The sum of sixth powers of six integers minus $1$ is six times greater than the product of these six integers. Prove that one of them is $1$ or $-1$ and all others are $0$s. (LD Kurliandchik)

Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$

Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$. [i]Mircea Fianu[/i]

$A, B$, 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 $\sin(\angle CBP)$?

Circles $\text{A}$ and $\text{B}$ are concentric, with the radius of $\text{A}$ being $\sqrt{17}$ times the radius of $B$. The largest line segment that can be draw in the region bounded by the two circles has length $32$. Compute the radius of circle $B$. [center][img]https://cdn.artofproblemsolving.com/attachments/7/4/6bc4aed9842cdfbeb95853d508a22b61a10c9c.png[/img][/center]

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$, respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$. The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$, respectively (where $E$ and $F$ are both different from $D$). Express the length of the segment $EF$ in terms of $a$ and $b$.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Given a sequence $a_n$: \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots \] (one '1', two '2' and so on) and another sequence $b_n$ such that $a_{b_n}=b_{a_n}$ for all positive integers $n$. It is known that $b_k=1$ for some $k>100$. Prove that $b_m=1$ for all $m>k$.

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Given an even positive integer $m$, find all positive integers $n$ for which there exists a bijection $f:[n]\to [n]$ so that, for all $x,y\in [n]$ for which $n\mid mx-y$, $$(n+1)\mid f(x)^m-f(y).$$ Note: For a positive integer $n$, we let $[n] = \{1,2,\dots, n\}$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.

Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$. (A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$. An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$. The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

$K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \subseteq [0, 1]$. $\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\int_{0}^{1} g(y) K(x, y) dy = f(x)$ for all $x \subseteq [0, 1]$. Show that $f = g$ on $[0, 1]$.

Let $\theta$ be an obtuse angle with $\sin{\theta}=\frac{3}{5}$. If an ant starts at the origin and repeatedly moves $1$ unit and turns by an angle of $\theta$, there exists a region $R$ in the plane such that for every point $P\in R$ and every constant $c>0$, the ant is within a distance $c$ of $P$ at some point in time (so the ant gets arbitrarily close to every point in the set). What is the largest possible area of $R$? [i]2020 CCA Math Bonanza Individual Round #15[/i]

Let $n\geqslant 2$ be an integer. Determine all complex numbers $z{}$ which satisfy \[|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.\]

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

The two roots of the equation $ a(b\minus{}c)x^2\plus{}b(c\minus{}a)x\plus{}c(a\minus{}b)\equal{}0$ are $ 1$ and: $ \textbf{(A)}\ \frac{b(c\minus{}a)}{a(b\minus{}c)} \qquad \textbf{(B)}\ \frac{a(b\minus{}c)}{c(a\minus{}b)} \qquad \textbf{(C)}\ \frac{a(b\minus{}c)}{b(c\minus{}a)} \qquad \textbf{(D)}\ \frac{c(a\minus{}b)}{a(b\minus{}c)} \qquad \textbf{(E)}\ \frac{c(a\minus{}b)}{b(c\minus{}a)}$

$ \mathcal F$ is a family of 3-subsets of set $ X$. Every two distinct elements of $ X$ are exactly in $ k$ elements of $ \mathcal F$. It is known that there is a partition of $ \mathcal F$ to sets $ X_1,X_2$ such that each element of $ \mathcal F$ has non-empty intersection with both $ X_1,X_2$. Prove that $ |X|\leq4$.

Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose that for every $\varepsilon >0$ , there exists a function $g:\mathbb{R}\to (0,\infty)$ such that for every pair $(x,y)$ of real numbers, if $|x-y|<\text{min}\{g(x),g(y)\}$, then $|f(x)-f(y)|<\varepsilon$ Prove that $f$ is pointwise limit of a squence of continuous $\mathbb{R}\to \mathbb{R}$ functions i.e., there is a squence $h_1,h_2,...,$ of continuous $\mathbb{R}\to \mathbb{R}$ such that $\lim_{n\to \infty}h_n(x)=f(x)$ for every $x\in \mathbb{R}$

Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$. (i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$. (ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.

During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as \[\dfrac{a-b\sqrt c}d,\] where all four variables are positive integers, $c$ is a multple of no perfect square greater than $1$, $a$ is coprime with $d$, and $b$ is coprime with $d$. Find the value of $a+b+c+d$.

Baron Munchausen claims that he has drawn a polygon and chosen a point inside the polygon in such a way that any line passing through the chosen point divides the polygon into three polygons. Could the Baron’s claim be correct?

Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.

$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.