Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

You are at one vertex of a equilateral triangle with side length $ 1 $. All of the edges of the equilateral triangle will reflect the laser beam perfectly (angle of incidence is equal to angle of reflection). Given that the laser beam bounces off exactly $ 137 $ edges and returns to the original vertex without touching any other vertices, let $ M $ be the maximum possible distance the beam could have traveled, and $ m $ be the minimum possible distance the beam could have traveled. Find $ M^2 - m^2 $.

Let $ABCD$ be a parallelogram and $P$ a point in the plane. The line $BP$ intersects the circumcircle of $ABC$ again at $X$ and the line $DP$ intersects the circumcircle of $DAC$ again at $Y$. Let $M$ be the midpoint of the side $AC$. The point $N$ lies on the circumcircle of $PXY$ so that $MN$ is a tangent to this circle. Prove that the segments $MN$ and $AM$ have the same length. [i]Proposed by David Anghel[/i]

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$

In a right triangle $ABC$, $K$ is the midpoint of the hypotenuse $AB$ and $M$ such a point on the $BC$ that $| B M | = 2 | MC |$. Prove that $\angle MAB = \angle MKC$.

The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$

We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof. The coordinates of all vertices of a given rectangle are Fibonacci numbers. Suppose that the rectangle is not such that one of its vertices is on the $x$-axis and another on the $y$-axis. Prove that either the sides of the rectangle are parallel to the axes, or make an angle of $45^{\circ}$ with the axes.

Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.

Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^{o}$.

Find all pairs of real numbers $(x,y)$ satisfying both equations \[ 3x^2+3xy+2y^2 =2 \] \[ x^2+2xy+2y^2 =1. \] [i]2020 CCA Math Bonanza Team Round #5[/i]

Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that: \[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]

In triangle $ ABC$, $ AB \equal{} 10$, $ BC \equal{} 14$, and $ CA \equal{} 16$. Let $ D$ be a point in the interior of $ \overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \triangle BPC$ can be expressed in the form $ a\minus{}b\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a\plus{}b\plus{}c$.

If $ab(a+b)$ divides $a^2 + ab+ b^2$ for different integers $a$ and $b$, prove that \[|a-b|>\sqrt[3]{ab}.\]

Given $8$ coins, at most one of them is counterfeit. A counterfeit coin is lighter than a real coin. You have a free weight balance. What is the minimum number of weighings necessary to determine the identity of the counterfeit coin if it exists

Pete claims that he can draw $4$ segments of length $1$ and a circle of radius less than $\sqrt3 /3 $ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $4$ segments. Is Pete right?

A permutation f of the set of integers is called bounded if | x - f (x) | is bounded. Bounded permutations with permutation multiplication form a group W. Show that the additive group of rational numbers is not isomorphic to any subgroup of W.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a bounded function in some neighbourhood of $ 0, $ such that there are three real numbers $ a>0, b>1, c $ with the property that $$ f(ax)=bf(x)+c,\quad\forall x\in\mathbb{R} . $$ Show that $ f $ is continuous at $ 0 $ if and only if $ c=0. $

In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5,$ point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=3,$ $AB=8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

$a$ is irrational , but $a$ and $a^3-6a$ are roots of square polynomial with integer coefficients.Find $a$

Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 24$

One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth. Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid? Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either: $x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.

The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.

Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

Let $a_1, a_2, ..., a_{n-1}$ be $n - 1$ consecutive positive integers in increasing order such that $k$ ${n \choose k}$ $\equiv 0$ (mod $a_k$), for all $k \in \{1, 2, ... , n - 1\}$. Find the possible values of $a_1$.