Found problems: 6530
1998 Korea - Final Round, 1
Let $ x,y,z$ be positive real numbers satisfying $ x\plus{}y\plus{}z\equal{}xyz$. Prove that:
\[\frac1{\sqrt{1+x^2}}+\frac1{\sqrt{1+y^2}}+\frac1{\sqrt{1+z^2}}\leq\frac{3}{2}\]
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
2007 Croatia Team Selection Test, 7
Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]
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 $2b^3+ 9a^2 d - 7abc \le 0$.
2024 Vietnam Team Selection Test, 4
Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that
(i) $P(0) = 1$.
(ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$.
Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.
2018 Turkey Junior National Olympiad, 4
For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing
$$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$
2022 Germany Team Selection Test, 1
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
1972 IMO Longlists, 32
If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$,
show that
\[max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},\]
where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.
2018 Brazil Team Selection Test, 4
Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$
be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.
1978 Czech and Slovak Olympiad III A, 1
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive numbers. Show that
\[\sqrt{\left(a_1+\cdots+a_n\right)\left(b_1+\cdots+b_n\right)}\ge\sqrt{a_1b_1}+\cdots+\sqrt{a_nb_n}\]
and prove that equality holds if and only if
\[\frac{a_1}{b_1}=\cdots=\frac{a_n}{b_n}.\]
1972 Kurschak Competition, 1
A triangle has side lengths $a, b, c$. Prove that
$$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$
1976 All Soviet Union Mathematical Olympiad, 225
Given $4$ vectors $a,b,c,d$ in the plane, such that $a+b+c+d=0$. Prove the following inequality: $$|a|+|b|+|c|+|d| \ge |a+d|+|b+d|+|c+d|$$
1997 Junior Balkan MO, 3
Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$.
[i]Greece[/i]
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 $Q(3^k) \ge Q(3^{k+1})$.
2016 Costa Rica - Final Round, F1
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.
2019 Benelux, 1
[list=a]
[*]Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that
$$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$[/*]
[*]Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality.
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1986 Tournament Of Towns, (114) 1
For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?
2015 Romania Team Selection Tests, 2
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.
IV Soros Olympiad 1997 - 98 (Russia), 11.7
Solve the inequality $$\log_{\frac12} x\ge 16^x$$
2014 Estonia Team Selection Test, 2
Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$
2012 Balkan MO Shortlist, N3
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
1964 Polish MO Finals, 1
Prove that the inequality $$ \frac{1}{3} \leq \frac{\tan 3\alpha}{\tan \alpha} \leq 3 $$ is not true for any value of $ \alpha $.
2011 ISI B.Stat Entrance Exam, 1
Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that
\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]
2010 IFYM, Sozopol, 2
If $a,b,c>0$ and $abc=3$,find the biggest value of:
$\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$
1968 Vietnam National Olympiad, 1
Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$.
i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$.
ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$.
Show that $f(x)$ has two roots that are in between $-1$ and $1$.