Found problems: 6530
1976 IMO Longlists, 8
In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.
2000 India National Olympiad, 6
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that
(i) $f(1999) > f (1996)$;
(ii) $f(2000) = f(1997)$.
2007 Purple Comet Problems, 7
There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$.
2019 JBMO Shortlist, A4
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
2001 Singapore Senior Math Olympiad, 2
Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$
Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.
1970 Polish MO Finals, 6
Find the smallest real number $A$ such that, for every quadratic polynomial $f(x)$ satisfying $ | f(x)| \le 1$ for $0 \le x \le 1$, it holds that $f' (0) \le A$.
1984 Polish MO Finals, 2
Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$.
Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$
2010 Contests, 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
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2021 Moldova Team Selection Test, 9
Positive real numbers $a$, $b$, $c$ satisfy $a+b+c=1$. Find the smallest possible value of
$$E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}.$$
2007 Hanoi Open Mathematics Competitions, 2
Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.
2014 Math Prize For Girls Problems, 15
There are two math exams called A and B. 2014 students took the A exam and/or the B exam. Each student took one or both exams, so the total number of exam papers was between 2014 and 4028, inclusive. The score for each exam is an integer from 0 through 40. The average score of all the exam papers was 20. The grade for a student is the best score from one or both exams that she took. The average grade of all 2014 students was 14. Let $G$ be the [i]greatest[/i] possible number of students who took both exams. Let $L$ be the [i]least[/i] possible number of students who took both exams. Compute $G - L$.
1988 All Soviet Union Mathematical Olympiad, 472
$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$
2017 Tournament Of Towns, 3
From given positive numbers, the following infinite sequence is defined: $a_1$ is the sum of all
original numbers, $a_2$ is the sum of the squares of all original numbers, $a_3$ is the sum of the
cubes of all original numbers, and so on ($a_k$ is the sum of the $k$-th powers of all original
numbers).
a) Can it happen that $a_1 > a_2 > a_3 > a_4 > a_5$ and $a_5 < a_6 < a_7 < \ldots$? (4 points)
b) Can it happen that $a_1 < a_2 < a_3 < a_4 < a_5$ and $a_5 > a_6 > a_7 > \ldots$? (4 points)
[i](Alexey Tolpygo)[/i]
1992 Kurschak Competition, 1
Define for $n$ given positive reals the [i]strange mean[/i] as the sum of the squares of these numbers divided by their sum. Decide which of the following statements hold for $n=2$:
a) The strange mean is never smaller than the third power mean.
b) The strange mean is never larger than the third power mean.
c) The strange mean, depending on the given numbers, can be larger or smaller than the third power mean.
Which statement is valid for $n=3$?
2003 All-Russian Olympiad, 2
Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality
\[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.\]
2019-2020 Fall SDPC, 5
Is there a function $f$ from the positive integers to themselves such that $$f(a)f(b) \geq f(ab)f(1)$$ with equality [b]if and only if[/b] $(a-1)(b-1)(a-b)=0$?
2001 Austria Beginners' Competition, 3
Find all real numbers $x$ such that $(x-1)^2(x-4)^2<(x-2)^2$.
2016 Dutch IMO TST, 1
Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$
2002 Tournament Of Towns, 4
$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that:
\[ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} \]
2022 ISI Entrance Examination, 8
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.
2015 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality:
$$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$
2011 National Olympiad First Round, 19
For which inequality, there exists a line such that the region defined by the inequality and the line intersect in exactly two distinct points?
$\textbf{(A)}\ x^2+y^2\leq 1 \qquad\textbf{(B)}\ |x+y|+|x-y| \leq 1 \qquad\textbf{(C)}\ |x|^3+|y|^3 \leq 1 \\ \textbf{(D)}\ |x|+|y| \leq 1 \qquad\textbf{(E)}\ |x|^{1/2} + |y|^{1/2} \leq 1$
2010 Balkan MO, 1
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
2014 Turkey Team Selection Test, 3
Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$
\[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]
2000 Tournament Of Towns, 1
Can the product of $2$ consecutive natural numbers equal the product of $2$ consecutive even natural numbers?
(natural means positive integers)