Found problems: 85335
2019 LIMIT Category C, Problem 1
Which of the following are true?
$\textbf{(A)}~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R)$
$\textbf{(B)}~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible}$
$\textbf{(C)}~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC)$
$\textbf{(D)}~\text{None of the above}$
2022 MOAA, 5
Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.
2013 International Zhautykov Olympiad, 2
Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).
2020 Indonesia MO, 7
Determine all real-coefficient polynomials $P(x)$ such that
\[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.
2021 Sharygin Geometry Olympiad, 12
Suppose we have ten coins with radii $1, 2, 3, \ldots , 10$ cm. We can put two of them on the table in such a way that they touch each other, after that we can add the coins in such a way that each new coin touches at least two of previous ones. The new coin cannot cover a previous one. Can we put several coins in such a way that the centers of some three coins are collinear?
2024 Euler Olympiad, Round 1, 1
Using each of the ten digits exactly once, construct two five-digit numbers such that their difference is minimized. Determine this minimal difference.
[i]Proposed by Giorgi Arabidze, Georgia [/i]
2010 Romania Team Selection Test, 1
Given a positive integer number $n$, determine the minimum of
\[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\]
as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$.
[i]Kvant Magazine[/i]
1996 Argentina National Olympiad, 1
$100$ numbers were written around a circle. The sum of the $100$ numbers is equal to $100$ and the sum of six consecutive numbers is always less than or equal to $6$. The first number is $6$. Find all the numbers.
1955 AMC 12/AHSME, 34
A $ 6$-inch and $ 18$-inch diameter pole are placed together and bound together with wire. The length of the shortest wire that will go around them is:
$ \textbf{(A)}\ 12\sqrt{3}\plus{}16\pi \qquad
\textbf{(B)}\ 12\sqrt{3}\plus{}7\pi \qquad
\textbf{(C)}\ 12\sqrt{3}\plus{}14\pi \\
\textbf{(D)}\ 12\plus{}15\pi \qquad
\textbf{(E)}\ 24\pi$
2000 APMO, 3
Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.
Prove that $QO$ is perpendicular to $BC$.
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
2000 Flanders Math Olympiad, 4
Solve for $x \in [0,2\pi[$: \[\sin x < \cos x < \tan x < \cot x\]
2019 USAJMO, 5
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
[i]Proposed by Ricky Liu[/i]
2016 Azerbaijan Balkan MO TST, 4
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$
1996 Brazil National Olympiad, 2
Does there exist a set of $n > 2, n < \infty$ points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set?
2017 Kyiv Mathematical Festival, 1
Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have two new neighbours and exactly one third of the dwarves have only one new neighbour, if the number of the dwarves is a) 9; b) 12?
II Soros Olympiad 1995 - 96 (Russia), 9.3
Is there a convex pentagon in which each diagonal is equal to some side?
PEN H Problems, 46
Let $a, b, c, d, e, f$ be integers such that $b^{2}-4ac>0$ is not a perfect square and $4acf+bde-ae^{2}-cd^{2}-fb^{2}\neq 0$. Let \[f(x, y)=ax^{2}+bxy+cy^{2}+dx+ey+f\] Suppose that $f(x, y)=0$ has an integral solution. Show that $f(x, y)=0$ has infinitely many integral solutions.
2021 LMT Fall, 1
Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?
2021 Indonesia MO, 2
Let $ABC$ be an acute triangle. Let $D$ and $E$ be the midpoint of segment $AB$ and $AC$ respectively. Suppose $L_1$ and $L_2$ are circumcircle of triangle $ABC$ and $ADE$ respectively. $CD$ intersects $L_1$ and $L_2$ at $M (M \not= C)$ and $N (N \not= D)$. If $DM = DN$, prove that $\triangle ABC$ is isosceles.
2011 Romanian Masters In Mathematics, 1
Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing.
[i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]
2023 Taiwan TST Round 2, N
Find all polynomials $P$ with real coefficients satisfying that there exist infinitely many pairs $(m, n)$ of coprime positives integer such that $P(\frac{m}{n})=\frac{1}{n}$.
[i]
Proposed by usjl[/i]
2011 China Girls Math Olympiad, 6
Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?
2019 BMT Spring, 5
What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$?
2015 Latvia Baltic Way TST, 8
Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that
$$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$
i) Prove that one can find a $q$ such that all positive rational numbers are charismatic.
ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?