Found problems: 85335
2016 Fall CHMMC, 3
A gambler offers you a $2$ dollar ticket to play the following game: First, you pick a real number $0 \leq p \leq 1$, then you are given a weighted coin that comes up heads with probability $p$. If you receive $1$ dollar the [i]first[/i] time you flip a tail, and if you receive $2$ dollars [i]first[/i] time you flip a head, what is the optimal expected net winning of flipping the coin twice?
2014 Harvard-MIT Mathematics Tournament, 10
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.
2014 Contests, 2
Find the least natural number $n$, which has at least 6 different divisors
$1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.
2012 NIMO Problems, 1
Compute the largest integer $N \le 2012$ with four distinct digits.
[i]Proposed by Evan Chen[/i]
2002 Italy TST, 3
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that
$(\text{i})$ $x$ and $y$ are relatively prime;
$(\text{ii})$ $x$ divides $y^2+m;$
$(\text{iii})$ $y$ divides $x^2+m.$
Fractal Edition 2, P1
Viorel claims that for any natural number $n$ greater than $2024$, the number $2024^n + 1$ is prime. Is Viorel's statement true?
2018 BMT Spring, Tie 1
Every face of a cube is colored one of $3$ colors at random. What is the expected number of edges that lie along two faces of different colors?
1988 China Team Selection Test, 2
Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.
1993 AMC 12/AHSME, 5
Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is
$ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $
2001 China Team Selection Test, 2
Find the largest positive real number \( c \) such that for any positive integer \( n \), satisfies \(\{ \sqrt{7n} \} \geq \frac{c}{\sqrt{7n}}\).
2017 Saudi Arabia IMO TST, 3
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
2022 District Olympiad, P3
$a)$ Solve over the positive integers $3^x=x+2.$
$b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.
1987 IMO Longlists, 12
Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers?
[i]Proposed by Finland.[/i]
1997 IMO Shortlist, 7
The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that:
\[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}.
\]
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2012 Princeton University Math Competition, B2
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
2001 AIME Problems, 5
A set of positive numbers has the $\text{triangle property}$ if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n$?
1991 National High School Mathematics League, 14
$O$ is the vertex of a parabola, $F$ is its focus. $PQ$ is a chord of the parabola. If $|OF|=a,|PQ|=b$, find the area of $\triangle OPQ$.
2017 Online Math Open Problems, 22
Given a sequence of positive integers $a_1, a_2, a_3, \dots, a_{n}$, define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let $b_1, b_2, b_3, \dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 2017 inclusive, \[f(a_1, a_2, a_3, \dots, a_i, \dots, a_{2017})\equiv f(a_1, a_2, a_3, \dots, a_i+b_i, \dots, a_{2017}) \pmod{2017}\] for all sequences $a_1, a_2, a_3, \dots, a_{2017}$ of positive integers greater than 2017. Find the smallest possible value of $b_1+b_2+b_3+\dots+b_{2017}$.
[i]Proposed by Yannick Yao
2019 AIME Problems, 2
Lily pads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly and independently with probability $\tfrac12$. The probability that the frog visits pad $7$ is $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
1988 Tournament Of Towns, (199) 2
Prove that $a^2pq + b^2qr + c^2rp \le 0$, whenever $a, b$ and $c$ are the lengths of the sides of a triangle and $p + q + r = 0$ .
( J. Mustafaev , year 12 student, Baku)
2021 Yasinsky Geometry Olympiad, 2
Given a rectangle $ABCD$, which is located on the line $\ell$ They want it "turn over" by first turning around the vertex $D$, and then as point $C$ appears on the line $\ell$ - by making a turn around the vertex $C$ (see figure). What is the length of the curve along which the vertex $A$ is moving , at such movement, if $AB = 30$ cm, $BC = 40$ cm?
(Alexey Panasenko)
[img]https://cdn.artofproblemsolving.com/attachments/d/9/3cca36b08771b1897e385d43399022049bbcde.png[/img]
2006 MOP Homework, 1
$ ABC$ is an acute triangle. The points $ B'$ and $ C'$are the reflections
of $ B$ and $ C$ across the lines $ AC$ and $ AB$ respectively. Suppose
that the circumcircles of triangles$ ABB$' and $ ACC'$ meet at $ A$
and $ P$. Prove that the line $ PA$ passes through the circumcenter
of triangle$ ABC.$
2002 IMC, 1
A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis.
Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.
2007 Putnam, 3
Let $ k$ be a positive integer. Suppose that the integers $ 1,2,3,\dots,3k \plus{} 1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $ 3$ ? Your answer should be in closed form, but may include factorials.