Found problems: 329
2014 USAMTS Problems, 3:
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
2014 AMC 8, 18
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
$ \textbf{(A) }\text{all 4 are boys}$\\ $\textbf{(B) }\text{all 4 are girls}$\\$ \textbf{(C) }\text{2 are girls and 2 are boys}$\\ $\textbf{(D) }\text{3 are of one gender and 1 is of the other gender}$\\ $\textbf{(E) }\text{all of these outcomes are equally likely} $
1987 China Team Selection Test, 2
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
2005 Germany Team Selection Test, 1
In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word.
A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word.
For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$.
Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.
2012 NIMO Problems, 2
A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$.
[i]Proposed by Aaron Lin[/i]
2003 Putnam, 1
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.
2005 Manhattan Mathematical Olympiad, 1
Is there a whole number which becomes exactly $57$ times less than itself when one crosses out its first digit?
2013 Tuymaada Olympiad, 7
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2003 AMC 8, 11
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40$ dollars on Thursday?
$\textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 39.60 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 40.40 \qquad
\textbf{(E)}\ 44$
2008 ITest, 55
Let $\triangle XOY$ be a right-angled triangle with $\angle XOY=90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Find the length $XY$ given that $XN=22$ and $YM=31$.
2014 Online Math Open Problems, 15
Let $\phi = \frac{1+\sqrt{5}}{2}$. A [i]base-$\phi$ number[/i] $(a_n a_{n-1} ... a_1 {a_0})_{\phi}$, where $0 \le a_n, a_{n-1}, ..., a_0 \le 1$ are integers, is defined by \[ (a_n a_{n-1} ... a_1 {a_0})_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + ... + a_1 \cdot \phi^1 + a_0. \]
Compute the number of base-$\phi$ numbers $(b_jb_{j-1}... b_1{b_0})_\phi$ which satisfy $b_j \ne 0$ and
\[ (b_jb_{j-1}... b_1{b_0})_\phi = \underbrace{(100 ... 100)_\phi}_{\text{Twenty}\ 100's}. \][i]Proposed by Yang Liu[/i]
2014 AMC 12/AHSME, 12
Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
$\textbf{(A) }2\qquad
\textbf{(B) }1+\sqrt3\qquad
\textbf{(C) }3\qquad
\textbf{(D) }2+\sqrt3\qquad
\textbf{(E) }4\qquad$
1992 Poland - First Round, 7
Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \in 0,1,2,3)$ be the point determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$. Find the volume of the smallest convex polyhedron which contains all the points $P_{ij}$.
2013 HMNT, 7
Find the largest number $\lambda$ such that $a^2+b^2+c^2+d^2 \geq ab + \lambda bc + cd$ for all real numbers $a,b,c,d$
2012 AIME Problems, 6
The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < n$. Find $n$.
2003 USAMO, 3
Let $n \neq 0$. For every sequence of integers \[ A = a_0,a_1,a_2,\dots, a_n \] satisfying $0 \le a_i \le i$, for $i=0,\dots,n$, define another sequence \[ t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n) \] by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.
2011 AMC 10, 18
Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
[asy]
pathpen = linewidth(.7); pointpen = black;
pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy]
${
\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad
\textbf{(B)}\ \frac{\pi}{2} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{3\pi}{4} \qquad
\textbf{(E)}\ 1+\frac{\pi}{2}} $
2005 Germany Team Selection Test, 1
In the following, a [i]word[/i] will mean a finite sequence of letters "$a$" and "$b$". The [i]length[/i] of a word will mean the number of the letters of the word. For instance, $abaab$ is a word of length $5$. There exists exactly one word of length $0$, namely the empty word.
A word $w$ of length $\ell$ consisting of the letters $x_1$, $x_2$, ..., $x_{\ell}$ in this order is called a [i]palindrome[/i] if and only if $x_j=x_{\ell+1-j}$ holds for every $j$ such that $1\leq j\leq\ell$. For instance, $baaab$ is a palindrome; so is the empty word.
For two words $w_1$ and $w_2$, let $w_1w_2$ denote the word formed by writing the word $w_2$ directly after the word $w_1$. For instance, if $w_1=baa$ and $w_2=bb$, then $w_1w_2=baabb$.
Let $r$, $s$, $t$ be nonnegative integers satisfying $r + s = t + 2$. Prove that there exist palindromes $A$, $B$, $C$ with lengths $r$, $s$, $t$, respectively, such that $AB=Cab$, if and only if the integers $r + 2$ and $s - 2$ are coprime.
1985 IMO Longlists, 6
On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars.
[b](a)[/b] What is the probability that the pedestrian crosses the street uninjured?
[b](b)[/b] Can he improve this probability by crossing the road in a direction other than perpendicular?
2009 AMC 12/AHSME, 2
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
1996 Vietnam National Olympiad, 3
Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying
1) There exist s, t such that 1<=s<t<=k and a_s>a_t.
2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2.
P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...
2007 Stanford Mathematics Tournament, 8
If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.
1985 IMO Longlists, 13
Find the average of the quantity
\[(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2\]
taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$
2009 AMC 8, 23
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $ 400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
$ \textbf{(A)}\ 26 \qquad
\textbf{(B)}\ 28 \qquad
\textbf{(C)}\ 30 \qquad
\textbf{(D)}\ 32 \qquad
\textbf{(E)}\ 34$
2002 AMC 8, 1
A circle and two distinct lines are drawn on a sheet of paper. What is the largest possible number of points of intersection of these figures?
$ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $