Prove that all real roots of the polynomial $$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$ are positive.

Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(xf(x)+y^2) = x^2+yf(y) \] for any positive reals $x,y$.

The price of a gold ring in a certain universe is proportional to the square of its purity and the cube of its diameter. The purity is inversely proportional to the square of the depth of the gold mine and directly proportional to the square of the price, while the diameter is determined so that it is proportional to the cube root of the price and also directly proportional to the depth of the mine. How does the price vary solely in terms of the depth of the gold mine?

Let $A\subseteq \mathbb{R}$ such that: i) If $a,b\in A$ then $\sqrt{ab} \in A$ ii) $1\in A$ and $2\in A$ Prove that $\sqrt[\displaystyle 2^{1453}]{2^{1821}}\in A$.

The nonzero real numbers $a, b, c$ satisfy the following system: $$\begin{cases} a+ab=c\\ b+bc=a\\ c+ca=b \end{cases}$$ Find all possible values of the $abc$.

Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.

Let $M$ be the set of positive integers which do not have a prime divisor greater than 3. For any infinite family of subsets of $M$, say $A_1,A_2,\ldots $, prove that there exist $i\ne j$ such that for each $x\in A_i$ there exists some $y\in A_j $ such that $y\mid x$.

Vasya has a numeric expression $$ \Box \cdot \Box +\Box \cdot \Box $$ and 4 cards with numbers that can be put on 4 free places in the expression. Vasya tried to put cards in all possible ways and all the time obtained the same value as a result. Prove that equal numbers are written on three of his cards.

Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.

Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial \[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n. \]

Compare the numbers: $P = 888...888 \times 333 .. 333$ ($2010$ digits of $8$ and $2010$ digits of $3$) and $Q = 444...444\times 666...6667$ ($2010$ digits of $4$ and $2009$ digits of $6$) (A): $P = Q$, (B): $P > Q$, (C): $P < Q$.

Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime? [img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img] p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$ 3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$. ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once. p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$. a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$. b) Is it always true that $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer. p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows: $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of $N$ ?

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

We calculate the sum of $7$ natural consecutive pairs (e.g. $2+4+6+8+10+12+14$) and we will call the result $A$, then the sum of the next $7$ consecutive pairs (in the example, $16+ 18+...$) and its result we will call $B$, and finally we calculate the sum of the following $7$ consecutive pairs and its result we will call $C$. Can the product $ABC$ be $2002^3$?

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Find all positive integers $a, b, c$ such that $3abc+11(a+b+c)=6(ab+bc+ac)+18.$

Compute $$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0}{i + j \choose i} 3^{-(i+j)}.$$

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

Prove that $$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) ... (1 + x^{2^{k-1}} ) = 1 + x + x^2 + x^3 +... + x^{2^{k-1}}$$

At Zoom University, people’s faces appear as circles on a rectangular screen. The radius of one’s face is directly proportional to the square root of the area of the screen it is displayed on. Haydn’s face has a radius of $2$ on a computer screen with area $36$. What is the radius of his face on a $16 \times 9$ computer screen?

Solve the equation $$(2^x-4)^3 +(4^x-2)^3=(4^x+2^x-6)^3$$ where $x$ is a real number.

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $.

Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices. Proposed by Mohammad Ahmadi