Found problems: 85335
2018 Taiwan TST Round 2, 2
There are $n$ sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep $A$:
(a) If a friend of $A$ is originally a friend of the wolf, it un-friends the wolf.
(b) If a friend of $A$ is originally not a friend of the wolf, it becomes a friend of the wolf.
Repeat the procedure until the wolf has no friend left.
Find the largest integer $m$ in terms of $n$ satisfying the following: There exists an initial friendsheep structure such that the wolf has $m$ different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.
1950 Polish MO Finals, 6
Prove that if a natural number $n$ is greater than $4$ and is not a prime number, then the produxt of the consecutive natural numbers from $1$ to $n-1$ is divisible by $ n$.
2018 Iran MO (3rd Round), 3
Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that:
i)each number appears at least once
ii)In each row or column,there are no two numbers having difference 0 or 1
1997 Rioplatense Mathematical Olympiad, Level 3, 5
Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$.
Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$.
1979 Putnam, A3
Let $x_1,x_2,x_3, \dots$ be a sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.$$ Establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n.$
2024 Brazil Team Selection Test, 1
Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps.
[list=1]
[*]select a $2\times 2$ square in the grid;
[*]flip the coins in the top-left and bottom-right unit squares;
[*]flip the coin in either the top-right or bottom-left unit square.
[/list]
Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves.
[i]Thanasin Nampaisarn, Thailand[/i]
1912 Eotvos Mathematical Competition, 2
Prove that for every positive integer $n$, the number $A_n = 5^n + 2 \cdot 3^{n-1} + 1$ is a multiple of $8$.
2008 Purple Comet Problems, 18
The diagram below contains eight line segments, all the same length. Each of the angles formed by the intersections of two segments is either a right angle or a $45$ degree angle. If the outside square has area $1000$, find the largest integer less than or equal to the area of the inside square.
[asy]
size(130);
real r = sqrt(2)/2;
defaultpen(linewidth(0.8));
draw(unitsquare^^(r,0)--(0,r)^^(1-r,0)--(1,r)^^(r,1)--(0,1-r)^^(1-r,1)--(1,1-r));
[/asy]
2020 Romanian Master of Mathematics, 3
Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$.
Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
2002 Moldova National Olympiad, 2
Five parcels of land are given. In each step, we divide one parcel into three or four smaller ones. Assume that, after several steps, the number of obtained parcels equals four times the number of steps made. How many steps were performed?
2015 IFYM, Sozopol, 8
A cross with length $p$ (or [i]p-cross[/i] for short) will be called the figure formed by a unit square and 4 rectangles $p-1$ x $1$ on its sides. What’s the least amount of colors one has to use to color the cells of an infinite table, so that each [i]p-cross[/i] on it covers cells, no two of which are in the same color?
2015 Poland - Second Round, 3
Let $ABC$ be a triangle. Let $K$ be a midpoint of $BC$ and $M$ be a point on the segment $AB$. $L=KM \cap AC$ and $C$ lies on the segment $AC$ between $A$ and $L$. Let $N$ be a midpoint of $ML$. $AN$ cuts circumcircle of $\Delta ABC$ in $S$ and $S \neq N$. Prove that circumcircle of $\Delta KSN$ is tangent to $BC$.
2016 ASDAN Math Tournament, 11
Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?
2018 India IMO Training Camp, 1
Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$ $$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$
Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.
2007 USA Team Selection Test, 1
Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.
2022 BMT, 10
Each box in the equation
$$\square \times \square \times \square - \square \times \square \times \square = 9$$
is filled in with a different number in the list 2, $3, 4, 5, 6, 7, 8$ so that the equation is true. Which number in the list is not used to fill in a box?
1978 AMC 12/AHSME, 15
If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is
$\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$
$\textbf{(E) }\text{not completely determined by the given information}$
2006 Junior Balkan Team Selection Tests - Romania, 1
Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.
2007 ITest, 8
Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of $50$ miles per hour, and Joe can run at a speed of $10$ miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?
$\textbf{(A) }7.2\hspace{14em}\textbf{(B) }14.4\hspace{14em}\textbf{(C) }36$
$\textbf{(D) }10\hspace{14.3em}\textbf{(E) }12\hspace{14.8em}\textbf{(F) }2.4$
$\textbf{(G) }25.2\hspace{13.5em}\textbf{(H) }123456789$
2023 Thailand Online MO, 3
Let $a$ and $n$ be positive integers such that the greatest common divisor of $a$ and $n!$ is $1$. Prove that $n!$ divides $a^{n!}-1$.
2019-IMOC, A3
Find all $3$-tuples of positive reals $(a,b,c)$ such that
$$\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}$$
2014 National Olympiad First Round, 15
What is the sum of distinct real numbers $x$ such that $(2x^2+5x+9)^2=56(x^3+1)$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ \dfrac{7}{4}
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \dfrac{9}{2}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1999 AMC 8, 5
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
$ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $
2004 India IMO Training Camp, 3
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
2017 Saudi Arabia JBMO TST, 4
Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$.
1. Prove that $CK = BH$.
2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.