Let $M \ge 1$ be a real number. Determine all natural numbers $n$ for which there exist pairwise distinct natural numbers $a, b, c \gt M$, such that
$$
n = (a, b) \cdot (b, c) + (b, c) \cdot (c, a) + (c, a) \cdot (a, b)
$$
(where $(x, y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
Let $ABC$ be an acute angled triangle with $\angle B \gt \angle C$. Consider the points $D, E, J, K, S$ on its circumcircle $C(O)$, such that $A, E, J$ and $K$ are on the same side of the line $BC$, the diameter $DE$ and the line $BC$ are orthogonal, $S \in \widehat{EK}$ and $\widehat{AE} = \widehat{BJ} = \widehat{CK} = \frac{1}{4}\widehat{CE}$. Let $F, M, P, Q$ be the intersection points of the lines $AC$ and $DE$, $BK$ and $AD$, $BK$ and $AC$, $CJ$ and $BF$, respectively. If $\angle SMK = 30^\circ$ and $\angle AQP = 90^\circ$, prove that the line $MS$ is tangent to the circumcircle of the triangle $AOF$.
Dana Heuberger
Given the trapezoid $ABCD$ with the smaller base $AB$, squares $ADEF$ and $BCGH$ are constructed externally to the trapezoid. Prove that the perpendicular bisector of $AB$ passes through the midpoint of $FH$.
Fix an integer $n \ge 3$. Let $S$ be a set of $n$ points in the plane, no three of which are collinear. Given pairwise distinct points $A, B, C$ in $S$, the triangle $ABC$ is suitable for $AB$ if area $ABC \le$ area $ABX$ for all $X$ in $S$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several suitable triangles.) A triangle is adequate if its vertices are all in $S$ and it is suitable for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ adequate triangles.
Bulgaria, Alexadar Ivanov
Let $n \ge 2$ be an integer, and let $f$ be a $4n$-variable polynomial with real coefficients, such that, for any $2n$ points $(x_1, y_1), \dots, (x_{2n}, y_{2n})$ in the Cartesian plane,
$$
f(x_1, y_1, \dots, x_{2n}, y_{2n}) = 0
$$
if and only if they form the vertices of a regular $2n$-gon in some order, or are all equal. Determine the smallest possible degree of $f$.
Given a triangle $ABC$, let $H$ and $O$ be its orthocentre and circumcentre, respectively. Let $K$ be the midpoint of the line segment $AH$. Let further $\ell$ be a line through $O$, and let $P$ and $Q$ be the orthogonal projections of $B$ and $C$ onto $\ell$, respectively. Prove that $KP + KQ \ge BC$.
Let $P(x)$, $Q(x)$, $R(x)$ and $S(x)$ be non-constant polynomials with real coefficients such that $P(Q(x)) = R(S(x))$. Prove that, if the degree of $P(x)$ is divisible by the degree of $R(x)$, then $P(x) = R(T(x))$ for some polynomial $T(x)$ with real coefficients.
Let $\Gamma$ be a connected graph on $r + g + b + 1$ vertices. The edges of $\Gamma$ bear three colours: red, green, and blue. It turns out that $\Gamma$ has a spanning tree with exactly $r$ red edges, a spanning tree with exactly $g$ green edges, and a spanning tree with exactly $b$ blue edges. Prove that $\Gamma$ has a spanning tree with exactly $r$ red edges, exactly $g$ green edges, and exactly $b$ blue edges.
Find all four-digit numbers $\overline{abcd}$, so that $\overline{ab}$, $\overline{cb}$ and $d$ are prime numbers, and $\overline{ab}^2 + \overline{cb}^2 + d^2 = 2022$.
A positive integer is called *special* if all of its digits are nonzero and any two adjacent digits are consecutive (not necessarily in ascending order).
a) Find the largest special number for which the sum of its digits is $2023$.
b) Find the smallest special number for which the sum of its digits is $2022$.
A store sold 235 robots over the course of twelve months. Each month, 16, 20, or 25 robots were sold. Find the number of months in which exactly 20 robots were sold.
Find the natural numbers $m, n$ so that
$$
n \cdot (n + 1) = 3^m + s(n) + 1182,
$$
where $s(n)$ represents the sum of the digits of the positive integer $n$.
On a blackboard there are written, one after another, all positive integers from $1$ to $30000$, forming the following sequence of digits:
$123456789101112\ldots30000$.
Find the number of occurrences of $2023$ in this sequence.
We say that a natural number $n \ge 2$ is *nice*, if in its prime factor decomposition, one of the primes has exponent 3.
a) Determine the smallest natural number $N$, such that no matter how we choose $N$ consecutive natural numbers, at least one of them is nice.
b) Find the smallest 15 consecutive natural numbers $a_1, a_2, \dots, a_{15}$ that are not nice, but for which the sum $5a_1 + 5a_2 + \dots + 5a_{15}$ is nice.
Find all the sequences of equal ratios of the form $\frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}$ fulfilling the conditions:
- the set $\{a_1, a_2, \dots, a_8\}$ is the set of the positive divisors of $24$;
- the common value of the ratios is an integer.
The elements of a set $A$ are 13 consecutive positive integers, the elements of a set $B$ are 12 consecutive positive integers, and the elements of the set $A \cup B$ are 15 consecutive integers.
a) Find the number of elements of $A \setminus B$.
b) If, moreover, the sum of the elements of the set $A$ is equal to the sum of the elements of the set $B$, find the two sets.
The triangle $ABC$ is isosceles, with $AB = AC$ and $\angle ABC = 72^\circ$. The point $D$ is taken on the line $BC$, so that $C$ is on the segment $BD$ and $CD = AB$.
a) Prove that $AC$ is the bisector of the angle $\angle BAD$.
b) The point $E$ is taken on the parallel to $AB$ through $D$, on the same side of $BD$ as $A$, so that $DE = DB$. Let $F$ be the common point of the lines $AD$ and $BE$. Prove that the lines $AC$ and $AE$ are perpendicular and $AF = FC = BC$.