Found problems: 85335
2000 National Olympiad First Round, 24
Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. What is the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$?
$ \textbf{(A)}\ \frac{\sqrt 2}2
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ \sqrt 2
\qquad\textbf{(D)}\ \frac32
\qquad\textbf{(E)}\ 2
$
2008 Greece Junior Math Olympiad, 2
If $x,y,z$ are positive real numbers with $x^2+y^2+z^2=3$, prove that
$\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3$
2003 Romania National Olympiad, 2
In a meeting there are 6 participants. It is known that among them there are seven pairs of friends and in any group of three persons there are at least two friends. Prove that:
(a) there exists a person who has at least three friends;
(b) there exists three persons who are friends with each other.
[i]Valentin Vornicu[/i]
2009 Today's Calculation Of Integral, 420
Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane.
(1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis.
(2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.
2020 Purple Comet Problems, 13
There are relatively prime positive integers $s$ and $t$ such that $$\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}$$ Find $s + t$.
1991 IMO Shortlist, 27
Determine the maximum value of the sum
\[ \sum_{i < j} x_ix_j (x_i \plus{} x_j)
\]
over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$
2005 Miklós Schweitzer, 12
Let $x_1,x_2,\cdots,x_n$ be iid rv. $S_n=\sum x_k$
(a) let $P(|x_1|\leq 1)=1$ , $E[x_1]=0$ , $E[x_1^2]=\sigma^2>0$
Prove that $\exists C>0$ , $\forall u\geq 2n\sigma^2$
$P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}$
(b) let $P(x_1=1)=P(x_1=-1)=\sigma^2/2$ , $P(x_1=0)=1-\sigma^2$
Prove that $\exists B_1<1,B_2>1,B_3>0$ , $\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2$
$P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}$
2012 Online Math Open Problems, 29
How many positive integers $a$ with $a\le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?
[i]Author: Ray Li[/i]
2021 IMO Shortlist, G6
Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)
2006 Harvard-MIT Mathematics Tournament, 7
Find all positive real numbers $c$ such that the graph of $f\text{ : }\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3-cx$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.
STEMS 2021 Phy Cat C, Q2
[b]Little Mario and the Cylindrical Beam[/b]
Little Mario wishes to jump over a very long (practically infinite) cylindrical beam of radius $r$ whose axis is at a height $h$ from the ground. With what minimum initial speed must he launch himself if:
[list=1]
[*] Mario is allowed to touch the beam (neglect frictional effects)? [/*]
[*] Mario is not allowed to touch the beam? [/*]
[/list]
Approximate Little Mario by a point particle for convenience. Acceleration due to gravity is $g$.
2022 Iran Team Selection Test, 11
Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.)
Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli
1992 Vietnam National Olympiad, 3
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.
2022 Princeton University Math Competition, B2
The [i]base factorial[/i] number system is a unique representation for positive integers where the $n$th digit from the right ranges from $0$ to $n$ inclusive and has place value $n!$ for all $n \ge 1.$ For instance, $71$ can be written in base factorial as $2321_{!} = 2 \cdot 4! + 3 \cdot 3! + 2 \cdot 2! + 1 \cdot 1!.$ Let $S_{!}(n)$ be the base $10$ sum of the digits of $n$ when $n$ is written in base factorial. Compute $\sum_{n=1}^{700} S_{!}(n)$ (expressed in base $10$).
2020-21 KVS IOQM India, 3
Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?
2001 SNSB Admission, 6
There are $ n\ge 1 $ ordered bulbs controlled by $ n $ ordered switches such that the $ k\text{-th} $ switch controls the $ k\text{-th} $ bulb and also the $ j\text{-th} $ bulb if and only if the $ j\text{-th} $ switch controls the $ k\text{-th} $ bulb, for any $ 1\le k,j\le n. $ If all bulbs are off, show that it can be chosen some switches such that, if pushed simmultaneously, the bulbs turn all on.
LMT Team Rounds 2021+, 8
The $53$-digit number
$$37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719$$
can be expressed as $n^21$ where $n$ is a positive integer. Find $n$.
2022 Sharygin Geometry Olympiad, 9.2
Let circles $s_1$ and $s_2$ meet at points $A$ and $B$. Consider all lines passing through $A$ and meeting the circles for the second time at points $P_1$ and $P_2$ respectively. Construct by a compass and a ruler a line such that $AP_1.AP_2$ is maximal.
2018 Online Math Open Problems, 11
Let an ordered pair of positive integers $(m, n)$ be called [i]regimented[/i] if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016}-N$.
[i]Proposed by Ashwin Sah[/i]
2004 National High School Mathematics League, 3
The solution set to the inequality $\sqrt{\log_2 x-1}+\frac{1}{2}\log_{\frac{1}{2}}x^3+2>0$ is
$\text{(A)}[2,3)\qquad\text{(B)}(2,3]\qquad\text{(C)}[2,4)\qquad\text{(D)}(2,4]$
1956 Moscow Mathematical Olympiad, 340
a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
1958 Polish MO Finals, 3
Prove that if $ n $ is a natural number greater than $ 1 $, then
$$
\cos \frac{2\pi}{n} + \cos \frac{4\pi}{n} + \cos \frac{6\pi}{n} + \ldots + \cos \frac{2n \pi}{n} = 0.$$
2009 Cuba MO, 1
Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.
2020 Jozsef Wildt International Math Competition, W54
Consider two parallel lines $a$ and $b$.
The circles $C,C_1$ are tangent to each other and to the line $a$.
The circles $C,C_2$ are tangent to each other and to the line $b$.
The circles $C_1,C_2$ are tangent to each other, have radii $R_1=9,R_2=16$.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy82L2E1Y2NjMTY0NTNjMzExYmRmOTZjYmZlMWMwNzE4YmNlM2I0YTNkLnBuZw==&rn=Sm9zZXBoIDIwMjAucG5n[/img]
What is the radius $R$ of the circle $C$?
[i]Proposed by Ovidiu Bagdasar[/i]
1967 AMC 12/AHSME, 30
A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is:
$\textbf{(A)}\ 18\qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 15\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 11$