Found problems: 85335
2016 USAMTS Problems, 4:
Let ${A_1, \dots , A_n }$ and ${B_1, \dots , B_n}$ be sets of points in the plane. Suppose that for all points $x$,
$$D \left( x , A_1 \right) + D \left( x , A_2 \right) + \cdots + D \left( x , A_n \right) \ge D \left( x , B_1 \right) + D \left( x , B_2 \right) + \cdots + D \left( x , B_n \right)$$
where $D \left( x , y \right)$ denotes the distance between $x$ and $y$. Show that the $A_i$'s and the $B_i$'s share the same center of mass.
Dumbest FE I ever created, 5.
Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$ . Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . [hide=original]Find all non decreasing functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$
$$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(x))+y$$ .[/hide]
2018 Costa Rica - Final Round, A2
Determine the sum of the real roots of the equation $$x^2-8x+20=2\sqrt{x^2-8x+30}$$
2013 China National Olympiad, 1
Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition
\[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\]
Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.
2017 Tuymaada Olympiad, 8
Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$.
(K. Tyschu)
2023 Korea - Final Round, 4
Find all positive integers $n$ satisfying the following.
$$2^n-1 \text{ doesn't have a prime factor larger than } 7$$
2019 USAJMO, 6
Two rational numbers \(\tfrac{m}{n}\) and \(\tfrac{n}{m}\) are written on a blackboard, where \(m\) and \(n\) are relatively prime positive integers. At any point, Evan may pick two of the numbers \(x\) and \(y\) written on the board and write either their arithmetic mean \(\tfrac{x+y}{2}\) or their harmonic mean \(\tfrac{2xy}{x+y}\) on the board as well. Find all pairs \((m,n)\) such that Evan can write 1 on the board in finitely many steps.
[i]Proposed by Yannick Yao[/i]
2024 Serbia Team Selection Test, 2
Let $n$ be a positive integer. Initially a few positive integers are written on the blackboard. On one move Igor chooses two numbers $a, b$ of the same parity on the blackboard and writes $\frac{a+b} {2}$. After a few moves the numbers on the blackboard were exactly $1, 2, \ldots, n$. Find the smallest possible number of positive integers that were initially written on the blackboard.
2002 AIME Problems, 2
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$
[asy]
size(250);real x=sqrt(3);
int i;
draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle);
for(i=0; i<7; i=i+1) {
draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1));
}
for(i=0; i<6; i=i+1) {
draw(Circle((2*i+2,1+x), 1));
}[/asy]
2000 ITAMO, 5
A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
1974 IMO Longlists, 25
Let $f : \mathbb R \to \mathbb R$ be of the form $f(x) = x + \epsilon \sin x,$ where $0 < |\epsilon| \leq 1.$ Define for any $x \in \mathbb R,$
\[x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x).\]
Show that for every $x \in \mathbb R$ there exists an integer $k$ such that $\lim_{n\to \infty } x_n = k\pi.$
2021 Alibaba Global Math Competition, 13
Let $M_n=\{(u,v) \in S^n \times S^n: u \cdot v=0\}$, where $n \ge 2$, and $u \cdot v$ is the Euclidean inner product of $u$ and $v$. Suppose that the topology of $M_n$ is induces from $S^n \times S^n$.
(1) Prove that $M_n$ is a connected regular submanifold of $S^n \times S^n$.
(2) $M_n$ is Lie Group if and only if $n=2$.
2011 Akdeniz University MO, 5
For all $n \in {\mathbb Z^+}$ we define
$$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$
infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$.
[b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$
[b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$
2013 Tournament of Towns, 3
Each of $11$ weights is weighing an integer number of grams. No two weights are equal. It is known that if all these weights or any group of them are placed on a balance then the side with a larger number of weights is always heavier. Prove that at least one weight is heavier than $35$ grams.
1995 French Mathematical Olympiad, Problem 5
Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.
2002 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.
1990 IMO Longlists, 59
Given eight real numbers $a_1 \leq a_2 \leq \cdots \leq a_7 \leq a_8$. Let $x = \frac{ a_1 + a_2 + \cdots + a_7 + a_8}{8}$, $y = \frac{ a_1^2 + a_2^2 + \cdots + a_7^2 + a_8^2}{8}$. Prove that
\[2 \sqrt{y-x^2} \leq a_8 - a_1 \leq 4 \sqrt{y-x^2}.\]
2010 Hanoi Open Mathematics Competitions, 9
Let $x,y$ be the positive integers such that $3x^2 +x = 4y^2 + y$.
Prove that $x - y$ is a perfect (square).
2019 Sharygin Geometry Olympiad, 21
An ellipse $\Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\Gamma$.
2009 Junior Balkan Team Selection Tests - Romania, 2
Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.
2017 Romania National Olympiad, 1
[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation.
$$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$
[b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.
2010 Germany Team Selection Test, 1
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.
Prove that $GR=GS$.
[i]Proposed by Hossein Karke Abadi, Iran[/i]
KoMaL A Problems 2022/2023, A. 850
Prove that there exists a positive real number $N$ such that for arbitrary real numbers $a,b>N$ it is possible to cover the perimeter of a rectangle with side lengths $a$ and $b$ using non-overlapping unit disks (the unit disks can be tangent to each other).
[i]Submitted by Benedek Váli, Budapest[/i]
2023 AMC 12/AHSME, 24
Integers $a, b, c, d$ satisfy the following:
$abcd=2^6\cdot 3^9\cdot 5^7$
$\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
$\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
$\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
$\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
$\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$
Find $\text{gcd}(a,b,c,d)$
$\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$
2010 Paenza, 1
a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct.
$\tab$ $\tab$ $ABC$
$\tab$ $\tab$ $DEF$
[u]$+GHI$[/u]
$\tab$ $\tab$ $\tab$ $J J J$
Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$.
b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).