Found problems: 6530
2011 Austria Beginners' Competition, 3
Let $x, y$ be positive real numbers with $x + y + xy= 3$. Prove that$$x + y\ge 2.$$ When does equality holds?
(K. Czakler, GRG 21, Vienna)
2002 India IMO Training Camp, 15
Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality
\[
\frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots +
\frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
\]
2013 Costa Rica - Final Round, A1
Let the real numbers $x, y, z$ be such that $x + y + z = 0$. Prove that $$6(x^3 + y^3 + z^3)^2 \le (x^2 + y^2 + z^2)^3.$$
2005 Germany Team Selection Test, 1
Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.
PEN O Problems, 53
Suppose that the set $M=\{1,2,\cdots,n\}$ is split into $t$ disjoint subsets $M_{1}$, $\cdots$, $M_{t}$ where the cardinality of $M_i$ is $m_{i}$, and $m_{i} \ge m_{i+1}$, for $i=1,\cdots,t-1$. Show that if $n>t!\cdot e$ then at least one class $M_z$ contains three elements $x_{i}$, $x_{j}$, $x_{k}$ with the property that $x_{i}-x_{j}=x_{k}$.
2006 Irish Math Olympiad, 4
Find the greatest value and the least value of $x+y$ where $x,y$ are real numbers, with $x\ge -2$, $y\ge -3$ and $$x-2\sqrt{x+2}=2\sqrt{y+3}-y$$
2023-IMOC, A5
We can conduct the following moves to a real number $x$: choose a positive integer $n$, and positives reals $a_1,a_2,\cdots, a_n$ whose reciprocals sum up to $1$. Let $x_0=x$, and $x_k=\sqrt{x_{k-1}a_k}$ for all $1\leq k\leq n$. Finally, let $y=x_n$. We said $M>0$ is "tremendous" if for any $x\in \mathbb{R}^+$, we can always choose $n,a_1,a_2,\cdots, a_n$ to make the resulting $y$ smaller than $M$. Find all tremendous numbers.
[i]Proposed by ckliao914.[/i]
1940 Moscow Mathematical Olympiad, 069
Let $a_1, ...,, a_n$ be positive numbers. Prove the inequality:
$$\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+ ... +\frac{a_{n-1}}{a_n}+ \frac{a_n}{a_1} \ge n$$
2008 Ukraine Team Selection Test, 3
For positive $ a, b, c, d$ prove that
$ (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)$
2008 Germany Team Selection Test, 1
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that
\[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}.
\]
[i]Author: Marcin Kuzma, Poland[/i]
2006 Germany Team Selection Test, 2
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.
Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .
[i]Proposed by Hojoo Lee, Korea[/i]
1989 Cono Sur Olympiad, 3
Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.
2013 Estonia Team Selection Test, 3
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
2023 Korea - Final Round, 6
For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following.
$$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$
($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)
MIPT Undergraduate Contest 2019, 1.5 & 2.5
Prove the inequality
$$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$
for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$
2005 MOP Homework, 2
Find all real numbers $x$ such that
$\lfloor x^2-2x \rfloor+2\lfloor x \rfloor=\lfloor x \rfloor^2$.
(For a real number $x$, $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$.)
2002 Federal Math Competition of S&M, Problem 2
Let $O$ be a point inside a triangle $ABC$ and let the lines $AO,BO$, and $CO$ meet sides $BC,CA$, and $AB$ at points $A_1,B_1$, and $C_1$, respectively. If $AA_1$ is the longest among the segments $AA_1,BB_1,CC_1$, prove that
$$OA_1+OB_1+OC_1\le AA_1.$$
1988 Romania Team Selection Test, 14
Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.
2018 Brazil National Olympiad, 1
We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.
1967 Putnam, B2
Let $0\leq p,r\leq 1$ and consider the identities
$$a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.$$
Show that
$$ a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.$$
1971 Poland - Second Round, 2
Prove that if $ A, B, C $ are angles of a triangle, then
$$
1 < \cos A + \cos B + \cos C \leq \frac{3}{2}.$$
2008 National Olympiad First Round, 31
If the inequality
\[
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
\]
is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 18
\qquad\textbf{(E)}\ 20
$
Kvant 2019, M2544
Let $P(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n$ be a polynomial of degree $n$ and $n$ real roots, all of them in the interval $(0,1)$. Prove that for all $k=\overline{1,n}$ the following inequality holds:
\[(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.\]
[i]Proposed by N. Safaei (Iran)[/i]
1997 Austrian-Polish Competition, 7
(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, .
(b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$
(c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.
2021 Peru IMO TST, P1
Suppose positive real numers $x,y,z,w$ satisfy $(x^3+y^3)^4=z^3+w^3$. Prove that
$$x^4z+y^4w\geq zw.$$