Found problems: 6530
2005 Cuba MO, 9
Let $x_1, x_2, …, x_n$ and $y_1, y_2, …,y_n$ be positive reals such that $$x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i$$ for all $i = 1, 2, …, n$. Prove that
$$\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.$$
2016 Danube Mathematical Olympiad, 1
1.Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the
side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the
lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one
quarter the perimeter of the triangle $ABC$.
2004 Croatia National Olympiad, Problem 3
Prove that for any three real numbers $x,y,z$ the following inequality holds:
$$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$
2018 Greece JBMO TST, 1
Let $a,b,c,d$ be positive real numbers such that $a^2+b^2+c^2+d^2=4$.
Prove that exist two of $a,b,c,d$ with sum less or equal to $2$.
1990 Federal Competition For Advanced Students, P2, 3
In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that:
$ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$
2020 Azerbaijan Senior NMO, 1
$x,y,z\in\mathbb{R^+}$. If $xyz=1$, then prove the following: $$\sum\frac{x^6+2}{x^3}\geq3(\frac{x}{y}+\frac{y}{z}+\frac{z}{x})$$
2003 India National Olympiad, 3
Show that $8x^4 - 16x^3 + 16x^2 - 8x + k = 0$ has at least one real root for all real $k$. Find the sum of the non-real roots.
2018 Moldova EGMO TST, 5
Let $a$ and $b$ be real numbers such that $a + b = 1$. Prove the inequality
$$\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.$$
[i]Proposed by Baasanjav Battsengel[/i]
1984 IMO Longlists, 35
Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions
\[\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}\]
where $\pi$ is the ratio between a circle and its diameter.
1999 Romania National Olympiad, 2
For $a, b > 0$, denote by $t(a,b)$ the positive root of the equation $$(a+b)x^2-2(ab-1)x-(a+b) = 0.$$
Let $M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}$
Determine, for $(a, b)\in M$, the mmimum value of $t(a,b)$.
2021 Alibaba Global Math Competition, 6
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
1987 IMO Longlists, 77
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$
\[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
2023 Czech-Polish-Slovak Match, 2
Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.
2015 Junior Balkan Team Selection Test, 3
Prove inequallity :
$$1+\frac{1}{2^3}+...+\frac{1}{2015^3}<\frac{5}{4}$$
2006 Korea Junior Math Olympiad, 6
For all reals $a, b, c,d $ prove the following inequality:
$$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$
2010 Contests, 3
let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let
$A_k=\frac{\sum_{i=1}^{k}a_i}{k}$
prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.
2012 South africa National Olympiad, 6
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that
$f(km)+f(kn)-f(k)f(mn)\ge 1$
for all $k,m,n\in\mathbb{N}$.
2013 China Team Selection Test, 2
Let $k\ge 2$ be an integer and let $a_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n$ be non-negative real numbers. Prove that\[\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i\equal{}1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i\equal{}1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.\]
2007 Pre-Preparation Course Examination, 3
$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]
2003 Putnam, 4
Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]
2007 USA Team Selection Test, 2
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that
\[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \]
and
\[ a_1 + \dots + a_n = b_1 + \dots + b_n. \]
Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.
2022 Balkan MO Shortlist, A3
Let $a, b, c, d$ be non-negative real numbers such that \[\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.\]
Prove that \[3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.\][i]Vasile Cîrtoaje and Leonard Giugiuc[/i]
2008 ITest, 90
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.
2006 District Olympiad, 1
Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that
\[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\]
Prove that
\[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\]
[i]Cezar Lupu[/i]
1998 VJIMC, Problem 4-M
A function $f:\mathbb R\to\mathbb R$ has the property that for every
$x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such
that $0<t<1$ and
$$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$
Does it imply that
$$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$
for every $x,y\in\mathbb R$?