Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$

Given positive integers $a \ge 2$ and $k$, let $m_a(k)$ denote the remainder when $k$ is divided by $a$. Compute the number of positive integers, $n$, less than 500 such that $m_2(m_5(m_{11}(n))) = 1$.

Show that the inequality \[3(1 + a^2 + a^4)\geq  (1 + a + a^2)^2\] holds for all real numbers $a.$

Show that there do not exist more than $27$ half-lines (or rays) emanating from the origin in the $3$-dimensional space, such that the angle between each pair of rays is $\geq \frac{\pi}{4}$.

Let $d$ be a positive integer. The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ . Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression. Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.

7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$) 8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probability that $\text{gcd}(n, 2015) = 1$? 9. How many $3$-element subsets of the set $\{1, 2, 3, . . . , 19\}$ have sum of elements divisible by $4$?

Consider the grid of points $X = \{(m,n) | 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $( c = a \pm 2$ and $d = b \pm 1)$ or $( c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:

A circle with diameter $20$ has points $A, B, C, D, E,$ and $F$ equally spaced along its circumference. A second circle is tangent to the lines $AB$ and $AF$ and internally tangent to the circle. If the second circle has diameter $\sqrt{m}+n$ for integers $m$ and $n$, find $m + n.$ [asy] import geometry; size(180); draw(circle((0,0),5)); pair[] p; string[] l={"A","B","C","D","E","F"}; for (int i=0; i<6; ++i){ p.append(new pair[]{dir(i*60+180)*5}); dot(p[i]); label(l[i],p[i],p[i]/3); } draw(p[0]--p[1]^^p[0]--p[5]); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(-60)*90,p[3]--p[3]+(0,-100))}); p.append(new pair[]{intersectionpoint(p[0]--p[0]+dir(+60)*90,p[3]--p[3]+(0,+100))}); draw(incircle(p[0],p[6],p[7]));[/asy]

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function with the property that for any $a,b \in \mathbb{R},$ $a<b,$ there are $c_1,c_2 \in [a,b],$ $c_1 \le c_2$ such that $f(c_1)= \min_{x \in [a,b]} f(x)$ and $f(c_2)= \max_{x \in [a,b]} f(x).$ Prove that $f$ is increasing.

$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive?

Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following: The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation. Lisa repeats this until she gets a number for the first time was on the board. (a) Show that the sequence of moves always ends. (b) If Lisa begins with the number $53^{2022} - 1$, what is the last number on the board? Example: If Lisa starts with the number $2022$, she gets $202 + 4\cdot 2 = 210$ in the first move and overall the result $$2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21$$. Since Lisa gets $21$ for the second time, the turn order ends. [i](Stephan Pfannerer)[/i]

Let $ABCD$ be an inscribed trapezoid such that the sides $[AB]$ and $[CD]$ are parallel. If $m(\widehat{AOD})=60^\circ$ and the altitude of the trapezoid is $10$, what is the area of the trapezoid?

$100$ distinct natural numbers $a_1, a_2, a_3, \ldots, a_{100}$ are written on the board. Then, under each number $a_i$, someone wrote a number $b_i$, such that $b_i$ is the sum of $a_i$ and the greatest common factor of the other $99$ numbers. What is the least possible number of distinct natural numbers that can be among $b_1, b_2, b_3, \ldots, b_{100}$?

Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.) [asy]unitsize(10mm); defaultpen(linewidth(1.2pt)+fontsize(10pt)); dotfactor=4; pair A=(1,0), B=(4.24,0), C=(5.24,3.08), D=(2.62,4.98), E=(0,3.08); dot (A); dot (B); dot (C); dot (D); dot (E); label("$A$",A,S); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$E$",E,W); guide squiggly(path g, real stepsize, real slope=45) { real len = arclength(g); real step = len / round(len / stepsize); guide squig; for (real u = 0; u < len; u += step){ real a = arctime(g, u); real b = arctime(g, u + step / 2); pair p = point(g, a); pair q = point(g, b); pair np = unit( rotate(slope) * dir(g,a)); pair nq = unit( rotate(0 - slope) * dir(g,b)); squig = squig .. p{np} .. q{nq}; } squig = squig .. point(g, length(g)){unit(rotate(slope)*dir(g,length(g)))}; return squig; } pen pp = defaultpen + 2.718; draw(squiggly(A--B, 4.04, 30), pp); draw(squiggly(A--D, 7.777, 20), pp); draw(squiggly(A--E, 5.050, 15), pp); draw(squiggly(B--C, 5.050, 15), pp); draw(squiggly(B--D, 4.04, 20), pp); draw(squiggly(C--D, 2.718, 20), pp); draw(squiggly(D--E, 2.718, -60), pp); [/asy] $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18 $

Let $f(x)$ is an odd function on $R$ , $f(1)=1$ and $f(\frac{x}{x-1})=xf(x)$ $(\forall x<0)$. Find the value of $f(1)f(\frac{1}{100})+f(\frac{1}{2})f(\frac{1}{99})+f(\frac{1}{3})f(\frac{1}{98})+\cdots +f(\frac{1}{50})f(\frac{1}{51}).$

Let $A_1A_2...A_{99}$ be a regular $99-$gon and point $A_{100}$ be its center. find the smallest possible natural number $n$ , such that Parsa can color all segments $A_iA_j$ ( $1 \le i < j \le 100$ ) with one of $n$ colors in such a way that no two homochromatic segments intersect each other or share a vertex. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

For any real value of $ x$ the maximum value of $ 8x \minus{} 3x^2$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac83 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \frac{16}{3}$

Compute all real numbers $a$ such that the polynomial $x^4 + ax^3 + 1$ has exactly one real root.

Two men set out at the same time to walk towards each other from $ M$ and $ N$, $ 72$ miles apart. The first man walks at the rate of $ 4$ mph. The second man walks $ 2$ miles the first hour, $ 2\frac {1}{2}$ miles the second hour, $ 3$ miles the third hour, and so on in arithmetic progression. Then the men will meet: $ \textbf{(A)}\ \text{in 7 hours} \qquad \textbf{(B)}\ \text{in }{8\frac {1}{4}}\text{ hours}\qquad \textbf{(C)}\ \text{nearer }{M}\text{ than }{N}\qquad \\ \textbf{(D)}\ \text{nearer }{N}\text{ than }{M}\qquad \textbf{(E)}\ \text{midway between }{M}\text{ and }{N}$

For each real coefficient polynomial $f(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}$, let $$\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}.$$ Let be given polynomial $P(x)=(x+1)(x+2)\ldots (x+2020).$ Prove that there exists at least $2019$ pairwise distinct polynomials ${{Q}_{k}}(x)$ with $1\le k\le {{2}^{2019}}$ and each of it satisfies two following conditions: i) $\deg {{Q}_{k}}(x)=2020.$ ii) $\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right)$ for all positive initeger $n$.

Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$ $$\sin^2{y} + \cos^2{x} = x^2. $$ [i]A. Khrabov[/i]

Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]