Found problems: 6530
2015 Iran Team Selection Test, 6
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2017 China National Olympiad, 6
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
2018-IMOC, A5
Show that for all reals $x,y,z$, we have
$$\left(x^2+3\right)\left(y^2+3\right)\left(z^2+3\right)\ge(xyz+x+y+z+4)^2.$$
2014 District Olympiad, 2
Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$
Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$
1970 IMO Longlists, 31
Prove that for any triangle with sides $a, b, c$ and area $P$ the following inequality holds:
\[P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.\]
Find all triangles for which equality holds.
2014 Mexico National Olympiad, 6
Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.
2011 Stars Of Mathematics, 3
For a given integer $n\geq 3$, determine the range of values for the expression
\[ E_n(x_1,x_2,\ldots,x_n) := \dfrac {x_1} {x_2} + \dfrac {x_2} {x_3} + \cdots + \dfrac {x_{n-1}} {x_n} + \dfrac {x_n} {x_1}\]
over real numbers $x_1,x_2,\ldots,x_n \geq 1$ satisfying $|x_k - x_{k+1}| \leq 1$ for all $1\leq k \leq n-1$.
Do also determine when the extremal values are achieved.
(Dan Schwarz)
1991 Vietnam National Olympiad, 3
Prove that:
$ \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2}$
where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$
2009 Bulgaria National Olympiad, 6
Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$
are positive real numbers, than
$ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.
1981 Vietnam National Olympiad, 2
Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$. Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$, respectively. Prove that
\[\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.\]
1996 China Team Selection Test, 2
Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that
\[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\]
Define
\begin{align*}
A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\
B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\
W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2.
\end{align*}
Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.
2010 ISI B.Stat Entrance Exam, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
1999 Romania National Olympiad, 2a
let $x_i,y_i 1 \le i \le n$ be positive numbers such that :
$\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$
Prove :
$\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$
2009 Cuba MO, 7
Let $x_1, x_2, ..., x_n$ be positive reals. Prove that
$$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$
In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.
2017 Iran MO (3rd round), 3
Let $a,b$ and $c$ be positive real numbers. Prove that
$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
2014 China Girls Math Olympiad, 7
Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$
2003 Federal Math Competition of S&M, Problem 2
Let $ f : [0, 1] \to\ R $ be a function such that :-
$1.)$ $f(x) \ge 0$ for all $x$ in $[0,1]$ .
$2.)$ $f(1) = 1$ .
$3.)$ If $x_1 , x_2$ are in $[0,1]$ such that $x_1 + x_2 \le 1$ , then $f(x_1) + f(x_2) \le f(x_1 + x_2)$ .
Show that $f(x) \le 2x $ for all $x$ in $ [0,1] $.
2023 Kyiv City MO Round 1, Problem 1
Find all positive integers $n$ that satisfy the following inequalities:
$$-46 \leq \frac{2023}{46-n} \leq 46-n$$
2010 Spain Mathematical Olympiad, 1
Let $a,b,c$ be three positive real numbers. Show that
\[ \frac {a+b+3c}{3a+3b+2c}+\frac {a+3b+c}{3a+2b+3c}+\frac {3a+b+c}{2a+3b+3c} \ge \frac {15}{8}\]
2006 Mediterranean Mathematics Olympiad, 2
Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that
\[ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)\]
2012 Thailand Mathematical Olympiad, 9
Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.
2014 Turkey MO (2nd round), 3
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
2009 USAMO, 4
For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that
\[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2.
\]
Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
1976 Miklós Schweitzer, 3
Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that
a) $ n! \in H$ for all sufficiently large $ n$,
b)$ H$ has density $ 0$.
[i]P. Erdos[/i]
2011 APMO, 1
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.