Found problems: 85335
1991 India Regional Mathematical Olympiad, 1
Let $P$ be an interior point of a triangle $ABC$ and $AP,BP,CP$ meet the sides $BC,CA,AB$ in $D,E,F$ respectively. Show that \[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]
[hide="Remark"]This is known as [i]Van Aubel's[/i] Theorem.[/hide]
1969 IMO Shortlist, 32
$(GDR 4)$ Find the maximal number of regions into which a sphere can be partitioned by $n$ circles.
2022 HMNT, 5
An apartment building consists of 20 rooms number 1, 2,..., 20 arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1)\pmod{20}$) or walk across the center to the opposite room (i.e. from room $i$ to room $(i+10)\pmod{20}$). Find the number of ways to move from room 10 to room 20 without visiting the same room twice.
2010 Bulgaria National Olympiad, 1
A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
2006 China Team Selection Test, 2
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
2004 Germany Team Selection Test, 2
Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.
1998 Croatia National Olympiad, Problem 3
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
2018 Korea National Olympiad, 7
Let there be a figure with $9$ disks and $11$ edges, as shown below.
We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.
2024 Austrian MO National Competition, 4
A positive integer is called [i]powerful [/i]if all exponents in its prime factorization are $\ge 2$. Prove that there are infinitely many pairs of powerful consecutive positive integers.
[i](Walther Janous)[/i]
1949 Moscow Mathematical Olympiad, 171
* Prove that a number of the form $2^n$ for a positive integer $n$ may begin with any given combination of digits.
2021 AMC 12/AHSME Fall, 12
For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n$. For example,
\[f(14) = (1 + 2 + 7 + 14) \div 14 = \frac{12}{7}.\]
What is $f(768) - f(384)?$
$\textbf{(A) }\frac{1}{768}\qquad\textbf{(B) }\frac{1}{192}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad\textbf{(E) }\frac{8}{3}$
2014 Baltic Way, 16
Determine whether $712! + 1$ is a prime number.
1993 IMO Shortlist, 2
Show that there exists a finite set $A \subset \mathbb{R}^2$ such that for every $X \in A$ there are points $Y_1, Y_2, \ldots, Y_{1993}$ in $A$ such that the distance between $X$ and $Y_i$ is equal to 1, for every $i.$
2022 MMATHS, 4
How many ways are there to choose three digits $A,B,C$ with $1 \le A \le 9$ and $0 \le B,C \le 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \ge 10$?
1995 Tournament Of Towns, (449) 5
Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained.
(AV Shapovalov)
2015 Sharygin Geometry Olympiad, 4
A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point.
(R. Krutovsky, A. Yakubov)
1986 ITAMO, 7
On a long enough highway, a passenger in a bus observes the traffic. He notes that, during an hour, the bus going with a constant velocity overpasses $a$ cars and gets overpassed by $b$ cars, while $c$ cars pass in the opposite direction. Assuming that the traffic is the same in both directions, is it possible to determine the number of cars that pass along the highway per hour? (You may assume that the velocity of a car can take only two values.)
2013 Balkan MO, 3
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:
(a) $xf(x,y,z) = zf(z,y,x)$,
(b) $f(x, ky, k^2z) = kf(x,y,z)$,
(c) $f(1, k, k+1) = k+1$.
([i]United Kingdom[/i])
2002 Iran Team Selection Test, 3
A "[i]2-line[/i]" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.
2011 Math Prize for Girls Olympiad, 2
Let $\triangle ABC$ be an equilateral triangle. If $0 < r < 1$, let $D_r$ be the point on $\overline{AB}$ such that $AD_r = r \cdot AB$, let $E_r$ be the point on $\overline{BC}$ such that $BE_r = r \cdot BC$, and let $P_r$ be the point where $\overline{AE_r}$ and $\overline{CD_r}$ intersect. Prove that the set of points $P_r$ (over all $0 < r < 1$) lie on a circle.
2010 Hong kong National Olympiad, 2
Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]
2018 Irish Math Olympiad, 9
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
2023 AMC 10, 16
In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
$\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66$
1986 AIME Problems, 1
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?
2016 Auckland Mathematical Olympiad, 1
It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?