Found problems: 6530
2011 National Olympiad First Round, 35
Which of these has the smallest maxima on positive real numbers?
$\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$
1963 German National Olympiad, 3
It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds
$$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$
Under what conditions does equality occur?
2010 Belarus Team Selection Test, 3.1
Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that
a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$
b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$
(D. Pirshtuk)
2017 Junior Balkan MO, 2
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2020 EGMO, 2
Find all lists $(x_1, x_2, \ldots, x_{2020})$ of non-negative real numbers such that the following three conditions are all satisfied:
[list]
[*] $x_1 \le x_2 \le \ldots \le x_{2020}$;
[*] $x_{2020} \le x_1 + 1$;
[*] there is a permutation $(y_1, y_2, \ldots, y_{2020})$ of $(x_1, x_2, \ldots, x_{2020})$ such that $$\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.$$
[/list]
[i]A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example, $(2, 1, 2)$ is a permutation of $(1, 2, 2)$, and they are both permutations of $(2, 2, 1)$. Note that any list is a permutation of itself.[/i]
2021 China Team Selection Test, 3
Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that
$$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.
2002 National Olympiad First Round, 5
The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude?
$
\textbf{a)}\ 4
\qquad\textbf{b)}\ 7
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ 23
$
2010 Cuba MO, 7
Let $x, y, z$ be positive real numbers such that $xyz = 1$. Prove that:
$$\frac{x^3 + y^3}{x^2 + xy + y^2} +\frac{ y^3 + z^3}{y^2 + yz + z^2} + \frac{z^3 + x^3}{z^2 + zx + x^2}
\ge 2.$$
1956 Putnam, B4
Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$
1999 Croatia National Olympiad, Problem 2
How do I prove that, for every $a, b, c$ positive real numbers such that $a+b+c = 1$ the following inequality holds: $\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}$?
2017 China Team Selection Test, 3
Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that:
$(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$;
$(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$;
$(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$
$(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$.
Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.
2021 Brazil Team Selection Test, 4
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2019 Jozsef Wildt International Math Competition, W. 58
In the $[ABCD]$ tetrahedron having all the faces acute angled triangles, is denoted by $r_X$, $R_X$ the radius lengths of the circle inscribed and circumscribed respectively on the face opposite to the $X \in \{A,B,C,D\}$ peak, and with $R$ the length of the radius of the sphere circumscribed to the tetrahedron. Show that inequality occurs$$8R^2 \geq (r_A + R_A)^2 + (r_B + R_B)^2 + (r_C + R_C)^2 + (r_D + R_D)^2$$
2002 India IMO Training Camp, 5
Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that
\[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]
2013 IPhOO, 10
Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$.
[i](Trung Phan, 10 points)[/i]
2012 Baltic Way, 13
Let $ABC$ be an acute triangle, and let $H$ be its orthocentre. Denote by $H_A$, $H_B$, and $H_C$ the second intersection of the circumcircle with the altitudes from $A$, $B$, and $C$ respectively. Prove that the area of triangle $H_A H_B H_C$ does not exceed the area of triangle $ABC$.
2019 India Regional Mathematical Olympiad, 3
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
$$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$
2011 Morocco National Olympiad, 4
The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that
\[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\]
Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.
2018 Poland - Second Round, 2
Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers
$1 = k_1 < k_2 < ... < k_m = n$
are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.
2017 Moldova Team Selection Test, 6
Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that
$$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$
1975 Czech and Slovak Olympiad III A, 2
Show that the system of equations
\begin{align*}
\lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\
3x+y &=2,
\end{align*}
has infinitely many solutions and all these solutions satisfy bounds
\begin{align*}
0<\ &x <4, \\
-9\le\ &y\le 1.
\end{align*}
2019 Tuymaada Olympiad, 4
A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.
2018 Romania National Olympiad, 2
Let $x>0.$ Prove that $$2^{-x}+2^{-1/x} \leq 1.$$
1981 All Soviet Union Mathematical Olympiad, 311
It is known about real $a$ and $b$ that the inequality $$a \cos x + b \cos (3x) > 1$$ has no real solutions. Prove that $|b|\le 1$.
2019 Romania National Olympiad, 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x+y)\leq f(x^2+y)$$ for all $x,y$.