Found problems: 85335
2004 National Olympiad First Round, 18
How many consequtive numbers are there in the set of positive integers in which powers of all prime factors in their prime factorizations are odd numbers?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 15
$
2011 NZMOC Camp Selection Problems, 4
Let a point $P$ inside a parallelogram $ABCD$ be given such that $\angle APB +\angle CPD = 180^o$. Prove that $AB \cdot AD = BP \cdot DP + AP \cdot CP$.
2008 Princeton University Math Competition, B6
Circles $A, B$, and $C$ each have radius $r$, and their centers are the vertices of an equilateral triangle of side length $6r$. Two lines are drawn, one tangent to $A$ and $C$ and one tangent to $B$ and $C$, such that $A$ is on the opposite side of each line from $B$ and $C$. Find the sine of the angle between the two lines.
[img]http://4.bp.blogspot.com/-IZv8q-3NYZg/XXmrroy2PnI/AAAAAAAAKxg/jSOcOOQ8Kyw0EwHUifXJ1jOd2ENAo1FfACK4BGAYYCw/s200/2008%2Bpumac%2Bb6.png[/img]
1963 AMC 12/AHSME, 26
[b]Form 1[/b]
Consider the statements:
$\textbf{(1)}\ p\text{ } \wedge\sim q\wedge r \qquad
\textbf{(2)}\ \sim p\text{ } \wedge\sim q\wedge r\qquad
\textbf{(3)}\ p\text{ } \wedge\sim q\text{ }\wedge \sim r \qquad
\textbf{(4)}\ \sim p\text{ } \wedge q\text{ }\wedge r $,
where $p,q,$ and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
[b]Form 2[/b]
Consider the statements $(1)$ $p$ and $r$ are true and $q$ is false $(2)$ $r$ is true and $p$ and $q$ are false $(3)$ $p$ is true and $q$ and $r$ are false $(4)$ $q$ and $r$ are true and $p$ is false. How many of these imply the truth of the statement
"$r$ is implied by the statement that $p$ implies $q$"?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 4$
2006 MOP Homework, 2
Prove that $\frac{a}{(a + 1)(b + 1)} +\frac{ b}{(b + 1)(c + 1)} + \frac{c}{(c + 1)(a + 1)} \ge \frac34$ where $a, b$ and $c$ are positive real numbers satisfying $abc = 1$.
2023/2024 Tournament of Towns, 6
A table $2 \times 2024$ is filled with positive integers. Specifically, the first row is filled with numbers from the set $\{1, \ldots, 2023\}$. It turned out that for any two columns the difference of numbers from the first row is divisible by the difference of numbers from the second row, while all numbers in the second row are pairwise different. Is it true for sure that the numbers in the first row are equal?
Ivan Kukharchuk
2012 Princeton University Math Competition, A2 / B5
How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?
1967 IMO Shortlist, 2
If $x$ is a positive rational number show that $x$ can be uniquely expressed in the form $x = \sum^n_{k=1} \frac{a_k}{k!}$ where $a_1, a_2, \ldots$ are integers, $0 \leq a_n \leq n - 1$, for $n > 1,$ and the series terminates. Show that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^6.$
2012 Korea National Olympiad, 3
Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of
\[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]
1994 AMC 12/AHSME, 7
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
[asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle);
draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle);
label("A", (0,0), W);
label("B", (10,0), E);
label("C", (10,10), NE);
label("D", (0,10), NW);
label("G", (5,5), N);
label("F", (12,-2), E);
label("E", (5,-9), S);
label("H", (-2,-2), W);
dot((-2,-2));
dot((5,-9));
dot((12,-2));
dot((0,0));
dot((10,0));
dot((10,10));
dot((0,10));
dot((5,5));
[/asy]
$ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 125 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 175 $
2019 Saudi Arabia IMO TST, 1
Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers
2008 South East Mathematical Olympiad, 3
Captain Jack and his pirate men plundered six chests of treasure $(A_1,A_2,A_3,A_4,A_5,A_6)$. Every chest $A_i$ contains $a_i$ coins of gold, and all $a_i$s are pairwise different $(i=1,2,\cdots ,6)$. They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?
