Let $n \geq 3$ be an integer. Let $f$ be a function from the set of all integers to itself with the following property: If the integers $a_1,a_2,\ldots,a_n$ form an arithmetic progression, then the numbers $$f(a_1),f(a_2),\ldots,f(a_n)$$ form an arithmetic progression (possibly constant) in some order. Find all values for $n$ such that the only functions $f$ with this property are the functions of the form $f(x)=cx+d$, where $c$ and $d$ are integers.

Prove that for every positive integer $n$, there exist integers $a$ and $b$ such that $4a^2 + 9b^2 - 1$ is divisible by $n$.

Consider a sequence of words each consisting of two letters, $A$ and $B$ . The first word is "$A$" , while the second word is "$B$" . The $k$-th word is obtained from the ($k -2$)-nd by writing after it the ($k -1$)th one. (So the first few elements of the sequence are "$A$" , "$B$" ,"$AB$" , "$BAB$" , "$ABBAB$" . ) Does there exist in this sequence a "periodical" word, i.e. a word of the form $P P P ... P$ , where $P$ is a word , repeated at least once? (Remark: For instance, the word $BABBBABB$ is of the form $PP$ , in which $P$ is repeated exactly once . ) (A. Andjans, Riga)

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

$A, B$, and $C$ are vertices of a triangle, and $P$ is a point within the triangle. If angles $\angle BAP$, $\angle BCP$, and $\angle ABP$ are all $30^o$ and angle $\angle ACP$ is $45^o$, what is $\sin(\angle CBP)$?

Let there be $3399$ numbers arbitrarily chosen among the first $6798$ integers $1, 2, \ldots , 6798$ in such a way that none of them divides another. Prove that there are exactly $1982$ numbers in $\{1, 2, \ldots, 6798\}$ that must end up being chosen.

We build a modified version of Pascal’s triangle as follows: in the first row we write a $2$ and a $3$, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a $2$ and end with a $3$). In the $13$th row, what is the $5$th number from the left? [img]https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png[/img]

Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.

Find a triangle $ABC$ with a point $D$ on side $AB$ such that the measures of $AB, BC, CA$ and $CD$ are all integers and $\frac{AD}{DB}=\frac{9}{7}$, or prove that such a triangle does not exist.

For every positive $ \alpha$, natural number $ n$, and at most $ \alpha n$ points $ x_i$, construct a trigonometric polynomial $ P(x)$ of degree at most $ n$ for which \[ P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,\] where the constant $ c$ depends only on $ \alpha$. [i]G. Halasz[/i]

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

Find $k \in \mathbb{N}$ such that [b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)$ forms an arithmetic progression. [b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)$ forms an arithmetic progression. Find all $n$ which satisfies part [b]b.)[/b]

Let $ABC$ be a triangle with $BC=9$, $CA=8$, and $AB=10$. Let the incenter and incircle of $ABC$ be $I$ and $\gamma$, respectively, and let $N$ be the midpoint of major arc $BC$ of the cirucmcircle of $ABC$. Line $NI$ meets the circumcircle of $ABC$ a second time at $P$. Let the line through $I$ perpendicular to $AI$ meet segments $AB$, $AC$, and $AP$ at $C_1$, $B_1$, and $Q$, respectively. Let $B_2$ lie on segment $CQ$ such that line $B_1B_2$ is tangent to $\gamma$, and let $C_2$ lie on segment $BQ$ such that line $C_1C_2$ tangent to $\gamma$. The length of $B_2C_2$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $100m+n$. [i]Proposed by Vincent Huang[/i]

Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

Estimate the number of times a one-digit answer (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) has been submitted as an answer for any question by any team in the first 4 sets of this competition's lightning round. An estimate $E$ earns $\frac{2}{1+|log_2(A)-log_2(E)|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.3[/i]

Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.

The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? $\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 $

(a) Find two real numebrs $a,b$ such that $|ax+b-\sqrt{x}| \le \frac{1}{24}$ for $1 \le x \le 4$. (b) Prove that the constant $\frac{1}{24}$ cannot be replaced by a smaller one.

For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.