Found problems: 85335
2018 BMT Spring, 6
Let $x,y,z \in \mathbb{R}$ and $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$. The minimum value of $x^2 + y^2 + z^2$ can be expressed as $\dfrac{a}{b}$ where $a,b \in \mathbb{Z}, \gcd(a,b) = 1$. What is $a + b$?
2013 Costa Rica - Final Round, 6
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.
2014 BMT Spring, 1
Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
2020 JHMT, 12
Circle $O$ is inscribed inside a non-isosceles trapezoid $JHMT$, tangent to all four of its sides. The longer of the two parallel sides of $JHMT$ is $\overline{JH}$ and has a length of $24$ units. Let $P$ be the point where $O$ is tangent to $\overline{JH}$, and let $Q$ be the point where $O$ is tangent to $\overline{MT}$. The circumcircle of $\vartriangle JQH$ intersects $O$ a second time at point $R$. $\overleftrightarrow{QR}$ intersects $\overleftrightarrow{JH}$ at point $S$, $35$ units away from $P$. The points inside $JHMT$ at which $\overline{JQ}$ and $\overline{HQ}$ intersect $O$ lie $\frac{63}{4}$ units apart. The area of $O$ can be expressed as $\frac{m\pi}{n}$ , where $\frac{m}{n}$ is a common fraction. Compute $m + n$.
2004 Croatia National Olympiad, Problem 2
If $a,b,c$ are the sides and $\alpha,\beta,\gamma$ the corresponding angles of a triangle, prove the inequality
$$\frac{\cos\alpha}{a^3}+\frac{\cos\beta}{b^3}+\frac{\cos\gamma}{c^3}\ge\frac3{2abc}.$$
2017 Romania Team Selection Test, P1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
2010 Hong kong National Olympiad, 2
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
2008 IberoAmerican Olympiad For University Students, 2
Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.
2022 Sharygin Geometry Olympiad, 9.1
Let $BH$ be an altitude of right angled triangle $ABC$($\angle B = 90^o$). An excircle of triangle $ABH$ opposite to $B$ touches $AB$ at point $A_1$; a point $C_1$ is defined similarly. Prove that $AC // A_1C_1$.
2001 China Team Selection Test, 1
Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$):
$\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$.
$\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.
2015 Bundeswettbewerb Mathematik Germany, 2
In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma.
Show $n>1250$.
[i]Example:[/i] I mean something like $0.7143$.
IMSC 2023, 6
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
2016 Regional Olympiad of Mexico Southeast, 3
Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]
2010 Germany Team Selection Test, 2
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
2019 PUMaC Algebra B, 5
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.
1996 Moldova Team Selection Test, 5
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify the properties:
[b]a)[/b] $P(-x)=P(x), \forall x\in\mathbb{R};$
[b]b)[/b] $P(x)\geq0, \forall x\in\mathbb{R};$
[b]c)[/b] $P(0)=1;$
[b]d)[/b] $P(X)$ has exactly two local minimums $x_1$ and $x_2$ such that $|x_1-x_2|=2.$
2001 Moldova Team Selection Test, 8
A group of $n{}$ $(n>1)$ people each visited $k{}$ $(k>1)$ citites. Each person makes a list of these $k$ cities in the order they want to visit them. A permutation $(a_1,a_2,\ldots,a_k)$ is called $m-prefered$ $(m\in\mathbb{N})$, if for every $i=1,2,\ldots,k$ there are at least $m$ people that would prefer to visit the city $a_i$ before the city $a_{i+1}$, $(a_{k+1}=a_1)$. Prove that there exists an m-prefered permutation if and only if $km\leq n(k-1)$.
2014 Oral Moscow Geometry Olympiad, 2
Is it possible to cut a regular triangular prism into two equal pyramids?
2015 IMO Shortlist, A6
Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences
\begin{eqnarray*}
P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\
Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)
\end{eqnarray*}
are permutations of each other.
(a) Prove that there exist distinct block-similar polynomials of degree $n + 1$.
(b) Prove that there do not exist distinct block-similar polynomials of degree $n$.
[i]Proposed by David Arthur, Canada[/i]
2020 BMT Fall, 9
For any point $(x, y)$ with $0\le x < 1$ and $0 \le y < 1$, Jenny can perform a shuffle on that point, which takes the point to $(\{3x + y\} ,\{x + 2y\})$ where $\{a\}$ denotes the fractional or decimal part of $a$ (so for example,$\{\pi\} = \pi - 3 = 0.1415...$). How many points $p$ are there such that after $3$ shuffles on $p$, $p$ ends up in its original position?
KoMaL A Problems 2023/2024, A. 881
We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
1984 Austrian-Polish Competition, 8
The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.
2005 Tournament of Towns, 3
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)
[i](6 points)[/i]
VMEO III 2006 Shortlist, A3
For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that
$$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$
2021 Vietnam TST, 3
Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. Similarly, we define $B_c, C_b, A_c, C_a$. Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$. Define similarly $m_b, m_c$.
a) Assume that $N$ is the orthocenter of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line.
b) Assume that $N$ is the nine-point center of $\triangle ABC$, show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.