Found problems: 85335
Denmark (Mohr) - geometry, 2006.1
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.
[img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]
2020 Online Math Open Problems, 15
Let $ABC$ be a triangle with $AB = 20$ and $AC = 22$. Suppose its incircle touches $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at $D$, $E$, and $F$ respectively, and $P$ is the foot of the perpendicular from $D$ to $\overline{EF}$. If $\angle BPC = 90^{\circ}$, then compute $BC^2$.
[i]Proposed by Ankan Bhattacharya[/i]
2019 IFYM, Sozopol, 6
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that:
$xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$
1971 IMO Longlists, 39
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.
2013 National Olympiad First Round, 28
In the beginning, there is a pair of positive integers $(m,n)$ written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs $(m,n)$ from the pairs $(7,79)$, $(17,71)$, $(10,101)$, $(21,251)$, $(50,405)$, can Alice guarantee to win when she makes the first move?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of above}
$
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.)
2012 Paraguay Mathematical Olympiad, 3
Let $ABC$ be a triangle (right in $B$) inscribed in a semi-circumference of diameter $AC=10$. Determine the distance of the vertice $B$ to the side $AC$ if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
2017 CCA Math Bonanza, T4
The $12$ islands of the Bonanza archipelago are labeled $A,B,C,\dots,K,L$. Some of the islands are connected by bridges, as indicated in the diagram below. Tristan wants be able to walk from island to island crossing each bridge exactly once (he doesn't care if he visits a given island more than once, or whether he starts and ends on the same island). Submit a pair of unconnected islands such that if they are connected by a bridge, Tristan can accomplish his goal.
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC80L2M0MTU1ZDVmZTRlNjQ5MmQ5ZTNhN2U3NTQwZDRhMzRmNjk1YTk4LnBuZw==&rn=bWJncmFwaHB1enpsZS5wbmc=[/img]
[i]2017 CCA Math Bonanza Team Round #4[/i]
Russian TST 2022, P3
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
2021 BMT, Tie 3
For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.
2012 Czech-Polish-Slovak Match, 1
Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.
1997 Israel National Olympiad, 3
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
2023 Olympic Revenge, 2
Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$
2014 JHMMC 7 Contest, 21
Kelvin the Frog and Alex the Kat play a game. Kelvin the Frog goes first, and they alternate rolling a standard $6\text{-sided die.} If they roll an even number or a number that was previously rolled, they win. What is the probability that Alex
wins?
2016 239 Open Mathematical Olympiad, 7
Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$ for all $x>0, y>x+1$.
2010 Dutch BxMO TST, 3
Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying
$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$
Is $N$ even or odd?
Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.
2016 Junior Balkan Team Selection Tests - Moldova, 5
Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$.
2011 Iran MO (3rd Round), 3
Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$?
[i]Proposed by Mahyar Sefidgaran[/i]
2019 Oral Moscow Geometry Olympiad, 2
On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.
1978 Chisinau City MO, 158
Five points are selected on the plane so that no three of them lie on one straight line. Prove that some four of these five points are the vertices of a convex quadrilateral.
2019 Romania Team Selection Test, 1
Let $ I,O $ denote the incenter, respectively, the circumcenter of a triangle $ ABC. $ The $ A\text{-excircle} $ touches the lines $ AB,AC,BC $ at $ K,L, $ respectively, $ M. $ The midpoint of $ KL $ lies on the circumcircle of $ ABC. $ Show that the points $ I,M,O $ are collinear.
[i]Павел Кожевников[/i]
1991 Baltic Way, 15
In each of the squares of a chessboard an arbitrary integer is written. A king starts to move on the board. Whenever the king moves to some square, the number in that square is increased by $1$. Is it always possible to make the numbers on the chessboard:
(a) all even;
(b) all divisible by $3$;
(c) all equal?
2025 Kyiv City MO Round 1, Problem 3
Point \( H \) is the orthocenter of the acute triangle \( ABC \), and \( AD \) is its altitude. Tangents are drawn from points \( B \) and \( C \) to the circle with center \( A \) and radius \( AD \), which do not coincide with the line \( BC \). These tangents intersect at point \( P \). Prove that the radius of the incircle of \( \triangle BCP \) is equal to \( HD \).
[i]Proposed by Danylo Khilko[/i]
Ukrainian From Tasks to Tasks - geometry, 2012.13
The sides of a triangle are consecutive natural numbers, and the radius of the inscribed circle is $4$. Find the radius of the circumscribed circle.
2015 QEDMO 14th, 8
There are many cities in penguin's land. A road runs between some of them, which either can be one or two lanes. When two streets meet outside of a city, it becomes prevent traffic chaos by building a bridge and avoiding any junctions. Now the penguin parliament has passed a new law, according to which every street is only a one-way street may be used. The Minister of Transport now liked the direction of each street stipulate that in each city at most one lane more or less leads in and out. He also knows that the streets of every city have odd number of tracks. Show that he can succeed in his endeavor.