Found problems: 85335
2012 Iran MO (3rd Round), 3
Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers;
[b]a)[/b] Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$.
[b]b)[/b] If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$.
[b]c)[/b] If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.
2024 Vietnam Team Selection Test, 3
Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.
2015 Denmark MO - Mohr Contest, 5
For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?
2022 Yasinsky Geometry Olympiad, 4
Let $BM$ be the median of triangle $ABC$. On the extension of $MB$ beyond $B$, the point $K$ is chosen so that $BK =\frac12 AC$. Prove that if $\angle AMB=60^o$, then $AK=BC$.
(Mykhailo Standenko)
2020-21 IOQM India, 4
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.
2013 Today's Calculation Of Integral, 871
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
1992 Brazil National Olympiad, 6
Given a set of n elements, find the largest number of subsets such that no subset is contained in any other
2005 Germany Team Selection Test, 2
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that
\[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]
2017 Singapore Senior Math Olympiad, 1
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
2013 Dutch IMO TST, 5
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
2005 AMC 12/AHSME, 4
A store normally sells windows at $ \$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
$ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$
2000 Putnam, 2
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
1966 Poland - Second Round, 6
Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.
2016 Harvard-MIT Mathematics Tournament, 3
Let $PROBLEMZ$ be a regular octagon inscribed in a circle of unit radius. Diagonals $MR$, $OZ$ meet at $I$. Compute $LI$.
1997 Miklós Schweitzer, 4
An elementary change in a 0-1 matrix is a change in an element and with it all its horizontal, vertical, and diagonal neighbors (0 to 1 or 1 to 0). Can any 1791 x 1791 0-1 matrix be transformed into a zero matrix with elementary changes?
1955 AMC 12/AHSME, 26
Mr. A owns a house worth $ \$10000$. He sells it to Mr. B at $ 10 \%$ profit. Mr. B sells the house back to Mr. A at a $ 10 \%$ loss. Then:
$ \textbf{(A)}\ \text{Mr. A comes out even} \qquad
\textbf{(B)}\ \text{Mr. A makes }\$100 \qquad
\textbf{(C)}\ \text{Mr. A makes }\$1000 \\
\textbf{(D)}\ \text{Mr. B loses }\$100 \qquad
\textbf{(E)}\ \text{none of the above is correct}$
2024 JHMT HS, 16
Let $N_{15}$ be the answer to problem 15.
For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.
Kyiv City MO Juniors 2003+ geometry, 2015.9.3
It is known that a square can be inscribed in a given right trapezoid so that each of its vertices lies on the corresponding side of the trapezoid (none of the vertices of the square coincides with the vertex of the trapezoid). Construct this inscribed square with a compass and a ruler.
(Maria Rozhkova)
2020 Azerbaijan National Olympiad, 2
$a,b,c$ are positive integer.
Solve the equation:
$ 2^{a!}+2^{b!}=c^3 $
2008 IMS, 9
Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$
\[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha
\]
in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$
2002 Federal Math Competition of S&M, Problem 2
Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.
1998 AMC 12/AHSME, 4
Define $[a,b,c]$ to mean $\frac{a+b}{c},$ where $c \neq 0$. What is the value of \[[[60,30,90],[2,1,3],[10,5,15]]?\]
$\text{(A)} \ 0 \qquad \text{(B)} \ 0.5 \qquad \text{(C)} \ 1 \qquad \text{(D)} \ 1.5 \qquad \text{(E)} \ 2$
2014 Regional Competition For Advanced Students, 1
Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$.
Prove that:
i) $A + A \ne A \cdot A$
ii) $(A + A) \cap N = (A \cdot A) \cap N$.
Vasile Pop
2023 Indonesia TST, C
There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.