Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .

Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.

Let $r$ be the result of doubling both the base and exponent of $a^b$, $b\neq 0$. If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals: ${{ \textbf{(A)} a\qquad\textbf{(B)}\ 2a \qquad\textbf{(C)}\ 4a \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 4} $

Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probablility that the selected number is prime? $\textbf{(A) } \dfrac{8}{99} \qquad\textbf{(B) } \dfrac{2}{5} \qquad\textbf{(C) } \dfrac{9}{20} \qquad\textbf{(D) } \dfrac{1}{2} \qquad\textbf{(E) } \dfrac{9}{16} $

A $6$-inch-wide rectangle is rotated $90$ degrees about one of its corners, sweeping out an area of $45\pi$ square inches, excluding the area enclosed by the rectangle in its starting position. Find the rectangle’s length in inches.

Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \[2^{s(n)} = s(n^2).\]

In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly $100$ roads. We call $10$ roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.

Find the remainder when $2024^{2023^{2022^{2021...^{3^{2}}}}} + 2025^{2021^{2017^{2013...^{5^{1}}}}}$ is divided by $19$.

An $n\times n$ table whose cells are numerated with numbers $1, 2,\cdots, n^2$ in some order is called [i]Naissus[/i] if all products of $n$ numbers written in $n$ [i]scattered[/i] cells give the same residue when divided by $n^2+1$. Does there exist a Naissus table for $(a) n = 8;$ $(b) n = 10?$ ($n$ cells are [i]scattered[/i] if no two are in the same row or column.) [i]Proposed by Marko Djikic[/i]

Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

Solve the equation $x^3 - [x] = 3$.

There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$

The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.

Assume that the equality $2BC=AB+AC$ holds in $\triangle ABC$. Prove that: (a) The vertex $A$, the midpoints $M$ and $N$ of $AB$ and $AC$ respectively, the incenter $I$, and the circumcenter $O$ belong to a circle $k$. (b) The line $GI$, where $G$ is the centroid of $\triangle ABC$ is a tangent to $k$.

Emily is at $(0,0)$, chilling, when she sees a spider located at $(1,0)$! Emily runs a continuous path to her home, located at $(\sqrt{2}+2,0)$, such that she is always moving away from the spider and toward her home. That is, her distance from the spider always increases whereas her distance to her home always decreases. What is the area of the set of all points that Emily could have visited on her run home? [i]Proposed by Thomas Lam[/i]

A particle moves along the $x$-axis. It collides elastically head-on with an identical particle initially at rest. Which of the following graphs could illustrate the momentum of each particle as a function of time? [asy] size(400); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(0,5)); draw((0,1.5)--(5,1.5)); label("$p$",(0,5),N); label("$t$",(5,1.5),E); label("$\mathbf{(A)}$",(2.5,-0.5)); draw((0,1.5)--(2.5,1.5)--(2.5,0.75)--(4,0.75),black+linewidth(2)); draw((0,3.5)--(2.5,3.5)--(2.5,4.25)--(4,4.25),black+linewidth(2)); draw((8,0)--(8,5)); draw((8,1.5)--(13,1.5)); label("$p$",(8,5),N); label("$t$",(13,1.5),E); label("$\mathbf{(B)}$",(10.5,-0.5)); draw((8,1.5)--(10.5,1.5)--(10.5,2.5)--(12,2.5),black+linewidth(2)); draw((8,3.5)--(10.5,3.5)--(10.5,4.5)--(12,4.5),black+linewidth(2)); draw((16,0)--(16,5)); draw((16,1.5)--(21,1.5)); label("$p$",(16,5),N); label("$t$",(21,1.5),E); label("$\mathbf{(C)}$",(18.5,-0.5)); draw((16,1.5)--(18.5,1.5)--(18.5,2.25)--(20,2.25),black+linewidth(2)); draw((16,3.5)--(18.5,3.5)--(18.5,2.75)--(20,2.75),black+linewidth(2)); draw((24,0)--(24,5)); draw((24,1.5)--(29,1.5)); label("$p$",(24,5),N); label("$t$",(29,1.5),E); label("$\mathbf{(D)}$",(26.5,-0.5)); draw((24,1.5)--(26.5,1.5)--(26.75,3.25)--(28,3.25),black+linewidth(2)); draw((24,3.25)--(26.5,3.25)--(26.75,1.5)--(28,1.5),black+linewidth(2)); draw((32,0)--(32,5)); draw((32,1.5)--(37,1.5)); label("$p$",(32,5),N); label("$t$",(37,1.5),E); label("$\mathbf{(E)}$",(34.5,-0.5)); draw((32,1.5)--(34.5,1.5)--(34.5,0.5)--(36,0.5),black+linewidth(2)); draw((32,3.5)--(34.5,3.5)--(34.5,2.75)--(36,2.75),black+linewidth(2)); [/asy]

Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$

Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$

If $ S \equal{} i^n \plus{} i^{\minus{}n}$, where $ i \equal{} \sqrt{\minus{}1}$ and $ n$ is an integer, then the total number of possible distinct values for $ S$ is: $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{more than 4}$

$n$ people are in the plane, so that the closest person is unique and each one shoot this closest person with a squirt gun. If $n$ is odd, prove that there exists at least one person that nobody shot. If $n$ is even, will there always be a person who escape? Justify that.

Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.

Find the smallest positive integer $k$ such that \[(16a^2 + 36b^2 + 81c^2)(81a^2 + 36b^2 + 16c^2) < k(a^2 + b^2 + c^2)^2,\] for some ordered triple of positive integers $(a,b,c)$.

In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.

Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$