Found problems: 85335
2005 Germany Team Selection Test, 2
Let n be a positive integer, and let $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ be positive real numbers such that $a_1\geq a_2\geq ...\geq a_n$ and $b_1\geq a_1$, $b_1b_2\geq a_1a_2$, $b_1b_2b_3\geq a_1a_2a_3$, ..., $b_1b_2...b_n\geq a_1a_2...a_n$.
Prove that $b_1+b_2+...+b_n\geq a_1+a_2+...+a_n$.
2006 Baltic Way, 11
The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?
2003 Federal Competition For Advanced Students, Part 1, 1
Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.
2007 Hong Kong TST, 3
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 3
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$.
2023 Bulgarian Autumn Math Competition, 9.4
Let $p, q$ be coprime integers, such that $|\frac{p} {q}| \leq 1$. For which $p, q$, there exist even integers $b_1, b_2, \ldots, b_n$, such that $$\frac{p} {q}=\frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\ldots}}}? $$
2019 Latvia Baltic Way TST, 2
Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that
$$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$
holds for all real numbers $x; y$
2021 German National Olympiad, 3
For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once.
Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.
2012 CHMMC Spring, 10
A convex polygon in the Cartesian plane has all of its vertices on integer coordinates. One of the sides of the polygon is $AB$ where $A = (0, 0)$ and $B = (51, 51)$, and the interior angles at $A$ and $B$ are both at most $45$ degrees. Assuming no $180$ degree angles, what is the maximum number of vertices this polygon can have?
2009 South East Mathematical Olympiad, 4
Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points .
Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .
2009 JBMO Shortlist, 2
In right trapezoid ${ABCD \left(AB\parallel CD\right)}$ the angle at vertex $B$ measures ${{75}^{{}^\circ }}$. Point ${H}$is the foot of the perpendicular from point ${A}$ to the line ${BC}$. If ${BH=DC}$ and${AD+AH=8}$, find the area of ${ABCD}$.
1997 Baltic Way, 14
In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.
1997 May Olympiad, 2
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
[img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]
2020 China Team Selection Test, 3
For a non-empty finite set $A$ of positive integers, let $\text{lcm}(A)$ denote the least common multiple of elements in $A$, and let $d(A)$ denote the number of prime factors of $\text{lcm}(A)$ (counting multiplicity). Given a finite set $S$ of positive integers, and $$f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.$$
Prove that, if $0 \le x \le 2$, then $-1 \le f_S(x) \le 0$.
1998 IMC, 2
$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.
2023 Math Prize for Girls Olympiad, 3
Let $m$ be the product of the first 100 primes, and let $S$ denote the set of divisors of $m$ greater than 1 (hence $S$ has exactly $2^{100} - 1$ elements). We wish to color each element of $S$ with one of $k$ colors such that
$\ \bullet \ $ every color is used at least once; and
$\ \bullet \ $ any three elements of $S$ whose product is a perfect square have exactly two different colors used among them.
Find, with proof, all values of $k$ for which this coloring is possible.
MOAA Gunga Bowls, 2023.23
For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$.
[i]Proposed by Harry Kim[/i]
1979 IMO Longlists, 32
Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation
\[|x_1| + |x_2| +\cdots + |x_k| = n\]
2019 AMC 10, 6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
[list]
[*] a square
[*]a rectangle that is not a square
[*] a rhombus that is not a square
[*] a parallelogram that is not a rectangle or a rhombus
[*] an isosceles trapezoid that is not a parallelogram
[/list]
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$
2000 IMO Shortlist, 2
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
2022 Latvia Baltic Way TST, P6
The numbers $1,2,3,\ldots ,n$ are written in a row. Two players, Maris and Filips, take turns making moves with Maris starting. A move consists of crossing out a number from the row which has not yet been crossed out. The game ends when there are exactly two uncrossed numbers left in the row. If the two remaining uncrossed numbers are coprime, Maris wins, otherwise Filips is the winner. For each positive integer $n\ge 4$ determine which player can guarantee a win.
2020 Harvard-MIT Mathematics Tournament, 3
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)
[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/asy]
[i]Proposed by Andrew Lin.[/i]
2023 Argentina National Olympiad, 2
Find all positive integers $n$ such that all prime factors of $2^n-1$ are less than or equal to $7$.
2021 AMC 12/AHSME Spring, 16
Let $g(x)$ be a polynomial with leading coefficient $1,$ whose three roots are the reciprocals of the three roots of $f(x)=x^3+ax^2+bx+c,$ where $1<a<b<c.$ What is $g(1)$ in terms of $a,b,$ and $c?$
$\textbf{(A) }\frac{1+a+b+c}c \qquad \textbf{(B) }1+a+b+c \qquad \textbf{(C) }\frac{1+a+b+c}{c^2}\qquad \textbf{(D) }\frac{a+b+c}{c^2} \qquad \textbf{(E) }\frac{1+a+b+c}{a+b+c}$
2019 LIMIT Category B, Problem 7
Find the number of ordered pairs of positive integers for which
$$\frac1a+\frac1b=\frac4{2019}$$
2021 The Chinese Mathematics Competition, Problem 4
Find the equation of cylinder that passes three straight lines
$L_1=
\begin{cases}
x=0\\
y-z=2
\end{cases},
L_2=
\begin{cases}
x=0\\
x+y-z+2=0
\end{cases},
L_3=
\begin{cases}
x=\sqrt{2}\\
y-z=0
\end{cases}$.