Found problems: 85335
1966 IMO Shortlist, 25
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
1982 Spain Mathematical Olympiad, 3
A rocket is launched and reaches $120$ m in height; in the fall he loses $60$ m, then it recovers $40$ m, loses $ 30 $ again, gains $24$, loses $20$, etc. If the process continues indefinitely, at what height does it tend to stabilize?
2021-2022 OMMC, 8
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$.
[i]Proposed by Isaac Chen[/i]
2014 Contests, Problem 3
Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be?
1984 Iran MO (2nd round), 7
Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)
2015 Thailand Mathematical Olympiad, 5
Let $n$ be an integer greater than $6$.Show that if $n+1$ is a prime number,than
$\left\lceil \frac{(n-1)!}{n(n+1)}\right \rceil$ is $ODD.$
2002 Polish MO Finals, 3
$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).
2014 Cono Sur Olympiad, 2
A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i].
a) Prove that there are infinite [i]non-charrua[/i] pairs.
b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].
1985 Bulgaria National Olympiad, Problem 6
Let $\alpha_a$ denote the greatest odd divisor of a natural number $a$, and let $S_b=\sum_{a=1}^b\frac{\alpha_a}a$ Prove that the sequence $S_b/b$ has a finite limit when $b\to\infty$, and find this limit.
1984 IMO Longlists, 35
Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions
\[\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}\]
where $\pi$ is the ratio between a circle and its diameter.
2002 IMO Shortlist, 6
Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.
2021 JBMO Shortlist, A3
Let $n$ be a positive integer. A finite set of integers is called $n$-divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$-divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$, or $\{1, 5, 7\}$
and $\{3, 4, 6\}$. Find all the integers $n > 0$ for which there exists a $n$-divided set.
Proposed by [i]Martin Rakovsky, France[/i]
2020 IberoAmerican, 5
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$
2019 Romania National Olympiad, 4
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$
2023 Brazil Team Selection Test, 4
Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.
2014 Contests, 1
Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]
1995 AMC 12/AHSME, 2
If $\sqrt{2 + \sqrt{x}} = 3$, then $x =$
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \sqrt{7} \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 49 \qquad
\textbf{(E)}\ 121$
2015 AMC 12/AHSME, 17
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
1999 Harvard-MIT Mathematics Tournament, 8
Let $C$ be a circle with two diameters intersecting at an angle of $30$ degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius $1$. Find the largest possible radius of $C$.
2011 National Olympiad First Round, 9
Let $ABCD$ be a convex quadrilateral with $m(\widehat{ADC}) = 90^{\circ}$. The line through $D$ which is parallel to $BC$ meets $AB$ at $E$. If $m(\widehat{DAC}) = m(\widehat{DAE})$, $|AB|=3$ and $|AC|=4$, then $|AE| = ?$
$\textbf{(A)}\ \frac56 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \frac34$
1963 Bulgaria National Olympiad, Problem 2
It is given the equation $x^2+px+1=0$, with roots $x_1$ and $x_2$;
(a) find a second-degree equation with roots $y_1,y_2$ satisfying the conditions $y_1=x_1(1-x_1)$, $y_2=x_2(1-x_2)$;
(b) find all possible values of the real parameter $p$ such that the roots of the new equation lies between $-2$ and $1$.
2022 Saint Petersburg Mathematical Olympiad, 3
Given is a trapezoid $ABCD$, $AD \parallel BC$. The angle bisectors of the two pairs of opposite angles meet at $X, Y$. Prove that $AXYD$ and $BXYC$ are cyclic.
2017 Regional Olympiad of Mexico Southeast, 1
Let $ABC$ a triangle and $C$ it´s circuncircle. Let $D$ a point in arc $AB$ that not contain $A$, diferent of $B$ and $C$ such that $CD$ and $AB$ are not parallel. Let $E$ the intersection of $CD$ and $AB$ and $O$ the circumcircle of triangle $DBE$. Prove that the measure of $\angle OBE$ does not depend of the choice of $D$.
2019 Durer Math Competition Finals, 2
Albrecht fills in each cell of an $8 \times 8$ table with a $0$ or a $1$. Then at the end of each row and column he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the $64$ digits in the table uniquely. Show that the $8 \times 8$ table contained either a row full of $0$’s or a column full of $1$’s
1981 All Soviet Union Mathematical Olympiad, 312
The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$. The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle. Prove that $KLMN$ area is a half of $ABCD$ area.