Found problems: 6530
2017 Korea Winter Program Practice Test, 1
Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that
\[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]
II Soros Olympiad 1995 - 96 (Russia), 10.2
Find the smallest value that the expression can take
$$|a-1|+|b-2|+c-3|+|3a+2b+c|$$
($a$, $b$ and $c$ are arbitrary numbers).
2018 Thailand TST, 3
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
2005 IMO, 3
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
2011 Grand Duchy of Lithuania, 2
Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality
$(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$
1975 Spain Mathematical Olympiad, 7
Consider the real function defined by $f(x) =\frac{1}{|x + 3| + |x + 1| + |x - 2| + |x -5|}$ for all $x \in R$.
a) Determine its maximum.
b) Graphic representation.
2002 India IMO Training Camp, 5
Let $a,b,c$ be positive reals such that $a^2+b^2+c^2=3abc$. Prove that
\[\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2} \geq \frac{9}{a+b+c}\]
2024 Bundeswettbewerb Mathematik, 3
Let $ABC$ be a triangle. For a point $P$ in its interior, we draw the threee lines through $P$ parallel to the sides of the triangle. This partitions $ABC$ in three triangles and three quadrilaterals.
Let $V_A$ be the area of the quadrilateral which has $A$ as one vertex. Let $D_A$ be the area of the triangle which has a part of $BC$ as one of its sides. Define $V_B, D_B$ and $V_C, D_C$ similarly.
Determine all possible values of $\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}$, as $P$ varies in the interior of the triangle.
2016 Hong Kong TST, 3
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Determine the smallest possible value of
$$\frac{a^3+8}{a^3(b+c)}+\frac{b^3+8}{b^3(a+c)}+\frac{c^3+8}{c^3(b+a)}$$
2014 AMC 12/AHSME, 22
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
$\textbf{(A) }278\qquad
\textbf{(B) }279\qquad
\textbf{(C) }280\qquad
\textbf{(D) }281\qquad
\textbf{(E) }282\qquad$
1983 IMO Longlists, 49
Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that
\[n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).\]
2018 Romania National Olympiad, 2
Let $a,b,c \geq 0$ and $a+b+c=3.$ Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}$$
2009 Princeton University Math Competition, 5
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 $\overline{ab}$.
2002 Korea - Final Round, 1
For $n \ge 3$, let $S=a_1+a_2+\cdots+a_n$ and $T=b_1b_2\cdots b_n$ for positive real numbers $a_1,a_2,\ldots,a_n, b_1,b_2 ,\ldots,b_n$, where the numbers $b_i$ are pairwise distinct.
(a) Find the number of distinct real zeroes of the polynomial
\[f(x)=(x-b_1)(x-b_2)\cdots(x-b_n)\sum_{j=1}^n \frac{a_j}{x-b_j}\]
(b) Prove the inequality
\[\frac1{n-1}\sum_{j=1}^n \left(1-\frac{a_j}{S}\right)b_j > \left(\frac{T}{S}\sum_{j=1}^{n} \frac{a_j}{b_j}\right)^{\frac1{n-1}}\]
2010 China Girls Math Olympiad, 8
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$
2020 Jozsef Wildt International Math Competition, W25
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
1997 Rioplatense Mathematical Olympiad, Level 3, 5
Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$.
Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.
2001 Czech-Polish-Slovak Match, 1
Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that
\[\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)\]
2022 Thailand TST, 3
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
2023 Ukraine National Mathematical Olympiad, 8.3
Positive integers $x, y$ satisfy the following conditions:
$$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$
Show that $x = y$.
Here $\{x\}$ denotes the fractional part of $x$. For example, $\{3.14\} = 0.14$.
[i]Proposed by Anton Trygub[/i]
2021 China National Olympiad, 3
Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$
2009 China Northern MO, 5
Assume : $x,y,z>0$ , $ x^2+y^2+z^2 = 3 $ . Prove the following inequality :
$${\frac{x^{2009}-2008(x-1)}{y+z}+\frac{y^{2009}-2008(y-1)}{x+z}+\frac{z^{2009}-2008(z-1)}{x+y}\ge\frac{1}{2}(x+y+z)}$$
2008 Singapore Junior Math Olympiad, 2
Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.
1999 Spain Mathematical Olympiad, 5
The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:
a) $g_a,g_b,g_c \ge \frac{2}{3}r$
b) $g_a+g_b+g_c \ge 3r$
2016 Romanian Masters in Mathematic, 4
Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$