Found problems: 15925
2018 Thailand Mathematical Olympiad, 4
Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of
$\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$
.
1997 Singapore Senior Math Olympiad, 1
Let $x_1,x_2,x_3,x_4, x_5,x_6$ be positive real numbers. Show that
$$\left( \frac{x_2}{x_1} \right)^5+\left( \frac{x_4}{x_2} \right)^5+\left( \frac{x_6}{x_3} \right)^5+\left( \frac{x_1}{x_4} \right)^5+\left( \frac{x_3}{x_5} \right)^5+\left( \frac{x_5}{x_6} \right)^5 \ge \frac{x_1}{x_2}+\frac{x_2}{x_4}+\frac{x_3}{x_6}+\frac{x_4}{x_1}+\frac{x_5}{x_3}+\frac{x_6}{x_5}$$
2012 National Olympiad First Round, 19
What is the sum of real roots of the equation $x^4-7x^3+14x^2-14x+4=0$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2024 Ukraine National Mathematical Olympiad, Problem 3
$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum?
[i]Proposed by Oleksiy Masalitin[/i]
2024 Canadian Mathematical Olympiad Qualification, 6
For certain real constants $ p, q, r$, we are given a system of equations
$$\begin{cases} a^2 + b + c = p \\
a + b^2 + c = q \\
a + b + c^2 = r \end{cases}$$
What is the maximum number of solutions of real triplets $(a, b, c)$ across all possible $p, q, r$? Give an example of the $p$, $q$, $r$ that achieves this maximum.
1997 Baltic Way, 1
Determine all functions $f$ from the real numbers to the real numbers, different from the zero function, such that $f(x)f(y)=f(x-y)$ for all real numbers $x$ and $y$.
2024 Indonesia TST, A
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
2016 Irish Math Olympiad, 3
Do there exist four polynomials $P_1(x), P_2(x), P_3(x), P_4(x)$ with real coefficients, such that the sum of any three of them always has a real root, but the sum of any two of them has no real root?
1996 India National Olympiad, 3
Solve the following system for real $a , b, c, d, e$: \[ \left\{ \begin{array}{ccc} 3a & = & ( b + c+ d)^3 \\ 3b & = & ( c + d +e ) ^3 \\ 3c & = & ( d + e +a )^3 \\ 3d & = & ( e + a +b )^3 \\ 3e &=& ( a + b +c)^3. \end{array}\right. \]
2018 China Team Selection Test, 6
Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .
2019 Polish Junior MO Second Round, 1.
Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.
2017 Pan-African Shortlist, A?
We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$
We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$.
Prove that for every $n>0$, $y_n$ is the square of an odd integer
2021 Chile National Olympiad, 3
Find all polynomials $p(x)$ with real coefficients that satisfy $$4p(x^2) = 4(p(x))^2 + 4p(x)- 1$$
2017-2018 SDPC, 5
Given positive real numbers $a,b,c$ such that $abc=1$, find the maximum possible value of $$\frac{1}{(4a+4b+c)^3}+\frac{1}{(4b+4c+a)^3}+\frac{1}{(4c+4a+b)^3}.$$
I Soros Olympiad 1994-95 (Rus + Ukr), 10.8
Find all $x$ for which the inequality holds
$$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$
1987 IMO Longlists, 74
Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i]
[i]Proposed by Vietnam.[/i]
2014 Kosovo National Mathematical Olympiad, 1
Let $a$ and $b$ be the solutions to $x^2-x+q=0$, find
$a^3+b^3+3(a^3b+ab^3)+6(a^3b^2+a^2b^3)$.
2016 IMO Shortlist, A4
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$
2023 Iran Team Selection Test, 5
Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root.
[i]Proposed by Hesam Rajabzadeh [/i]
2014 Contests, 1
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$
1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .
2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$
3)Describe the possible values of $a_{1435}$
4)Prove that the values that you got in (3) are correct
2003 Silk Road, 3
Let $0<a<b<1$ be reals numbers and
\[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\]
Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$.
($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$)
Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]
2024 Taiwan TST Round 2, 2
Let $n$ be a positive integer. Prove that the inequality
\[n \sum_{i=1}^n \sum_{j = 1}^n \sum_{k=1}^n \frac{3}{a_ja_k + a_ka_i + a_i a_j} \ge \left(\sum_{j=1}^n \sum_{k=1}^n \frac{2}{a_j + a_k}\right)^2 \]
holds for any positive real numbers $a_1$, $a_2$, $\dots$, $a_n$.
[i]Proposed by Li4 and Ming Hsiao.[/i]
1988 IMO Longlists, 77
A function $ f$ defined on the positive integers (and taking positive integers values) is given by:
$ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\
f(2 \cdot n) \equal{} f(n) \\
f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\
f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$
for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$
1987 Romania Team Selection Test, 11
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
2002 Federal Competition For Advanced Students, Part 2, 1
Find all polynomials $P(x)$ of the smallest possible degree with the following properties:
(i) The leading coefficient is $200$;
(ii) The coefficient at the smallest non-vanishing power is $2$;
(iii) The sum of all the coefficients is $4$;
(iv) $P(-1) = 0, P(2) = 6, P(3) = 8$.