Found problems: 85335
2014 Brazil Team Selection Test, 1
Let $n$ be a positive integer. A [i]partition [/i] of $n$ is a multiset (set with repeated elements) whose sum of elements is $n$. For example, the partitions of $3$ are $\{1, 1, 1\}, \{1, 2\}$ and $\{3\}$. Each partition of $n$ is written as a non-descending sequence. For example, for $n = 3$, the list is $(1, 1, 1)$, $(1, 2)$ and $(3)$. For each sequence $x = (x_1, x_2, ..., x_k)$, define $f(x)=\prod_{i=1}^{k-1} {x_{i+1} \choose x_ i}$ . Furthermore, the $f$ of partition $\{n\}$ is $f((n)) = 1$. Prove that the sum of all $f$'s in the list is $2^{n-1}.$
2022 VJIMC, 3
Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit
$$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$
2018 AMC 12/AHSME, 21
Which of the following polynomials has the greatest real root?
$\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 $
2018 IFYM, Sozopol, 6
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of $S$.
2022 Cono Sur, 6
On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$.
Determine the largest possible value of $k$ for which it is possible to do that coloring.
1981 Putnam, B5
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether
$$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$
is a rational number.
2002 AMC 12/AHSME, 21
For all positive integers $ n$ less than $ 2002$, let
\[ a_n \equal{} \begin{cases} 11 & \text{if }n\text{ is divisible by }13\text{ and }14 \\
13 & \text{if }n\text{ is divisible by }11\text{ and }14 \\
14 & \text{if }n\text{ is divisible by }11\text{ and }13 \\
0 & \text{otherwise} \end{cases}
\]Calculate $ \sum_{n \equal{} 1}^{2001} a_n$.
$ \textbf{(A)}\ 448 \qquad \textbf{(B)}\ 486 \qquad \textbf{(C)}\ 1560 \qquad \textbf{(D)}\ 2001 \qquad \textbf{(E)}\ 2002$
2019 IMO Shortlist, C9
For any two different real numbers $x$ and $y$, we define $D(x,y)$ to be the unique integer $d$ satisfying $2^d\le |x-y| < 2^{d+1}$. Given a set of reals $\mathcal F$, and an element $x\in \mathcal F$, we say that the [i]scales[/i] of $x$ in $\mathcal F$ are the values of $D(x,y)$ for $y\in\mathcal F$ with $x\neq y$. Let $k$ be a given positive integer.
Suppose that each member $x$ of $\mathcal F$ has at most $k$ different scales in $\mathcal F$ (note that these scales may depend on $x$). What is the maximum possible size of $\mathcal F$?
2018 Hanoi Open Mathematics Competitions, 3
The lines $\ell_1$ and \ell_2 are parallel. The points $A_1,A_2, ...,A_7$ are on $\ell_1$ and the points $B_1,B_2,...,B_8$ are on $\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure).
The [b]greatest number[/b] of intersection points is
[img]https://cdn.artofproblemsolving.com/attachments/4/9/92153dce5a48fcba0f5175d67e0750b7980e84.png[/img]
A. $580$ B. $585$ C. $588$ D. $590$ E. $593$
2019 Bosnia and Herzegovina Junior BMO TST, 3
$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if :
$a)$ $k=9$?
$b)$ $k=8$?
2017 Taiwan TST Round 3, 2
$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of
\[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]
2010 China Western Mathematical Olympiad, 3
Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.
1976 Chisinau City MO, 123
Five points are given on the plane. Prove that among all the triangles with vertices at these points there are no more than seven acute-angled ones.
2013 BMT Spring, 2
If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?
2017 CMIMC Individual Finals, 1
Let $\tau(n)$ denote the number of positive integer divisors of $n$. For example, $\tau(4) = 3$. Find the sum of all positive integers $n$ such that $2 \tau(n) = n$.
PEN P Problems, 17
Let $p$ be a prime number of the form $4k+1$. Suppose that $r$ is a quadratic residue of $p$ and that $s$ is a quadratic nonresidue of $p$. Show that $p=a^{2}+b^{2}$, where \[a=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-r)}{p}\right), b=\frac{1}{2}\sum^{p-1}_{i=1}\left( \frac{i(i^{2}-s)}{p}\right).\] Here, $\left( \frac{k}{p}\right)$ denotes the Legendre Symbol.
1995 Rioplatense Mathematical Olympiad, Level 3, 3
Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
$\bullet$ those that contain the faces of the tetrahedron, and
$\bullet$ each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.
2009 China Team Selection Test, 1
Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$
2011 AMC 10, 20
Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
$ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad
\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad
\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad
\textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad
\textbf{(E)}\ 2 $
1994 Bulgaria National Olympiad, 5
Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$.
Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.
1987 IMO Longlists, 51
The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.
2024 Mexico National Olympiad, 4
Let $ABC$ an acute triangle with orthocenter $H$. Let $M$ be a point on segment $BC$. The line through $M$ and perpendicular to $BC$ intersects lines $BH$ and $CH$ in points $P$ and $Q$ respectively. Prove that the orthocenter of triangle $HPQ$ lies on the line $AM$.
2012 Singapore Senior Math Olympiad, 3
If $46$ squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least $3$ of the squares are colored red.
2004 Tournament Of Towns, 2
What is the maximal number of checkers that can be placed on an $8\times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
2023-24 IOQM India, 27
A quadruple $(a,b,c,d)$ of distinct integers is said to be $balanced$ if $a+c=b+d$. Let $\mathcal{S}$ be any set of quadruples $(a,b,c,d)$ where $1 \leqslant a<b<d<c \leqslant 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}.$