Found problems: 1148
2007 Purple Comet Problems, 2
A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$.
PEN S Problems, 4
If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.
2013 Stars Of Mathematics, 3
Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$.
i) Prove there exist infinitely many primes, none dividing any term of the sequence.
ii) Prove there exist infinitely many primes, each dividing some term of the sequence.
[i](Dan Schwarz)[/i]
1957 AMC 12/AHSME, 10
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad \\
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad \\
\textbf{(E)}\ \text{highest point at } {(\minus{}1,1)}$
1995 Cono Sur Olympiad, 3
Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$.
1. Find the distance between the centers of the circles(using $a$ and $b$).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
1985 AIME Problems, 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
2011 Middle European Mathematical Olympiad, 1
Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that
\[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]
2001 China Western Mathematical Olympiad, 1
Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.
1993 India National Olympiad, 2
Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.
2006 MOP Homework, 3
Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.
2011 ELMO Shortlist, 4
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
1991 Hungary-Israel Binational, 4
Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.
1999 Harvard-MIT Mathematics Tournament, 5
Let $f(x)=x+\cfrac{1}{2x+\cfrac{1}{2x+\cfrac{1}{2x+\cdots}}}$. Find $f(99)f^\prime (99)$.
1996 AMC 12/AHSME, 25
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$
1958 AMC 12/AHSME, 33
For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows:
$ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad
\textbf{(B)}\ 2b^2 \equal{} 9ac\qquad
\textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\
\textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad
\textbf{(E)}\ 9b^2 \equal{} 2ac$
1951 AMC 12/AHSME, 26
In the equation $ \frac {x(x \minus{} 1) \minus{} (m \plus{} 1)}{(x \minus{} 1)(m \minus{} 1)} \equal{} \frac {x}{m}$ the roots are equal when
$ \textbf{(A)}\ m \equal{} 1 \qquad\textbf{(B)}\ m \equal{} \frac {1}{2} \qquad\textbf{(C)}\ m \equal{} 0 \qquad\textbf{(D)}\ m \equal{} \minus{} 1 \qquad\textbf{(E)}\ m \equal{} \minus{} \frac {1}{2}$
1985 USAMO, 2
Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.
1995 South africa National Olympiad, 1
Prove that there are no integers $m$ and $n$ such that
\[19m^2+95mn+2000n^2=1995.\]
2004 National Olympiad First Round, 6
For which of the following value of $n$, there exists integers $a,b$ such that $a^2 + ab-6b^2 = n$?
$
\textbf{(A)}\ 17
\qquad\textbf{(B)}\ 19
\qquad\textbf{(C)}\ 29
\qquad\textbf{(D)}\ 31
\qquad\textbf{(E)}\ 37
$
2012 USA TSTST, 6
Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]
1992 India Regional Mathematical Olympiad, 6
Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]
2008 Harvard-MIT Mathematics Tournament, 6
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
2008 Saint Petersburg Mathematical Olympiad, 1
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
2015 Saint Petersburg Mathematical Olympiad, 1
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
1991 Federal Competition For Advanced Students, 3
Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$