Found problems: 6530
1999 Bundeswettbewerb Mathematik, 2
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$.
Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.
2015 Thailand TSTST, 2
Let $a, b, c \geq 1$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\geq\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca}.$$
MathLinks Contest 7th, 1.3
We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least
\[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}.
\]
2024-IMOC, A2
Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of
\[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\]
[i]Proposed by snap7822[/i]
2011 Kosovo National Mathematical Olympiad, 3
If $a,b,c$ are real positive numbers prove that the inequality holds:
\[ \frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}} \]
2021 Nigerian Senior MO Round 2, 4
let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$
1988 China National Olympiad, 1
Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality
\[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\]
holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.
2024 Irish Math Olympiad, P8
Let $a,b,c$ be positive real numbers with $a \leq c$ and $b \leq c$. Prove that $$ (a +10b)(b +22c)(c +7a) \geq 2024
abc.$$
2022 Kyiv City MO Round 1, Problem 4
For any nonnegative reals $x, y$ show the inequality $$x^2y^2 + x^2y + xy^2 \le x^4y + x + y^4$$.
2017 Princeton University Math Competition, 4
Ayase chooses three numbers $a, b, c$ independently and uniformly from the interval $[-1, 1]$.
The probability that $0 < a + b < a < a + b + c$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
2002 Canada National Olympiad, 3
Prove that for all positive real numbers $a$, $b$, and $c$,
\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]
and determine when equality occurs.
2009 Grand Duchy of Lithuania, 2
Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $f(x)$ has three real positive roots and that $f(0) < 0$, prove that $2b^3+ 9a^2 d - 7abc \le 0$.
2001 Vietnam Team Selection Test, 1
Let’s consider the real numbers $a, b, c$ satisfying the condition
\[21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.\]
Find the minimal value of the expression
\[P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.\]
2006 Moldova National Olympiad, 9.1
Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression:
$$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$
2020 Azerbaijan IMO TST, 1
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
2011 N.N. Mihăileanu Individual, 2
Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and
$$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$
[i]Gabriela Constantinescu[/i]
2010 Romanian Master of Mathematics, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
2009 Turkey MO (2nd round), 2
Show that
\[ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 \]
for all positive real numbers $a, \: b , \: c.$
2017 Iran MO (2nd Round), 4
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that
$$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$
1996 Bosnia and Herzegovina Team Selection Test, 1
$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$
$b)$ Does inequality $a)$ holds for
$1)$ arbitrary real numbers $a$, $b$ and $c$
$2)$ any integer $m$
1981 Bundeswettbewerb Mathematik, 4
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
2014 Korea Junior Math Olympiad, 6
Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$
2002 Romania National Olympiad, 2
Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that:
\[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\]
Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that:
\[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]
2019 Teodor Topan, 4
Let be an odd natural number $ n, $ and $ n $ real numbers $ y_1\le y_2\le\cdots\le y_n $ whose sum is $ 0. $ Prove that
$$ (n+2)y_{\frac{n+1}{2}}^2\le y_1^2+y_2^2+\cdots +y_n^2, $$
and specify where equality is attained.
[i]Nicolae Bourbăcuț[/i]
2007 Cuba MO, 9
Let $O$ be the circumcircle of $\triangle ABC$, with $AC=BC$ end let $D=AO\cap BC$. If $BD$ and $CD$ are integer numbers and $AO-CD$ is prime, determine such three numbers.