Found problems: 85335
2002 China Western Mathematical Olympiad, 4
Let $ n$ be a positive integer, let the sets $ A_{1},A_{2},\cdots,A_{n \plus{} 1}$ be non-empty subsets of the set $ \{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $ \{1,2,\cdots,n \plus{} 1\}$: $ \{i_{1},i_{2},\cdots,i_{k}\}$ and $ \{j_{1},j_{2},\cdots,j_{m}\}$ such that $ A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} \equal{} A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$.
2015 AMC 8, 7
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$
2021 Harvard-MIT Mathematics Tournament., 1
Compute the sum of all positive integers $n$ for which the expression
\[\frac{n+7}{\sqrt{n-1}}\]
is an integer.
2004 Purple Comet Problems, 23
Let $a$ and $b$ be real numbers satisfying \[a^4 + 8b = 4(a^3 - 1) - 16 \sqrt{3}\] and \[b^4 + 8a = 4(b^3 - 1) + 16 \sqrt{3}.\] Find $a^4 + b^4$.
1990 Putnam, A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
2009 Argentina Iberoamerican TST, 2
There are $ m \plus{} 1$ horizontal lines and $ m$ vertical lines on the plane so that $ m(m \plus{} 1)$ intersections are made.
A mark is placed at one of the $ m$ points of the lowest horizontal line.
2 players play the game of the following rules on this lines and points.
1. Each player moves a mark from a point to a point along the lines in turns.
2. The segment is erased after a mark moved along it.
3. When a player cannot make a move, then he loses.
Prove that the lead always wins the game.
PS I haven't found a student who solved it. There can be no one.
2018 Bangladesh Mathematical Olympiad, 8
a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s.
prove that
1. $n$ is greater than or equal to $2^{k+1}-1$
2.Find all $n$ such that $2$ is a n-good
2000 APMO, 2
Find all permutations $a_1, a_2, \ldots, a_9$ of $1, 2, \ldots, 9$ such that \[ a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1 \]
and
\[ a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2 \]
1967 AMC 12/AHSME, 10
If $\frac{a}{10^x-1}+\frac{b}{10^x+2}=\frac{2 \cdot 10^x+3}{(10^x-1)(10^x+2)}$ is an identity for positive rational values of $x$, then the value of $a-b$ is:
$\textbf{(A)}\ \frac{4}{3} \qquad
\textbf{(B)}\ \frac{5}{3} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{11}{4} \qquad
\textbf{(E)}\ 3$
2019 MOAA, 8
Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)
2014 Dutch Mathematical Olympiad, 5
We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape.
a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$?
b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?
2002 Paraguay Mathematical Olympiad, 1
There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist .
2014 Switzerland - Final Round, 5
Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have:
$$\sum_{d | n} a_d = 2^n.$$
Show for every $n \in N$ that $n$ divides $a_n$.
Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$
2004 National Olympiad First Round, 4
What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$?
$
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 0
$
2014-2015 SDML (High School), 3
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$
2018 Rio de Janeiro Mathematical Olympiad, 3
Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$.
Prove that the number of good functions is a perfect square.
1967 IMO Shortlist, 6
On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?
2002 National Olympiad First Round, 28
How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 1001
\qquad\textbf{e)}\ 2002
$
1996 IMC, 12
i) Prove that for every sequence $(a_{n})_{n\in \mathbb{N}}$, such that $a_{n}>0$ for all $n \in \mathbb{N}$ and
$\sum_{n=1}^{\infty}a_{n}<\infty$, we have
$$\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.$$
ii) Prove that for every $\epsilon>0$ there exists a sequence $(b_{n})_{n\in \mathbb{N}}$ such that $b_{n}>0$ for all $n \in \mathbb{N}$ and
$\sum_{n=1}^{\infty}b_{n}<\infty$ and
$$\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.$$
2010 Romania National Olympiad, 1
Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that
\[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\]
Prove that $(a_n)_{n\ge0}$ is a geometric sequence.
[i]Lucian Dragomir[/i]
1977 USAMO, 5
If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that
\[ (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2\]
and determine when there is equality.
1985 AMC 8, 20
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $ 1$ fall that year?
\[ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Friday} \qquad
\textbf{(E)}\ \text{Saturday}
\]
2007 Purple Comet Problems, 9
The four sets A, B, C, and D each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets?
2004 APMO, 1
Determine all finite nonempty sets $S$ of positive integers satisfying
\[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \]
where $(i,j)$ is the greatest common divisor of $i$ and $j$.
2009 Paraguay Mathematical Olympiad, 2
In a triangle $ABC$ ($\angle{BCA} = 90^{\circ}$), let $D$ be the intersection of $AB$ with a circumference with diameter $BC$. Let $F$ be the intersection of $AC$ with a line tangent to the circumference. If $\angle{CAB} = 46^{\circ}$, find the measure of $\angle{CFD}$.