of positive integers such that
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
regions of a three-set Venn Diagram with 
colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.
satisfy that 
.
and 
Prove that
Let 
Prove that
of them have the same color. I want to reshuffle the cupcakes such that each plate is replaced by a different color cupcake. Determine how many ways to reshuffle when:
.
.
is called beautiful if the are at least 
integers 
such that in base 
we have that 
end in the same digit. Find the sum of the two smallest beautiful numbers.
be the Fibonacci numbers, defined by 
and 
.
,
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
and 
. 
where 
are primes, show that 
can never be a perfect square.
Y by Mango247
On the map, the Flower City has the form of a right triangle 
(see Fig.1). The length of each leg is 
meters. All the streets of the city run parallel to one of the legs at a distance of 
meter from each other. A river flows along the hypotenuse. From their houses that are located at points 
and 
, at the same time get the Cog and Tab. Each short moves to rivers according to the following rule: tosses his coin, and if the heads falls, he passes meter parallel to the leg 
to the north (up), and if tails, then passes meter parallel to the leg 
on east (right). If the Cog and the Tab meet at the same point, then they move together, tossing a coin.
a) Which is more likely: Cog and Tab will meet on the way to the river, or will they come to different points on the shore?
b) At what point near the river should the Stranger sit, if he wants the most did Gvintik and Shpuntik come to him together?
(see Fig.1). The length of each leg is 
meters. All the streets of the city run parallel to one of the legs at a distance of 
meter from each other. A river flows along the hypotenuse. From their houses that are located at points 
and 
, at the same time get the Cog and Tab. Each short moves to rivers according to the following rule: tosses his coin, and if the heads falls, he passes meter parallel to the leg 
to the north (up), and if tails, then passes meter parallel to the leg 
on east (right). If the Cog and the Tab meet at the same point, then they move together, tossing a coin.a) Which is more likely: Cog and Tab will meet on the way to the river, or will they come to different points on the shore?
b) At what point near the river should the Stranger sit, if he wants the most did Gvintik and Shpuntik come to him together?
This post has been edited 2 times. Last edited by parmenides51, Nov 29, 2020, 9:26 PM
