Found problems: 85335
2021 Durer Math Competition Finals, 16
Consider a table consisting of $2\times 7$ squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls?
[i]The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed.[/i]
[img]https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png[/img]
1991 AMC 12/AHSME, 24
The graph, $G$ of $y = \log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
$ \textbf{(A)}\ y = \log_{10}\left(\frac{x + 90}{9}\right)\qquad\textbf{(B)}\ y = \log_{x}10\qquad\textbf{(C)}\ y = \frac{1}{x + 1}\qquad\textbf{(D)}\ y = 10^{-x}\qquad\textbf{(E)}\ y = 10^{x} $
2013 India IMO Training Camp, 2
In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.
2019 AMC 10, 24
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$
2007 F = Ma, 22
Two rockets are in space in a negligible gravitational field. All observations are made by an observer in a reference frame in which both rockets are initially at rest. The masses of the rockets are $m$ and $9m$. A constant force $F$ acts on the rocket of mass m for a distance $d$. As a result, the rocket acquires a momentum $p$. If the same constant force $F$ acts on the rocket of mass $9m$ for the same distance $d$, how much momentum does the rocket of mass $9m$ acquire?
$ \textbf{(A)}\ p/9 \qquad\textbf{(B)}\ p/3 \qquad\textbf{(C)}\ p \qquad\textbf{(D)}\ 3p \qquad\textbf{(E)}\ 9p $
2018 Chile National Olympiad, 5
Consider the set $\Omega$ formed by the first twenty natural numbers, $\Omega = \{1, 2, . . . , 20\}$ . A nonempty subset $A$ of $\Omega$ is said to be [i]sumfree [/i ] if for all pair of elements$ x, y \in A$, the sum $(x + y)$ is not in $A$, ( $x$ can be equal to $y$). Prove that $\Omega$ has at least $2018$ sumfree subsets.
2012 IFYM, Sozopol, 1
A ticket for the tram costs 1 leva. On the queue in front of the ticket seller are standing $n$ people with a banknote of 1 leva and $m$ people with a banknote of 2 leva. The ticket seller has no money in his cash deck so he can only sell a ticket to a buyer with a banknote of 2 leva, if he has at least 1 banknote of 1 leva.
Determine the probability that the ticket seller could sell tickets to all of the people standing in the queue.
1998 Korea - Final Round, 1
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.
2021 Moldova Team Selection Test, 10
On a board there are written the integers from $1$ to $119$. Two players, $A$ and $B$, make a move by turn. A $move$ consists in erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins and his score is equal with the positive difference of the two remaining numbers. The player $A$ makes the first move. Find the highest integer $k$, such that the player $A$ can be sure that his score is not smaller than $k$.
2008 ITest, 37
A triangle has sides of length $48$, $55$, and $73$. Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the length of the shortest altitude of the triangle. Find the value of $a+b$.
2023 MOAA, 18
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$.
[i]Proposed by Anthony Yang and Andy Xu[/i]
PEN H Problems, 24
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.
2017 Lusophon Mathematical Olympiad, 3
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
1971 IMO Shortlist, 14
A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$
1966 IMO Shortlist, 61
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
2010 AMC 12/AHSME, 12
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in the swamp, and they make the following statements:
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
2008 Germany Team Selection Test, 3
Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$
2007 Baltic Way, 8
Call a set $A$ of integers [i]non-isolated[/i], if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.
2013 Today's Calculation Of Integral, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
2004 Unirea, 2
Find the maximum value of the expression $ x+y+z, $ where $ x,y,z $ are real numbers satisfying
$$ \left\{ \begin{matrix} x^2+yz\le 2 \\y^2+zx\le 2\\ z^2+xy\le 2 \end{matrix} \right. . $$
2009 Indonesia TST, 2
Let $ ABC$ be a triangle with $ \angle BAC\equal{}60^{\circ}$. The incircle of $ ABC$ is tangent to $ AB$ at $ D$. Construct a circle with radius $ DA$ and cut the incircle of $ ABC$ at $ E$. If $ AF$ is an altitude, prove that $ AE\ge AF$.
2009 Oral Moscow Geometry Olympiad, 5
Prove that any convex polyhedron has three edges that can be used to form a triangle.
(Barbu Bercanu, Romania)
2021 Romanian Master of Mathematics Shortlist, C1
Determine the largest integer $n\geq 3$ for which the edges of the complete graph on $n$ vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
2014 Grand Duchy of Lithuania, 1
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
1969 IMO Shortlist, 2
$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$
$(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$
$(c)$ Find the locus of the centers of these hyperbolas.
$(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$