Found problems: 122
2013 NIMO Problems, 11
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]
1999 Romania Team Selection Test, 7
Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that
\[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \]
Give an example of two such progressions having at least five terms.
[i]Mihai Baluna[/i]
2008 Harvard-MIT Mathematics Tournament, 11
Let $ f(r) \equal{} \sum_{j \equal{} 2}^{2008} \frac {1}{j^r} \equal{} \frac {1}{2^r} \plus{} \frac {1}{3^r} \plus{} \dots \plus{} \frac {1}{2008^r}$. Find $ \sum_{k \equal{} 2}^{\infty} f(k)$.
2013 Online Math Open Problems, 19
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]
1985 National High School Mathematics League, 7
In $\triangle ABC$, if $A,B,C$ are geometric series, and $b^2-a^2=ac$, then $B=$________.
2008 AIME Problems, 9
Ten identical crates each of dimensions $ 3$ ft $ \times$ $ 4$ ft $ \times$ $ 6$ ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $ \frac{m}{n}$ be the probability that the stack of crates is exactly $ 41$ ft tall, where $ m$ and $ n$ are relatively prime positive integers. Find $ m$.
2010 Princeton University Math Competition, 8
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
2024 CCA Math Bonanza, T2
Echo the gecko starts on the point $(0, 0)$ in the 2D coordinate plane. Every minute, starting at the end of the first minute, he'll teleport $1$ unit up, left, right, or down with equal probability. Echo dies the moment he lands on a point that is more than $1$ unit away from the origin. The average number of minutes he'll live can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[i]Team #2[/i]
2013 Macedonian Team Selection Test, Problem 6
Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.
1989 IMO Longlists, 17
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and
\[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\]
Determine $ f \left( \frac{1}{7} \right).$
2004 AMC 10, 18
A sequence of three real numbers forms an arithmetic progression with a first term of $ 9$. If $ 2$ is added to the second term and $ 20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 36\qquad
\textbf{(D)}\ 49\qquad
\textbf{(E)}\ 81$
1959 AMC 12/AHSME, 39
Let $S$ be the sum of the first nine terms of the sequence \[x+a, x^2+2a, x^3+3a, \cdots.\]
Then $S$ equals:
$ \textbf{(A)}\ \frac{50a+x+x^8}{x+1} \qquad\textbf{(B)}\ 50a-\frac{x+x^{10}}{x-1}\qquad\textbf{(C)}\ \frac{x^9-1}{x+1}+45a\qquad$$\textbf{(D)}\ \frac{x^{10}-x}{x-1}+45a\qquad\textbf{(E)}\ \frac{x^{11}-x}{x-1}+45a$
2006 Stanford Mathematics Tournament, 6
The expression $16^n+4^n+1$ is equiavalent to the expression $(2^{p(n)}-1)/(2^{q(n)}-1)$ for all positive integers $n>1$ where $p(n)$ and $q(n)$ are functions and $\tfrac{p(n)}{q(n)}$ is constant. Find $p(2006)-q(2006)$.
Today's calculation of integrals, 851
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$
Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$
2016 AIME Problems, 1
For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.
2000 National High School Mathematics League, 4
Give positive numbers $p,q,a,b,c$, if $p,a,q$ is a geometric series, $p,b,c,q$ is an arithmetic sequence. Then, wich is true about the equation $bx^2-ax+c=0$?
$\text{(A)}$ It has no real roots.
$\text{(B)}$ It has two equal real roots.
$\text{(C)}$ It has two different real roots, and their product is positive.
$\text{(D)}$ It has two different real roots, and their product is negative.
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
2014 Bulgaria National Olympiad, 2
Find all functions $f: \mathbb{Q}^+ \to \mathbb{R}^+ $ with the property:
\[f(xy)=f(x+y)(f(x)+f(y)) \,,\, \forall x,y \in \mathbb{Q}^+\]
[i]Proposed by Nikolay Nikolov[/i]
1956 Putnam, A1
Evaluate
$$ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x},$$
where $a>0$ and $a\ne 1.$
2010 Princeton University Math Competition, 7
The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
2013 NIMO Problems, 4
The infinite geometric series of positive reals $a_1, a_2, \dots$ satisfies
\[ 1 = \sum_{n=1}^\infty a_n = -\frac{1}{2013} + \sum_{n=1}^{\infty} \text{GM}(a_1, a_2, \dots, a_n) = \frac{1}{N} + a_1 \]
where $\text{GM}(x_1, x_2, \dots, x_k) = \sqrt[k]{x_1x_2\cdots x_k}$ denotes the geometric mean. Compute $N$.
[i]Proposed by Aaron Lin[/i]
1990 IMO Longlists, 35
Prove that if $|x| < 1$, then
\[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]
1989 National High School Mathematics League, 10
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
2008 National Olympiad First Round, 24
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$?
$
\textbf{(A)}\ 121
\qquad\textbf{(B)}\ 224
\qquad\textbf{(C)}\ 275
\qquad\textbf{(D)}\ 364
\qquad\textbf{(E)}\ 375
$
1990 National High School Mathematics League, 14
Here are $n^2$ numbers:
$a_{11},a_{12},a_{13},\cdots,a_{1n}\\
a_{21},a_{22},a_{23},\cdots,a_{2n}\\
\cdots\\
a_{n1},a_{n2},a_{n3},\cdots,a_{nn}$
Numbers in each line are arithmetic sequence, numbers in each column are geometric series.
If $a_{24}=1,a_{42}=\frac{1}{8},a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+\cdots+a_{nn}$.