Found problems: 6530
2011 AIME Problems, 13
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\angle O_1 P O_2 = 120^\circ$, then $AP=\sqrt{a}+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.
2010 Contests, 1
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
2001 Czech And Slovak Olympiad IIIA, 3
Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.
2011 Romania National Olympiad, 1
Find all real numbers $x, y,z,t \in [0, \infty)$ so that
$$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$
1987 Tournament Of Towns, (154) 5
We are given three non-negative numbers $A , B$ and $C$ about which it is known that $$A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)$$
(a) Prove that each of $A, B$ and $C$ is not greater than the sum of the others.
(b) Prove that $A^2 + B^2 + C^2 \le 2(AB + BC + CA)$ .
(c) Does the original inequality follow from the one in (b)?
(V.A. Senderov , Moscow)
1979 All Soviet Union Mathematical Olympiad, 278
Prove that for the arbitrary numbers $x_1, x_2, ... , x_n$ from the $[0,1]$ segment $$(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)$$
2007 Germany Team Selection Test, 3
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?
2005 Morocco TST, 3
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
1989 All Soviet Union Mathematical Olympiad, 496
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.
2007 Pre-Preparation Course Examination, 14
Find all $a,b,c \in \mathbb{N}$ such that
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\]
[PS: The original problem was this:
Find all $a,b,c \in \mathbb{N}$ such that
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\]
But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]
2018 Taiwan TST Round 1, 2
Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$
1980 Vietnam National Olympiad, 1
Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{ n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{ i})$ is an odd integer. Prove that
\[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]
2021-IMOC, A8
Find all functions $f : \mathbb{N} \to \mathbb{N}$ with
$$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$
holds for all positive integers $x,y.$
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 $f$.
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
1963 Miklós Schweitzer, 6
Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx
<\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]
2014 BMO TST, 1
Prove that for $n\ge 2$ the following inequality holds:
$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
2008 Hungary-Israel Binational, 1
Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.
2011 Olympic Revenge, 5
Let $n \in \mathbb{N}$ and $z \in \mathbb{C}^{*}$. Prove that
$\left | n\textrm{e}^{z} - \sum_{j=1}^{n}\left (1+\frac{z}{j^2}\right )^{j^2}\right | < \frac{1}{3}\textrm{e}^{|z|}\left (\frac{\pi|z|}{2}\right)^2$.
2006 Estonia National Olympiad, 1
Find the greatest possible value of $ sin(cos x) \plus{} cos(sin x)$ and determine all real
numbers x, for which this value is achieved.
2010 Abels Math Contest (Norwegian MO) Final, 2b
Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.
2016 India Regional Mathematical Olympiad, 5
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
2021 Indonesia TST, A
Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.
2011 Dutch BxMO TST, 4
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$.
Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.
2020 Greece JBMO TST, 2
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that
$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$
When is the equality valid?