Found problems: 85335
2016 BMT Spring, 3
A little boy takes a $ 12$ in long strip of paper and makes a Mobius strip out of it by tapping the ends together after adding a half twist. He then takes a $ 1$ inch long train model and runs it along the center of the strip at a speed of $ 12$ inches per minute. How long does it take the train model to make two full complete loops around the Mobius strip? A complete loop is one that results in the train returning to its starting point.
2012 Dutch Mathematical Olympiad, 3
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,
for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
1986 Tournament Of Towns, (122) 4
Consider subsets of the set $1 , 2,..., N$.
For each such subset we can compute the product of the reciprocals of each member.
Find the sum of all such products.
2015 239 Open Mathematical Olympiad, 2
Prove that $\binom{n+k}{n}$ can be written as product of $n$ pairwise coprime numbers $a_1,a_2,\dots,a_n$ such that $k+i$ is divisible by $a_i$ for all indices $i$.
2014 Middle European Mathematical Olympiad, 1
Determine the lowest possible value of the expression
\[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \]
where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities
\[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]
1972 IMO Longlists, 31
Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.
2006 China Team Selection Test, 2
Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$.
Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.
2019 Jozsef Wildt International Math Competition, W. 62
Prove that $$\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}$$for all $n\in \mathbb{N}^*$
1988 IMO Longlists, 19
Let $Z_{m,n}$ be the set of all ordered pairs $(i,j)$ with $i \in {1, \ldots, m}$ and $j \in {1, \ldots, n}.$ Also let $a_{m,n}$ be the number of all those subsets of $Z_{m,n}$ that contain no 2 ordered pairs $(i_1,j_1)$ and $(i_2,j_2)$ with $|i_1 - i_2| + |j_1 - j_2| = 1.$ Then show, for all positive integers $m$ and $k,$ that \[ a^2_{m, 2 \cdot k} \leq a_{m, 2 \cdot k - 1} \cdot a_{m, 2 \cdot k + 1}. \]
1955 Miklós Schweitzer, 1
[b]1.[/b] Let $a_{1}, a_{2}, \dots , a_{n}$ and $b_{1}, b_{2}, \dots , b_{m}$ be $n+m$ unit vectors in the $r$-dimensional Euclidean space $E_{r} (n,m \leq r)$; let $a_{1}, a_{2}, \dots , a_{n}$ as well as $b_{1}, b_{2}, \dots , b_{m}$ be mutually orthogonal. For any vector $x \in E_{r}$, consider
$Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}$
($(a,b)$ denotes the scalar product of $a$ and $b$). Show that the sequence $(T^{k}x)^{\infty}_{ k =0}$, where $T^{0} x= x$ and $T^{k} x = T(T^{k-1}x)$, is convergent and give a geometrical characterization of how the limit depends on $x$. [b](S. 14)[/b]
2006 Bulgaria National Olympiad, 3
The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times.
[i]Aleksandar Ivanov, Emil Kolev [/i]
1969 IMO Shortlist, 62
Which natural numbers can be expressed as the difference of squares of two integers?
2010 Math Prize For Girls Problems, 5
Find the smallest two-digit positive integer that is a divisor of 201020112012.
2020 Caucasus Mathematical Olympiad, 1
Determine if there exists a finite set $A$ of positive integers satisfying the following condition: for each $a\in{A}$ at least one of two numbers $2a$ and
$\frac{a}{3}$ belongs to $A$.
2006 Moldova Team Selection Test, 1
Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.
2009 APMO, 1
Consider the following operation on positive real numbers written on a blackboard:
Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board.
Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.
2014 India Regional Mathematical Olympiad, 6
For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$.
1988 AMC 12/AHSME, 22
For how many integers $x$ does a triangle with side lengths $10$, $24$ and $x$ have all its angles acute?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{more than } 7 $
2012 Today's Calculation Of Integral, 800
For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$
1997 Czech and Slovak Match, 3
Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$
.
2012 Singapore Senior Math Olympiad, 5
For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove
\[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]
2023 Bosnia and Herzegovina Junior BMO TST, 2.
Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.
2012 IMO, 4
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
[i]Proposed by Liam Baker, South Africa[/i]
2014 AMC 8, 15
The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?
[asy]
size(230);
defaultpen(linewidth(0.65));
pair O=origin;
pair[] circum = new pair[12];
string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"};
draw(unitcircle);
for(int i=0;i<=11;i=i+1)
{
circum[i]=dir(120-30*i);
dot(circum[i],linewidth(2.5));
label(let[i],circum[i],2*dir(circum[i]));
}
draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle);
label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6])));
label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8])));
label("$O$",O,dir(60));
[/asy]
$\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad \textbf{(E) }150$
2019 Saudi Arabia JBMO TST, 4
Let $a, b, c$ be positive reals. Prove that
$a/b+b/c+c/a=>(c+a)/(c+b) + (a+b)/(a+c) + (b+c)/(b+a)$