The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? $\text{(A) }6000\qquad\text{(B) }7500\qquad\text{(C) }24,000\qquad\text{(D) }30,000\qquad\text{(E) }37,500$

Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Prove that \[ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c} \] and determine when equality occurs.

Some natural numbers are marked. It is known that on every a segment of the number line of length $1999$ has a marked number. Prove that there is a pair of marked numbers, one of which is divisible by the other.

In a $k$-player tournament for $k > 1$, every player plays every other player exactly once. Find with proof the smallest value of $k$ such that it is possible that for any two players, there was a third player who beat both of them.

Let $ABC$ be a triangle , $AD$ an altitude and $AE$ a median . Assume $B,D,E,C$ lie in that order on the line $BC$ . Suppose the incentre of triangle $ABE$ lies on $AD$ and he incentre of triangle $ADC$ lies on $AE$ . Find ,with proof ,the angles of triangle $ABC$ .

Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

Which of the following cannot be equal to $x^2+y^2$, if $x^2 + xy + y^2 = 1$ where $x,y$ are real numbers? $ \textbf{a)}\ \dfrac{1}{\sqrt 2} \qquad\textbf{b)}\ \dfrac 12 \qquad\textbf{c)}\ \sqrt 2 \qquad\textbf{d)}\ 3-\sqrt 3 \qquad\textbf{e)}\ \text{None of above} $

The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event? [i]Alexey Zaslavsky[/i]

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

A regular tetrahedron with unit edge length is given. Prove that: (a) There exist four points on the surface $S$ of the tetrahedron, such that the distance from any point of the surface to one of these four points does not exceed $1/2$; (b) There do not exist three points with this property. The distance between two points on surface $S$ is defined as the length of the shortest polygonal line going over $S$ and connecting the two points.

Let $\Gamma = \{\varepsilon,0,00,\ldots\}$ be the set of all finite strings consisting of only zeroes. We consider $\textit{six-state unary DFAs}$ $D = (F,q_0,\delta)$ where $F$ is a subset of $Q = \{1,2,3,4,5,6\}$, not necessarily strict and possibly empty; $q_0\in Q$ is some $\textit{start state}$; and $\delta: Q\rightarrow Q$ is the $\textit{transition function}$. For each such DFA $D$, we associate a set $F_D\subseteq\Gamma$ as the set of all strings $w\in\Gamma$ such that \[\underbrace{\delta(\cdots(\delta(q_0))\cdots)}_{|w|\text{ applications}}\in F,\] We say a set $\mathcal D$ of DFAs is $\textit{diverse}$ if for all $D_1,D_2\in\mathcal D$ we have $F_{D_1}\neq F_{D_2}$. What is the maximum size of a diverse set?

How many diagonals does a regular undecagon ($11$-sided polygon) have?

Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$: \begin{itemize} \item If $a$ divides $b$ and $a \in U$ then $b \in U$. \item If $a,b \in U$ then $\gcd(a,b) \in U$. \item If $a,b \notin U$ then $\operatorname{lcm} (a,b) \notin U$. \end{itemize} How many such pseudo-ultrafilters are there? [i]Proposed by Evan Chen[/i]

For every $n$ positive integers we denote $$\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}$$ where $x_n, y_n$ are coprime positive integers. Prove that $y_n$ is not divisible by $2^n$ for any positive integers $n$. Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

Let for natural numbers $a,b,c$ and any natural $n$ we have that $(abc)^n$ divides $ ((a^n-1)(b^n-1)(c^n-1)+1)^3$. Prove that then $a=b=c$.

A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$

Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]

Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$

What is the number of ways the numbers from $1$ to $14$ can be split into $7$ pairs such that for each pair, the greater number is at least $2$ times the smaller number? $\textbf{(A) }108\qquad\textbf{(B) }120\qquad\textbf{(C) }126\qquad\textbf{(D) }132\qquad\textbf{(E) }144$

Let $n$ be a positive integer. We call a $n$-tuple $(a_1, . . . , a_n)$ of positive integers [i]nice [/i] if $\bullet$ $gcd (a_1, . . . , a_n) = 1$, and $\bullet$ $a_i|a_{i-1} + a_{i+1}$, for all $i = 1, . . . , n$ (we define $a_0 = a_n$ and $a_{n+1} = a1$ here). Find the maximal possible value of the sum $a_1 +...+ a_n$ if $(a_1, . . . , a_n)$ is a nice $n$-tuple.

Cassandra sets her watch to the correct time at noon. At the actual time of $ \text{1: \!\,00 PM}$, she notices that her watch reads $ \text{12: \!\,57}$ and $ 36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads $ \text{10: \!\,00 PM}$? $ \textbf{(A)}\ \text{10: \!\,22 PM and 24 seconds} \qquad\textbf{(B)}\ \text{10: \!\,24 PM}\qquad\textbf{(C)}\ \text{10: \!\,25 PM}$ $ \textbf{(D)}\ \text{10: \!\,27 PM}\qquad\textbf{(E)}\ \text{10: \!\,30 PM}$

Martin invented the following algorithm. Let two irreducible fractions $ \frac{s_1}{t_1}$ and $ \frac{s_2}{t_2}$ be given as inputs, with the numerators and denominators being positive integers. Divide $ s_1$ and $ s_2$ by their greatest common divisor $ c$ and obtain $ a_1$ and $ a_2$, respectively. Similarily, divide $ t_1$ and $ t_2$ by their greatest common divisor $ d$ and obtain $ b_1$ and $ b_2$, respectively. After that, form a new fraction $ \frac{a_1b_2 \plus{} a_2b_1}{t_1b_2}$, reduce it, and multiply the numerator of the result by $ c$. Martin claims that this algorithm always finds the sum of the original fractions as an irreducible fraction. Is his claim correct?

From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.