The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.

We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number $m$ whose sum of digits is equal to $2023$. b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.

Determine all natural numbers $m$ and $n$ such that \[ n \cdot (n + 1) = 3^m + s(n) + 1182, \] where $s(n)$ represents the sum of the digits of the natural number $n$.

We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$. a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$ b) Determine the smallest $15$ consecutive numbers $a_1, a_2, \ldots, a_{15}$ that do not have the property $(P),$ such that the sum of the numbers $5 a_1, 5 a_2, \ldots, 5 a_{15}$ is a number with the property $(P).$

Determine all sequences of equal ratios of the form \[ \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8} \] which simultaneously satisfy the following conditions: $\bullet$ The set $\{ a_1, a_2, \ldots , a_8 \}$ represents all positive divisors of $24$. $\bullet$ The common value of the ratios is a natural number.

Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations: \begin{align*} a^2 + a = b + c, \\ b^2 + b = a + c, \\ c^2 + c = a + b. \end{align*}

Determine all positive integers $n$ for which the number \[ N = \frac{1}{n \cdot (n + 1)} \]can be represented as a finite decimal fraction.

Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$

For natural number $n$ we define \[ a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}. \] a) Show that $a_1 > 0,2$. b) Show that $a_n < 0$ for infinity many values of $n$ and $a_n > 0$ for infinity values of natural numbers of $n$ as well. ( We denote by $\{ x \} $ the fractional part of $x.$)

In the parallelogram $ABCD$, $AC \cap BD = { O }$, and $M$ is the midpoint of $AB$. Let $P \in (OC)$ and $MP \cap BC = { Q }$. We draw a line parallel to $MP$ from $O$, which intersects line $CD$ at point $N$. Show that $A,N,Q$ are collinear if and only if $P$ is the midpoint of $OC$.

We consider triangle $ABC$ with $\angle BAC = 90^{\circ}$ and $\angle ABC = 60^{\circ}.$ Let $ D \in (AC) , E \in (AB),$ such that $CD = 2 \cdot DA$ and $DE $ is bisector of $\angle ADB.$ Denote by $M$ the intersection of $CE$ and $BD$, and by $P$ the intersection of $DE$ and $AM$. a) Show that $AM \perp BD$. b) Show that $3 \cdot PB = 2 \cdot CM$.

a) Show that there exist irrational numbers $a$, $b$, and $c$ such that the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational numbers. b) Show that if $a$, $b$, and $c$ are real numbers such that $a+b+c=1$, and the numbers $a+b\cdot c$, $b+a\cdot c$, and $c+a\cdot b$ are rational and non-zero, then $a$, $b$, and $c$ are rational numbers.

We consider real positive numbers $a,b,c$ such that $a + b + c = 3.$ Prove that $a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.$

Prove that: a) There are infinitely many pairs $(x,y)$ of real numbers from the interval $[0,\sqrt{3}]$ which satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$. b) There do not exist any pairs $(x,y)$ of rational numbers from the interval $[0,\sqrt{3}]$ that satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.

We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \]where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)

Let $ABCD$ be a tetrahedron and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that for every point $P \in (MN)$ with $P \neq M$ and $P \neq N$, there exist unique points $X$ and $Y$ on segments $AB$ and $CD$, respectively, such that $X,P,Y$ are collinear.

We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$. a) Show that the equation has $2$ distinct real solutions. b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$

Determine functions $f : \mathbb{R} \rightarrow \mathbb{R},$ with property that \[ f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)), \] for every $x$ and $y$ are real numbers.

Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.

Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$. a) Prove that $\frac{s}{r}$ is a natural number. b) Show that both $r$ and $s$ are natural numbers. Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.

Solve the following equation for real values of $x$: \[ 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x. \]

Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying: \[ \sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k). \]

We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure. a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$. b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.

In an art museum, $n$ paintings are exhibited, where $n \geq 33.$ In total, $15$ colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of $n \geq 33$ such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as $T_1, T_2, T_3,$ and $T_4,$ such that any color that is used in both $T_1$ and $T_2$ can also be found in either $T_3$ or $T_4$.

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

Let $A,B \in M_{n}(\mathbb{R}).$ Show that $rank(A) = rank(B)$ if and only if there exist nonsingular matrices $X,Y,Z \in M_{n}(\mathbb{R})$ such that \[ AX + YB = AZB. \]

Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$ a) Prove that $(AB - BA)^2 = O_{n}.$ b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$

We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that \[ g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. \] a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$ b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$

Let $(G, \cdot)$ a finite group with order $n \in \mathbb{N}^{*},$ where $n \geq 2.$ We will say that group $(G, \cdot)$ is arrangeable if there is an ordering of its elements, such that \[ G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}. \] a) Determine all positive integers $n$ for which the group $(Z_n, +)$ is arrangeable. b) Give an example of a group of even order that is arrangeable.

Let $p$ be a prime number, $n$ a natural number which is not divisible by $p$, and $\mathbb{K}$ is a finite field, with $char(K) = p, |K| = p^n, 1_{\mathbb{K}}$ unity element and $\widehat{0} = 0_{\mathbb{K}}.$ For every $m \in \mathbb{N}^{*}$ we note $ \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} $ and define the polynomial \[ f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. \] a) Show that roots of $f_1$ are $ \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}$. b) Let $m \in \mathbb{N}^{*}.$ Determine the set of roots from $\mathbb{K}$ of polynomial $f_{m}.$

Let $a,b \in \mathbb{R}$ with $a < b,$ 2 real numbers. We say that $f: [a,b] \rightarrow \mathbb{R}$ has property $(P)$ if there is an integrable function on $[a,b]$ with property that \[ f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b]. \] Show that for all real number $t$ there exist a unique function $f:[a,b] \rightarrow \mathbb{R}$ with property $(P),$ such that $\int_{a}^{b} f(x) \text{dx} = t.$

Let $f:[0,1] \rightarrow \mathbb{R}$ a non-decreasing function, $f \in C^1,$ for which $f(0) = 0.$ Let $g:[0,1] \rightarrow \mathbb{R}$ a function defined by \[ g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1]. \] a) Show that \[ \int_{0}^{1} g(x) \text{dx} = 0. \] b) Prove that for all functions $\phi :[0,1] \rightarrow [0,1],$ convex and differentiable with $\phi(0) = 0$ and $\phi(1) = 1,$ the inequality holds \[ \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0. \]