Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.

For integers $n \ge k \ge 0$ we define the [i]bibinomial coefficient[/i] $\left( \binom{n}{k} \right)$ by \[ \left( \binom{n}{k} \right) = \frac{n!!}{k!!(n-k)!!} .\] Determine all pairs $(n,k)$ of integers with $n \ge k \ge 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark: The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.

Let $ ABCD $ be a parallelogram with $ BC = 17 $. Let $ M $ be the midpoint of $ \overline{BC} $ and let $ N $ be the point such that $ DANM $ is a parallelogram. What is the length of segment $ \overline{NC} $?

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $E$. If $AB=BE=5$, $EC=CD=7$, and $BC=11$, compute $AE$.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

Every nonnegative periodic decimal fraction represents a rational number, also in the form $\frac{p}{q}$ can be represented ($p$ and $q$ are natural numbers and coprime, $p\ge 0$, $q > 0)$. Now let $a_1$, $a_2$, $a_3$ and $a_4$ be digits to represent numbers in the decadic system. Let $a_1 \ne a_3$ or $a_2 \ne a_4$.Prove that it for the numbers: $z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...$ $z_2 = 0, \overline{a_4a_1a_2a_3}$ $z_3 = 0, \overline{a_3a_4a_1a_2}$ $z_4 = 0, \overline{a_2a_3a_4a_1}$ In the above representation $p/q$ always have the same denominator. [hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1 $\ne$ a3 oder a2 $\ne$ a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.[/hide]

In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?

Let $ABCD$ be a square of side length $1$, and let $P$ be a variable point on $\overline{CD}$. Denote by $Q$ the intersection point of the angle bisector of $\angle APB$ with $\overline{AB}$. The set of possible locations for $Q$ as $P$ varies along $\overline{CD}$ is a line segment; what is the length of this segment?

Let $a_1,a_2,a_3, \ldots$ be an infinite sequence of nonnegative real numbers such that for all positive integers $k$ the following conditions hold: $i)$ $a_k-2a_{k+1}+a_{k+2} \geq 0$; $ii)$ $\sum_{j=1}^{k} a_j \leq 1$. Prove that for all positive integer $k$ holds: $0 \leq a_k - a_{k+1} < \frac{2}{k^2}$

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}\plus{}4y^{4}\plus{}5z^{3}\minus{}y^{4}z \equiv 0$ is $ p^2$

Consider equilateral triangle $ABC$ with side length $1$. Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$. Compute the sum of all possible values of $\kappa$. [i]2018 CCA Math Bonanza Lightning Round #3.4[/i]

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $. a) Prove that $ I$ lies between $ M $ and $ H $. b) Calculate $ AB ^ 2-AC ^ 2 $. c) Determine $ \dfrac {AB} {AC} $. d) Find the measure of all the sides and angles of the triangle.

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$

Find all triples $(x,y,z)$ of positive integers such that $(x+1)^{y+1}+1=(x+2)^{z+1}$.

let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$. let $CE$ be the perpendicular from $C$ on $AB$ prove that $ CE^2 = AB. CD $

Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$. a) By giving a concrete example, show that such a function exists. b) For each such function define the sum \[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\] Determine all possible values of $S_f$.

Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$ where $0 < nx < \pi/2$, $n \in N$.

Determine the number of pairs of positive integers \( (p, k) \) such that \( p \) is a prime number and \( p^2 + 2^k \) is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.

There are $n$ snails on the plane where the $i$ snail has $a_i$ attack and $d_i$ defense, where $a_i, d_i\in \mathbb{R}$ and each snail has a distinct attack and a distinct defense. We said a 4-tuple of subsets of snails $(S_1, S_2, S_3, S_4)$ is [b]balanced[/b] if $|S_1|+|S_3|$ is either $\lceil n/2\rceil$ or $\lfloor n/2\rfloor$ and there exist real numbers $A, D$ such that \begin{align*} S_1=\{i\ |\ a_i\geq A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_2=\{i\ |\ a_i<A\text{ and } d_i\geq D, 1\leq i\leq n\}\\ S_3=\{i\ |\ a_i< A\text{ and } d_i< D, 1\leq i\leq n\}\\ S_4=\{i\ |\ a_i\geq A\text{ and } d_i< D, 1\leq i\leq n\} \end{align*} Find the largest integer $k$ such that there is always at least $k$ [b]balanced[/b] 4-tuples of subsets. [i]Proposed by redshrimp[/i]

The pieces in a game are squares of side $1$ with their sides colored with $4$ colors: blue, red, yellow and green, so that each piece has one side of each color. There are pieces in every possible color arrangement, and the game has a million pieces of each kind. With the pieces, rectangular puzzles are assembled, without gaps or overlaps, so that two pieces that share a side have that side of the same color. Determine if with this procedure you can make a rectangle of $99\times 2007$ with one side of each color. And $100\times 2008$? And $99\times 2008$?