Let \(n\) be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

Let $ABCD$ be a convex quadrilateral such that $m \left (\widehat{ADB} \right)=15^{\circ}$, $m \left (\widehat{BCD} \right)=90^{\circ}$. The diagonals of quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC|=4, |EA|=8$ and $|EP|=2$. What is $m \left (\widehat{PBD} \right)$? $ \textbf{(A)}\ 15^{\circ} \qquad\textbf{(B)}\ 30^{\circ} \qquad\textbf{(C)}\ 45^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 75^{\circ} $

Let $m,n$ are natural numbers such that $GCD(m,n)=1$.Find all triplets $(x,y,n)$ which sastify $(x^2+y^2)^m=(xy)^n$

a) Prove that center of smallest sphere containing a finite subset of $\mathbb R^{n}$ is inside convex hull of the point that lie on sphere. b) $A$ is a finite subset of $\mathbb R^{n}$, and distance of every two points of $A$ is not larger than 1. Find radius of the largest sphere containing $A$.

Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.

For an integer $n > 0$, let $p(n)$ be the product of the digits of $n$. Compute the sum of all integers $n$ such that $n - p(n) = 52$.

In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$

In a triangle $ABC, P$ and $P'$ are points on side $BC, Q$ on side $CA$, and $R $ on side $AB$, such that $\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP'}{P'B}$ . Let $G$ be the centroid of triangle $ABC$ and $K$ be the intersection point of $AP'$ and $RQ$. Prove that points $P,G,K$ are collinear.

Given a family $C$ of circles of the same radius $R$, which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to $R\sqrt3$ .

Let $a,b,c$ be positive reals s.t. $a^2+b^2+c^2=1$. Prove the following inequality \[ \sum \frac{a}{a^3+bc} >3 . \] [i]Proposed by A. Khrabrov[/i]

An ant starts at the point $(0,0)$ in the coordinate plane. It can make moves from lattice point $(x_1,y_1)$ to lattice point $(x_2,y_2)$ whenever $x_2\geq x_1$, $y_2\geq y_1$, and $(x_1,y_1)\neq(x_2,y_2)$. For all nonnegative integers $m,n$, define $a_{m,n}$ to be the number of possible sequences of moves from $(0,0)$ to $(m,n)$ (e.g. $a_{0,0}=1$ and $a_{1,1}=3$). Compute \[ \sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{a_{m,n}}{10^{m+n}}. \] [i]2020 CCA Math Bonanza Individual Round #14[/i]

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? ${ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$

Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE \equal{} 5$. What is $ CD$? [asy]unitsize(2.5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair A=(0,0), Ep=(5,0), B=(5+40/3,0); pair M=midpoint(A--Ep); pair C=intersectionpoints(Circle(M,2.5),Circle(A,3))[1]; pair D=B+8*dir(180+degrees(C)); dot(A); dot(C); dot(B); dot(D); draw(C--D); draw(A--B); draw(Circle(A,3)); draw(Circle(B,8)); label("$A$",A,W); label("$B$",B,E); label("$C$",C,SE); label("$E$",Ep,SSE); label("$D$",D,NW);[/asy]$ \textbf{(A) } 13\qquad \textbf{(B) } \frac {44}{3}\qquad \textbf{(C) } \sqrt {221}\qquad \textbf{(D) } \sqrt {255}\qquad \textbf{(E) } \frac {55}{3}$

Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas. [i]Proposed by Tran Quang Hung[/i]

Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent: $ \text{(i)} P\circ Q $ is divisible by $ P. $ $ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $ [i]Marcel Țena[/i]

For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

The natural numbers $0, 1, 2, 3, . . .$ are written on the square table $2015\times 2015$ in a circular order (anti-clockwise) such that $0$ is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number $2015$ is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number $0$ in the center by $14$, after that, each time, we can add $1$ to each of $12$ numbers on $12$ consecutive unit squares in a row, or $12$ consecutive unit squares in a column, or $12$ unit squares in a rectangle $3\times 4$. After a finite number of steps, can we make all numbers on the table are multiples of $2016$? [img]https://cdn.artofproblemsolving.com/attachments/c/d/223b32c0e3f58f62d0d40fa78c09a2cd035ed5.png[/img]

Prove that for every positive integer $m$ there exists a positive integer $n_m$ such that for every positive integer $n \ge n_m$, there exist positive integers $a_1, a_2, \ldots, a_n$ such that $$\frac{1}{a_1^m}+\frac{1}{a_2^m}+\ldots+\frac{1}{a_n^m}=1.$$

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$? [i]Proposed by Morteza Saghafian[/i]

Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

A car goes along a straight highway at the speed of $60$ km per hour. A $100$ meter long fence is standing parallel to the highway. Every second, the passenger of the car measures the angle of vision of the fence. Prove that the sum of all angles measured by him is less than $1100$ degrees.

Show that the product of three consecutive positive integers is never a perfect square.

Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.

We say two vertices of a simple polygon are [i]visible[/i] from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers $n$ such that there exists a simple polygon with $n$ vertices in which every vertex is visible from exactly $4$ other vertices. (A simple polygon is a polygon without hole that does not intersect itself.) [i]Proposed by Morteza Saghafian[/i]