Found problems: 85335
1969 IMO Longlists, 39
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible.
2007 IMS, 6
Let $R$ be a commutative ring with 1. Prove that $R[x]$ has infinitely many maximal ideals.
1970 Spain Mathematical Olympiad, 8
There is a point $M$ inside a circle, at a distance $OM = d$ of the center $O$. Two chords $AB$ and $CD$ are traced through $M$ that form a right angle . Join $A$ with $C$ and $B$ with $D$. Determine the cosine of the angle that must form the chord $AB$ with $OM$ so that the sum of the areas of the triangles $AMC$ and $BMD$ be minimal.
1995 Balkan MO, 3
Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation \[ x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0 \] are natural numbers, none of which is a perfect square.
[i]Albania[/i]
1953 Moscow Mathematical Olympiad, 248
a) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}$
b) Solve the system $\begin{cases}
x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\
x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\
x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\
... \\
x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$
2002 HKIMO Preliminary Selection Contest, 6
Points $A$ and $B$ lie on a plane. A straight line passing through $A$ will divide the plane into 2 regions. A further straight line through $B$ will altogether divide the plane into 4 regions, and so on. If 1002 and 1000 straight lines are drawn passing through $A$ and $B$ respectively, what is the maximum number of regions formed?
2012 Baltic Way, 17
Let $d(n)$ denote the number of positive divisors of $n$. Find all triples $(n,k,p)$, where $n$ and $k$ are positive integers and $p$ is a prime number, such that
\[n^{d(n)} - 1 = p^k.\]
2007 Balkan MO Shortlist, G4
Points $M,N$ and $P$ on the sides $BC, CA$ and $AB$ of $\vartriangle ABC$ are such that $\vartriangle MNP$ is acute. Denote by $h$ and $H$ the lengths of the shortest altitude of $\vartriangle ABC$ and the longest altitude of $\vartriangle MNP$. Prove that $h \le 2H$.
2012 AMC 10, 17
Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2005 Estonia National Olympiad, 1
Punches in the buses of a certain bus company always cut exactly six holes into the ticket. The possible locations of the holes form a $3 \times 3$ table as shown in the figure. Mr. Freerider wants to put together a collection of tickets such that, for any combination of punch holes, he would have a ticket with the same combination in his collection. The ticket can be viewed both from the front and from the back. Find the smallest number of tickets in such a collection.
[img]https://cdn.artofproblemsolving.com/attachments/b/b/de5f09317a9a109fbecccecdc033de18217806.png[/img]
2017 Harvard-MIT Mathematics Tournament, 26
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at $0$. Every minute, he has a $\frac{1}{3}$ chance of moving $1$ unit left, a $\frac{1}{3}$ chance of moving $1$ unit right, and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to $0$ (not including the start) before he gets eaten.
2015 Grand Duchy of Lithuania, 2
Let $\omega_1$ and $\omega_2$ be two circles , with respective centres $O_1$ and $O_2$ , that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\omega_2$ in $A$ and $C$ and the line $O_2A$ inetersects $\omega_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\omega_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.
2016 Kosovo National Mathematical Olympiad, 3
The distance from $A$ to $B$ is $408km$ . From $A$ in direction of $B$ move motorcyclist , and from $B$ in direction of $A$ move a bicyclist . If a motorcyclist start to move $2$ hours earlier then byciclist , then they will meet $7$ hours after bicyclist start to move . If a bicyclist start to move $2$ hours earlier then motorcyclist , then they will meet $8$ hours after after motorcyclist start to move . Find the velocity of motorcyclist and bicyclist if we now that the velocity of them was constant all the time .
2006 Bulgaria Team Selection Test, 1
[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$.
[i]Nikolai Nikolov[/i]
2013 IFYM, Sozopol, 2
Find the perimeter of the base of a regular triangular pyramid with volume 99 and apothem 6.
2015 Peru MO (ONEM), 4
Let $b$ be an odd positive integer. The sequence $a_1, a_2, a_3, a_4$, is definedin the next way: $a_1$ and $a_2$ are positive integers and for all $k \ge 2$,
$$a_{k+1}= \begin{cases} \frac{a_k + a_{k-1}}{2} \,\,\, if \,\,\, a_k + a_{k-1} \,\,\, is \,\,\, even \\ \frac{a_k + a_{k-1+b}}{2}\,\,\, if \,\,\, a_k + a_{k-1}\,\,\, is \,\,\,odd\end{cases}$$
a) Prove that if $b = 1$, then after a certain term, the sequence will become constant.
b) For each $b \ge 3$ (odd), prove that there exist values of $a_1$ and $a_2$ for which the sequence will become constant after a certain term.
1986 AMC 12/AHSME, 30
The number of real solutions $(x,y,z,w)$ of the simultaneous equations \[2y = x + \frac{17}{x},\quad 2z = y + \frac{17}{y},\quad 2w = z + \frac{17}{z},\quad 2x = w + \frac{17}{w}\] is
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 16 $
2009 Moldova Team Selection Test, 2
[color=darkblue]Let $ M$ be a set of aritmetic progressions with integer terms and ratio bigger than $ 1$.
[b]a)[/b] Prove that the set of the integers $ \mathbb{Z}$ can be written as union of the finite number of the progessions from $ M$ with different ratios.
[b]b)[/b] Prove that the set of the integers $ \mathbb{Z}$ can not be written as union of the finite number of the progessions from $ M$ with ratios integer numbers, any two of them coprime.[/color]
2020 AMC 12/AHSME, 17
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 13$
2017 IMC, 9
Define the sequence $f_1,f_2,\ldots :[0,1)\to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on $(0,1)$}, \quad \text{and}\quad f_{n+1}(0)=1. $$
Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.
2005 France Team Selection Test, 6
Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$.
Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.
2004 IMC, 4
For $n\geq 1$ let $M$ be an $n\times n$ complex array with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$, with multiplicities $m_1,m_2,\ldots,m_k$ respectively. Consider the linear operator $L_M$ defined by $L_MX=MX+XM^T$, for any complex $n\times n$ array $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose matrix of $M$).
2011 National Olympiad First Round, 29
A circle passing through $B$ and $C$ meets the side $[AB]$ of $\triangle ABC$ at $D$, and $[AC]$ at $E$. The circumcircle of $\triangle ACD$ intersects with $BE$ at a point $F$ outside $[BE]$. If $|AD| = 4, |BD|= 8$, then what is $|AF|$?
$\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2\sqrt6 \qquad\textbf{(C)}\ 4\sqrt6 \qquad\textbf{(D)}\ \sqrt6 \qquad\textbf{(E)}\ \text{None}$
2021 Harvard-MIT Mathematics Tournament., 5
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AEF]}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
2023 CMIMC Combo/CS, 5
A BWM tree is defined recursively:
[list]
[*] An empty tree is a BWM tree of height 0 and size 0.
[*] A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
[*] The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.
[/list]
What is the minimum size of a height-10 BWM tree?
[i]Proposed by Jacob Weiner[/i]