Let $a, b, c$ be non-zero real numbers such that $a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}.$ Prove that at least two of $a, b, c$ are equal.

Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.

[b]p1.[/b] For what integers $n$ is $2^6 + 2^9 + 2^n$ the square of an integer? [b]p2.[/b] Two integers are chosen at random (independently, with repetition allowed) from the set $\{1,2,3,...,N\}$. Show that the probability that the sum of the two integers is even is not less than the probability that the sum is odd. [b]p3.[/b] Let $X$ be a point in the second quadrant of the plane and let $Y$ be a point in the first quadrant. Locate the point $M$ on the $x$-axis such that the angle $XM$ makes with the negative end of the $x$-axis is twice the angle $YM$ makes with the positive end of the $x$-axis. [b]p4.[/b] Let $a,b$ be positive integers such that $a \ge b \sqrt3$. Let $\alpha^n = (a + b\sqrt3)^n = a_n + b_n\sqrt3$ for $n = 1,2,3,...$. i. Prove that $\lim_{n \to + \infty} \frac{a_n}{b_n}$ exists. ii. Evaluate this limit. [b]p5.[/b] Suppose $m$ and $n$ are the hypotenuses are of Pythagorean triangles, i.e,, there are positive integers $a,b,c,d$, so that $m^2 = a^2 + b^2$ and $n^2= c^2 + d^2$. Show than $mn$ is the hypotenuse of at least two distinct Pythagorean triangles. Hint: you may not assume that the pair $(a,b)$ is different from the pair $(c,d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.

Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that: $$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$ for all reals $x, y $.

Defined all $f : \mathbb{R} \to \mathbb{R} $ that satisfied equation $$f(x)f(y)f(x-y)=x^2f(y)-y^2f(x)$$ for all $x,y \in \mathbb{R}$

Given are three numbers $a, b, c \in R-\{0\}$. The parabola with equation $y = ax^2+bx+c$ lies above the line $y = cx$. Prove that the parabola with equation $y = cx^2 - bx + a$ lies above the line $y = cx - b$.

Compute the sum of all positive integers $n$ for which the expression \[\frac{n+7}{\sqrt{n-1}}\] is an integer.

Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?

$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.

Find all polynomials $ p(x)$ satisfying the equation: $ (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x)$ for all $ x$.

Prove that for any positive integer $n\ge 2$ the following inequality holds: $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}>\frac{1}{2}$$

Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$

The sequence of real numbers $(a_n)_{n\geq 0}$ is such that $a_0 = 1$, $a_1 = a > 2$ and $\displaystyle a_{n+1} = \left(\left(\frac{a_n}{a_{n-1}}\right)^2 -2\right)a_n$ for every positive integer $n$. Prove that $\displaystyle \sum_{i=0}^k \frac{1}{a_i} < \frac{2+a-\sqrt{a^2-4}}{2}$ for every positive integer $k$.

If $ x\neq1$, $ y\neq1$, $ x\neq y$ and \[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\] show that both fractions are equal to $ x\plus{}y\plus{}z$.

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial?

Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.

Prove that if $ a \geq 0 $ and $ b \geq 0 $, then $$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.

[b]p1.[/b] A rectangular pool has diagonal $17$ units and area $120$ units$^2$. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs $5$ units/sec faster than Joey, how long does it take for her to catch him? [b]p2. [/b] Alice plays a game with her standard deck of $52$ cards. She gives all of the cards number values where Aces are $1$’s, royal cards are $10$’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value $v$, she places $12 - v$ cards on top of the flipped card. For example: if she flips the $3$ of diamonds then she places $9$ cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is $4$ and there are $5$ piles, what is the sum of the flipped over cards? [b]p3.[/b] There are $5$ people standing at $(0, 0)$, $(3, 0)$, $(0, 3)$, $(-3, 0)$, and $(-3, 0)$ on a coordinate grid at a time $t = 0$ seconds. Each second, every person on the grid moves exactly $1$ unit up, down, left, or right. The person at the origin is infected with covid-$19$, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as $(0, 1.5)$, they will not infect each other.) What is the maximum possible number of infected people after $t = 7$ seconds? [b]p4.[/b] Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is $2$ mm. If diamonds cost $\$100/ mm ^2$ and gold costs $\$50 /mm ^2$ , what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ x$,$ y$, and $ z$ be three positive real numbers whose sum is $ 1$. If no one of these numbers is more than twice any other, then the minimum possible value of the product $ xyz$ is $ \textbf{(A)}\ \frac{1}{32}\qquad \textbf{(B)}\ \frac{1}{36}\qquad \textbf{(C)}\ \frac{4}{125}\qquad \textbf{(D)}\ \frac{1}{127}\qquad \textbf{(E)}\ \text{none of these}$

Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then, $$S = \sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$ Find $\lfloor 80S \rfloor.$ (Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)

The real numbers $a_1, a_2, a_3$ are greater than $1$ and have the sum equal to $S$. If for any $i = 1, 2, 3$, holds the inequality $\frac{a_i^2}{a_i-1}>S$ , prove the inequality $$\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1$$