Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

Right triangle $XY Z$, with hypotenuse $Y Z$, has an incircle of radius $\frac38$ and one leg of length $3$. Find the area of the triangle.

Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.

Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.

Show that the number $122^n - 102^n - 21^n$ is always one less than a multiple of $2020$, for any positive integer $n$.

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

We say "$s$ grows to $r$" if there exists some integer $n>0$ such that $s^n = r.$ Call a real number $r$ "sparcs" if there are only finitely many real numbers $s$ that grow to $r.$ Find all real numbers that are sparse.

In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$. Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively. (a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$. (b) Prove the converse of (a).

The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying $$\lfloor x \rfloor \cdot \{x\} = a \cdot x^2$$ is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$ $\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

Let $C$ be a cube. Let $P$, $Q$, and $R$ be random vertices of $C$, chosen uniformly and independently from the set of vertices of $C$. (Note that $P$, $Q$, and $R$ might be equal.) Compute the probability that some face of $C$ contains $P$, $Q$, and $R$.

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)$. A [i]tourist attraction[/i] is a point where each of the entries of the associated array is either $1$, $2$, $4$, $8$ or $16$. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary? [i]Proposed by Michael Kural[/i]

In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point. What is the largest possible number of red points? [i]Proposed by Art Waeterschoot, Belgium[/i]

In $ \triangle ABC$, we have $ AC \equal{} BC \equal{} 7$ and $ AB \equal{} 2$. Suppose that $ D$ is a point on line $ AB$ such that $ B$ lies between $ A$ and $ D$ and $ CD \equal{} 8$. What is $ BD$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 2 \sqrt {3}\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 4 \sqrt {2}$

Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$

After swimming around the ocean with some snorkling gear, Joshua walks back to the beach where Alexis works on a mural in the sand beside where they drew out symbol lists. Joshua walks directly over the mural without paying any attention. "You're a square, Josh." "No, $\textit{you're}$ a square," retorts Joshua. "In fact, you're a $\textit{cube}$, which is $50\%$ freakier than a square by dimension. And before you tell me I'm a hypercube, I'll remind you that mom and dad confirmed that they could not have given birth to a four dimension being." "Okay, you're a cubist caricature of male immaturity," asserts Alexis. Knowing nothing about cubism, Joshua decides to ignore Alexis and walk to where he stashed his belongings by a beach umbrella. He starts thinking about cubes and computes some sums of cubes, and some cubes of sums: \begin{align*}1^3+1^3+1^3&=3,\\1^3+1^3+2^3&=10,\\1^3+2^3+2^3&=17,\\2^3+2^3+2^3&=24,\\1^3+1^3+3^3&=29,\\1^3+2^3+3^3&=36,\\(1+1+1)^3&=27,\\(1+1+2)^3&=64,\\(1+2+2)^3&=125,\\(2+2+2)^3&=216,\\(1+1+3)^3&=125,\\(1+2+3)^3&=216.\end{align*} Josh recognizes that the cubes of the sums are always larger than the sum of cubes of positive integers. For instance, \begin{align*}(1+2+4)^3&=1^3+2^3+4^3+3(1^2\cdot 2+1^2\cdot 4+2^2\cdot 1+2^2\cdot 4+4^2\cdot 1+4^2\cdot 2)+6(1\cdot 2\cdot 4)\\&>1^3+2^3+4^3.\end{align*} Josh begins to wonder if there is a smallest value of $n$ such that \[(a+b+c)^3\leq n(a^3+b^3+c^3)\] for all natural numbers $a$, $b$, and $c$. Joshua thinks he has an answer, but doesn't know how to prove it, so he takes it to Michael who confirms Joshua's answer with a proof. What is the correct value of $n$ that Joshua found?

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

During a school year $44$ competitions were held. Exactly $7$ students won in each of the competitions. For any two competitions, there exists exactly $1$ student who won in both competitions. Is it true that there exists a student who won all of the competitions?

The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.

Rosa dos Ventos, Aurora Boreal and Manuela do Norte organized a competition between them last weekend, consisting of several athletics events. The winner in each test obtained $x$ points, the second placed $y$ points and the third placed $z$ points ($x,y,z \in N$ and $x >y>z$). The final result of the competition, obtained by adding up the scores in each event, was Rosa had $22$ points, Manuela had $9$ points, Aurora had $9 $ points. In how many tests did they compete and who came second in the high jump knowing that the Manuela won the $100$ meters and no one gave up in any race? [hide=official wording]Rosa dos Ventos, a Aurora Boreal e a Manuela do Norte organizaram no passado fim de semana uma competi¸c˜ao entre elas, consistindo em v´arias provas de atletismo. A vencedora em cada prova obteve x pontos, a segunda classificada y pontos e a terceira classificada z pontos (x,y,z ∈ IN e x >y>z). O resultado final da competi¸c˜ao, obtido por soma das pontua¸c˜oes em cada prova, foi Rosa 22 pontos Manuela 9 pontos Aurora 9 pontos Em quantas provas competiram e quem ficou em segundo lugar no salto em altura sabendo que a Manuela ganhou os 100 metros e que ningu´em desistiu em nenhuma prova?[/hide]

Through a lens that inverts the image we look at the rearview mirror of our car. If it reflects the license plate of the car that follows us, $CS-3965-EN$, draw the image we receive. Also draw the one obtained by permuting previous transformations, that is, reflecting in the mirror the image that the license plate gives the lens. Is the product of both transformations , reflection in the mirror and refraction through the lens, commutative?

Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$. p2. If $a, b$, and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12$$ $$\frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$. p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of ​​the quadrilateral $PMCN$. [img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img] p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers. How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$