For $n \geq 3$, a [i]special n-triangle[/i] is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle: $$41$$ $$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$ $$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$ Note that a special $n$-triangle contains $3(n - 1)$ numbers. An infinite set $A$ of positive integers is a [i]special set[/i] if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle. Show that the set of positive integers that are not multiples of $2023$ is a special set.

The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$? Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.

Solve the equation $$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11$$

There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7\le k \le 2013$. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there? [i]Ray Li[/i]

Find all functions $f : \mathbb{R} \to \mathbb{R}$, such that for any real $x, y$ holds the following: $$f(x+yf(x+y)) = f(y^2) + xf(y) + f(x)$$ [i]Proposed by Vadym Koval[/i]

What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour? (A Shapovalov)

Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them and gets the same set of numbers as in the beginning. What were the starting numbers?

One of the excircles of triangle $ABC$ is tangent to the side $AB$ and to the extensions of sides $CA$ and $CB$ at points $C_1$, $B_1$ and $A_1$ respectively. Another excircle is tangent to side $AB$ and to the extensions of sides $BA$ and $BC$ at points $B_2$, $C_2$ and $A_2$ respectively. Line $A_1B_1$ and $A_2B_2$ intersect at point $P$,. lines $A_1C_1$ and $A_2C_2$ intersect at point $Q$. Prove that the points $A$, $P$, $Q$ are collinear [I]Proposed by S. Berlov[/i]

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

A [i]Georg Mohr[/i] cube is a cube with six faces printed respectively $G, E, O, R, M$ and $H$. Peter has nine identical Georg Mohr dice. Is it possible to stack them on top of each other for a tower there on each of the four pages in some order show the letters $G\,\, E \,\, O \,\, R \,\, G \,\, M \,\, O \,\, H \,\, R$?

Let $H$ be orthocenter of an acute $\Delta ABC$, $M$ is a midpoint of $AC$. Line $MH$ meets lines $AB, BC$ at points $A_1, C_1$ respectively, $A_2$ and $C_2$ are projections of $A_1, C_1$ onto line $BH$ respectively. Prove that lines $CA_2, AC_2$ meet at circumscribed circle of $\Delta ABC$. [i]Proposed by Anton Trygub[/i]

If both $ x$ and $ y$ are both integers, how many pairs of solutions are there of the equation $ (x\minus{}8)(x\minus{}10) \equal{} 2^y?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{more than 3}$

Find the sum of the arithmetic series \[ 20+20\frac{1}{5}+20\frac{2}{5}+\cdots+40 \] $ \textbf{(A)}\ 3000 \qquad\textbf{(B)}\ 3030 \qquad\textbf{(C)}\ 3150 \qquad\textbf{(D)}\ 4100 \qquad\textbf{(E)}\ 6000 $

Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively. a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint? Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.

In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$. Leonard Giugiuc

For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying \[a_1+a_2+\cdots+a_{10}=20.\] Compute the number of positive integer divisors of $N$. [i]2021 CCA Math Bonanza Individual Round #14[/i]

Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$

Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves: (a) those regions opening north or east; and (b) those regions opening south or west. [img]https://cdn.artofproblemsolving.com/attachments/b/e/aa20c9f9bc44bd1e5a9b9e86d49debf0f821b7.png[/img] (The figure above shows a path meeting the conditions of the problem on a $5 \times 8$ grid. The shaded regions are those opening north or east while the rest open south or west.)

Let be $f:\left[0,1\right]\rightarrow\left[0,1\right]$ a continuous and bijective function,such that : $f\left(0\right)=0$.Then the following inequality holds: $\left(\alpha+2\right)\cdotp\int_{0}^{1}x^{\alpha}\left(f\left(x\right)+f^{-1}\left(x\right)\right)\leq2,\forall\alpha\geq0 $

For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.

Find the area of the region of the $xy$-plane defined by the inequality $|x|+|y|+|x+y| \le 1$.

Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.