Found problems: 6530
2018 Harvard-MIT Mathematics Tournament, 9
Assume the quartic $x^4-ax^3+bx^2-ax+d=0$ has four real roots $\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2.$ Find the maximum possible value of $\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.$
1998 Balkan MO, 2
Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality
\[ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}. \]
[i]Bogdan Enescu, Romania[/i]
2005 Croatia National Olympiad, 2
Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.
2018 All-Russian Olympiad, 3
Suppose that $ a_1,\cdots , a_{25}$ are non-negative integers, and $ k$ is the smallest of them. Prove that
$$\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big].$$
(As usual, $[x]$ denotes the integer part of the number $x$ , that is, the largest integer not exceeding $x$.)
2017 Azerbaijan JBMO TST, 1
Let $x,y,z,t$ be positive numbers.Prove that
$\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
1983 Poland - Second Round, 2
There are three non-negative numbers $ a, b, c $ such that the sum of each two is not less than the remaining one. Prove that $$
\sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}.$$
2014 USAMO, 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
II Soros Olympiad 1995 - 96 (Russia), 9.6
Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0
\\ f(x)-f(y)\ge 0
\end{cases}$$
2019 EGMO, 5
Let $n\ge 2$ be an integer, and let $a_1, a_2, \cdots , a_n$ be positive integers. Show that there exist positive integers $b_1, b_2, \cdots, b_n$ satisfying the following three conditions:
$\text{(A)} \ a_i\le b_i$ for $i=1, 2, \cdots , n;$
$\text{(B)} \ $ the remainders of $b_1, b_2, \cdots, b_n$ on division by $n$ are pairwise different; and
$\text{(C)} \ $ $b_1+b_2+\cdots b_n \le n\left(\frac{n-1}{2}+\left\lfloor \frac{a_1+a_2+\cdots a_n}{n}\right \rfloor \right)$
(Here, $\lfloor x \rfloor$ denotes the integer part of real number $x$, that is, the largest integer that does not exceed $x$.)
2005 MOP Homework, 2
Find all real numbers $x$ such that
$\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$.
(For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)
2010 Today's Calculation Of Integral, 581
For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.
1967 Dutch Mathematical Olympiad, 4
The following applies: $$a, b, c, d > 0 , a + b < c + d$$
Prove that $$ac + bd > ab.$$
2007 Pre-Preparation Course Examination, 7
Let $p$ be a prime such that $p \equiv 3 \pmod 4$. Prove that we can't partition the numbers $a,a+1,a+2,\cdots,a+p-2$,($a \in \mathbb Z$) in two sets such that product of members of the sets be equal.
2008 IberoAmerican Olympiad For University Students, 5
Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy
\[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]
2014 Turkey Team Selection Test, 2
A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.
2019 JBMO Shortlist, A4
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
1960 Polish MO Finals, 1
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.
2018 Flanders Math Olympiad, 2
Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.
2025 Austrian MO National Competition, 1
Let $a$, $b$ and $c$ be pairwise distinct nonnegative real numbers. Prove that
\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\]
[i](Karl Czakler)[/i]
2009 Princeton University Math Competition, 7
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
1994 China National Olympiad, 3
Find all functions $f:[1,\infty )\rightarrow [1,\infty)$ satisfying the following conditions:
(1) $f(x)\le 2(x+1)$;
(2) $f(x+1)=\dfrac{1}{x}[(f(x))^2-1]$ .
1979 Romania Team Selection Tests, 3.
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations:
\[
\begin{cases}
x-\lambda y=a,\\
y-\lambda z=b,\\
z-\lambda x=c.
\end{cases}
\]
we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$.
[i]Radu Gologan[/i]
2015 India Regional MathematicaI Olympiad, 7
Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$. Prove that
\[ (1+x)(1+y)(1+z)\le 4+4xyz. \]
2009 Jozsef Wildt International Math Competition, W. 22
If $a_i >0$ ($i=1, 2, \cdots , n$), then $$\left (\frac{a_1}{a_2} \right )^k + \left (\frac{a_2}{a_3} \right )^k + \cdots + \left (\frac{a_n}{a_1} \right )^k \geq \frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots + \frac{a_n}{a_1}$$ for all $k\in \mathbb{N}$