Found problems: 85335
2016 CMIMC, 7
There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way to their horse, the mounted horse shouts angrily at the person, who then scurries home immediately. Otherwise, they get to their horse safely and mount it. The expected number of people who have scurried home after all eight people have attempted to reach their horse can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.
2019 HMNT, 9
For dinner, Priya is eating grilled pineapple spears. Each spear is in the shape of the quadrilateral $PINE$, with $PI = 6$ cm, $IN = 15$ cm, $NE = 6$ cm, $EP = 25$ cm, and $\angle NEP + \angle EPI = 60^o$: What is the area of each spear, in cm$^2$?
2014 Moldova Team Selection Test, 1
Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.
2020 Online Math Open Problems, 15
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)
[i]Proposed by Ankit Bisain[/i]
2013 Austria Beginners' Competition, 2
The following figure is given:
[img]https://cdn.artofproblemsolving.com/attachments/9/b/97a30e248fcd6f098a900c89721a2e1b3b3f0e.png[/img]
Determine the number of paths from the starting square $A$ to the target square $Z$, where a path consists of steps from a square to its top or right neighbor square .
(W. Janous, WRG Ursulinen, Innsbruck)
1962 Putnam, B6
Let
$$f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,$$
where $a_k$ and $b_k$ are constants. Show that if $|f(x)| \leq 1$ for $x \in [0, 2 \pi]$ and there exist $0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi$ with $|f(x_i )|=1,$ then $f(x)= \cos(nx +a)$ for some constant $a.$
2019 BMT Spring, 9
Let $ a_n $ be the product of the complex roots of $ x^{2n} = 1 $ that are in the first quadrant of the complex plane. That is, roots of the form $ a + bi $ where $ a, b > 0 $. Let $ r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} $. Find the smallest integer $ k $ such that $ r $ is a root of $ x^k = 1 $.
2006 Mexico National Olympiad, 5
Let $ABC$ be an acute triangle , with altitudes $AD,BE$ and $CF$. Circle of diameter $AD$ intersects the sides $AB,AC$ in $M,N$ respevtively. Let $P,Q$ be the intersection points of $AD$ with $EF$ and $MN$ respectively. Show that $Q$ is the midpoint of $PD$.
2015 Sharygin Geometry Olympiad, P4
In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.
Kyiv City MO 1984-93 - geometry, 1987.9.4
Inscribe a triangle in a given circle, if its smallest side is known, as well as the point of intersection of altitudes lying outside the circle.
2017 IMO Shortlist, C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
2017 QEDMO 15th, 8
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?
2009 Tournament Of Towns, 4
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
[i](5 points for Juniors and 4 points for Seniors)[/i]
2020 Ukrainian Geometry Olympiad - December, 3
Given convex $1000$-gon. Inside this polygon, $1020$ points are chosen so that no $3$ of the $2020$ points do not lie on one line. Polygon is cut into triangles so that these triangles have vertices only those specified $2020$ points and each of these points is the vertex of at least one of cutting triangles. How many such triangles were formed?
2010 Contests, 1
Solve in the integers the diophantine equation
$$x^4-6x^2+1 = 7 \cdot 2^y.$$
PEN N Problems, 14
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
1999 AMC 8, 14
In trapezoid $ABCD$ , the sides $AB$ and $CD$ are equal. The perimeter of $ABCD$ is
[asy]
draw((0,0)--(4,3)--(12,3)--(16,0)--cycle);
draw((4,3)--(4,0),dashed);
draw((3.2,0)--(3.2,.8)--(4,.8));
label("$A$",(0,0),SW);
label("$B$",(4,3),NW);
label("$C$",(12,3),NE);
label("$D$",(16,0),SE);
label("$8$",(8,3),N);
label("$16$",(8,0),S);
label("$3$",(4,1.5),E);[/asy]
$ \text{(A)}\ 27\qquad\text{(B)}\ 30\qquad\text{(C)}\ 32\qquad\text{(D)}\ 34\qquad\text{(E)}\ 48 $
2007 iTest Tournament of Champions, 1
Find the remainder when $3^{2007}$ is divided by $2007$.
1992 IMTS, 5
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$) and denote its area by $\triangle '$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle ''$. Given that $\triangle ' = 30$ and $\triangle '' = 20$, find $\triangle$.
2023 Putnam, B5
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $\pi(\pi(k)) \equiv m k(\bmod n)$ for all $k \in\{1,2, \ldots, n\}$.
2012 Turkey Team Selection Test, 3
Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?
2003 China Western Mathematical Olympiad, 4
Given that the sum of the distances from point $ P$ in the interior of a convex quadrilateral $ ABCD$ to the sides $ AB, BC, CD, DA$ is a constant, prove that $ ABCD$ is a parallelogram.
2012 Iran MO (3rd Round), 1
We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?
[i]Proposed by Morteza Saghafian[/i]
2019 Thailand Mathematical Olympiad, 8
Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle.
Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$.
Let the circle with diameter $AI$ meets $\omega$ again at $K$.
If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.