2001 Moldova Team Selection Test, 4
For every nonnegative integer $n{}$ let $f(n)$ be the smallest number of digits $1$ which can represent the number $n{}$ using the symbols $"+", "-", "\times", "(", ")"$. For example, $80=(1+1+1+1+1)\times(1+1+1+1)\times(1+1+1+1)$ and $f(80)\leq 13$. Prove that $2\log_3 n \leq f(n) < 5\log_3 n$ for every $n>1$.
2021 BMT, 5
Bill divides a $28 \times 30$ rectangular board into two smaller rectangular boards with a single straightcut, so that the side lengths of both boards are positive whole numbers. How many different pairs of rectangular boards, up to congruence and arrangement, can Bill possibly obtain? (For instance, a cut that is $1$ unit away from either of the edges with length $28$ will result in the same pair of boards: either way, one would end up with a $1 \times 28$ board and a $29 \times 28$ board.)
2014 NIMO Summer Contest, 11
Consider real numbers $A$, $B$, \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
[i]Proposed by Evan Chen[/i]
2010 Contests, 3
let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let
$A_k=\frac{\sum_{i=1}^{k}a_i}{k}$
prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.
LMT Accuracy Rounds, 2021 F2
A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[asy]
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue);
draw((0,0)--(4,0),black);
draw((0,0)--(0,2),black);
draw((4,0)--(4,2),black);
draw((4,2)--(0,2),black);
draw((0,1)--(4,1),black);
draw((1,0)--(1,2),black);
draw((2,0)--(2,2),black);
draw((3,0)--(3,2),black);
[/asy]
2010 Saint Petersburg Mathematical Olympiad, 3
$a$ is irrational , but $a$ and $a^3-6a$ are roots of square polynomial with integer coefficients.Find $a$
2020 BMT Fall, 7
A square has coordinates at $(0, 0)$, $(4, 0)$, $(0, 4)$, and $(4, 4)$. Rohith is interested in circles of radius $ r$ centered at the point $(1, 2)$. There is a range of radii $a < r < b$ where Rohith’s circle intersects the square at exactly $6$ points, where $a$ and $b$ are positive real numbers. Then $b - a$ can be written in the form $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.
1966 AMC 12/AHSME, 29
The number of postive integers less than $1000$ divisible by neither $5$ nor $7$ is:
$\text{(A)}\ 688 \qquad
\text{(B)}\ 686\qquad
\text{(C)}\ 684 \qquad
\text{(D)}\ 658\qquad
\text{(E)}\ 630$
2016 Putnam, A4
Consider a $(2m-1)\times(2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m,n\ge 4.$ The region is to be tiled using tiles of the two types shown:
\[
\begin{picture}(140,40)
\put(0,0){\line(0,1){40}}
\put(0,0){\line(1,0){20}}
\put(0,40){\line(1,0){40}}
\put(20,0){\line(0,1){20}}
\put(20,20){\line(1,0){20}}
\put(40,20){\line(0,1){20}}
\multiput(0,20)(5,0){4}{\line(1,0){3}}
\multiput(20,20)(0,5){4}{\line(0,1){3}}
\put(80,0){\line(1,0){40}}
\put(120,0){\line(0,1){20}}
\put(120,20){\line(1,0){20}}
\put(140,20){\line(0,1){20}}
\put(80,0){\line(0,1){20}}
\put(80,20){\line(1,0){20}}
\put(100,20){\line(0,1){20}}
\put(100,40){\line(1,0){40}}
\multiput(100,0)(0,5){4}{\line(0,1){3}}
\multiput(100,20)(5,0){4}{\line(1,0){3}}
\multiput(120,20)(0,5){4}{\line(0,1){3}}
\end{picture}
\]
(The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?
2020 Philippine MO, 3
Define the sequence $\{a_i\}$ by $a_0=1$, $a_1=4$, and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$. Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$.
2005 Taiwan TST Round 3, 3
The set $\{1,2,\dots\>,n\}$ is called $P$. The function $f: P \to \{1,2,\dots\>,m\}$ satisfies \[f(A\cap B)=\min (f(A), f(B)).\] What is the relationship between the number of possible functions $f$ with the sum $\displaystyle \sum_{j=1}^m j^n$?
There is a nice and easy solution to this. Too bad I did not think of it...
2021-IMOC qualification, C1
There are $3n$ $A$s and $2n$ $B$s in a string, where $n$ is a positive integer, prove that you can find a substring in this string that contains $3$ $A$s and $2$ $B$s.
1991 IMO, 1
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]