ for all $(a, b, c, d) \in S$.

Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the $\operatorname{arc} A B$. Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$. Prove that

(a) the measure of $\angle C E D$ is a constant;

(b) the circumcircle of triangle $C E D$ passes through a fixed point.

For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and

$$

(s(n))^2=m \quad \text { and } \quad(s(m))^2=n .

$$

Let $\Omega_1, \Omega_2$ be two intersecting circles with centres $O_1, O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A, C$ and $\Omega_2$ at points $B, D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $A B$ intersect $\Omega_1$ at points $P, Q$; and the perpendicular bisector of segment $C D$ intersect $\Omega_2$ at points $R, S$ such that $P, R$ are on the same side of $l$. Prove that the midpoints of $P R, Q S$ and $O_1 O_2$ are collinear.

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy

$$

\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n

$$

Consider a set of 16 points arranged in a $4 \times 4$ square grid formation. Prove that if any 7 of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

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Where are the solutions?

Solun

The diagonals AC and BD of a cyclic quadrilateral ABCD meet at P. The point Q is

chosen on the segment BC so that PQ is perpendicular to AC. Prove that the line joining

the centres of the circumcircles of triangles APD and BQD is parallel to AD.