Found problems: 85335
2009 Singapore MO Open, 4
find largest constant C st
$\sum_{i=1}^{4} (x_i+1/x_i)^3 \geq C$
for all positive real numbers $x_1,..,x_4$ st
$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1$
2015 AMC 8, 24
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a 76 game schedule. How many games does a team play within its own division?
$
\textbf{(A) } 36 \qquad
\textbf{(B) } 48 \qquad
\textbf{(C) } 54 \qquad
\textbf{(D) } 60 \qquad
\textbf{(E) } 72
$
2013 Switzerland - Final Round, 1
Find all triples $(a, b, c)$ of natural numbers such that the sets
$$\{ gcd (a, b), gcd(b, c), gcd(c, a), lcm (a, b), lcm (b, c), lcm (c, a)\}$$ and
$$\{2, 3, 5, 30, 60\}$$
are the same.
Remark: For example, the sets $\{1, 2013\}$ and $\{1, 1, 2013\}$ are equal.
2011 Math Prize For Girls Problems, 2
Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.
2023 AIME, 2
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292=444_{\text{eight}}$.
2022 JBMO TST - Turkey, 2
For a real number $a$, $[a]$ denotes the largest integer not exceeding $a$.
Find all positive real numbers $x$ satisfying the equation
$$x\cdot [x]+2022=[x^2]$$
2002 AMC 10, 10
Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$.
$\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$
2000 National High School Mathematics League, 10
In ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $F$ is its left focal point, $A$ is its right vertex, $B$ is its upper vertex. If the eccentricity of the ellipse is $\frac{\sqrt5-1}{2}$, then $\angle ABF=$________.
1952 Czech and Slovak Olympiad III A, 4
Let $p,q$ be positive integers. Consider a rectangle $ABCD$ with lengths of sides $p$ and $q$ that consists of $pq$ unital squares. How many of these squares are crossed by diagonal $AC$?
2020 LIMIT Category 2, 7
A circle $\mathfrak{D}$ is drawn through the vertices $A$ and $B$ of $\triangle ABC$. If $\mathfrak{D}$ intersects $AC$ at a point $M$ and $BC$ at $P$ and $MP$ contains the incenter of $\triangle ABC$, then the length $MP$ is (in standard notation, where $t=\frac{1}{a+b+c}$):
(A)$at(b+c)$
(B)$ct(b+a)$
(C)$bct$
(D)$abt$