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,
and 
.
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be triangle and point 
be 
altitude of 
.
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and 
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and 
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is the intersection of 
and 
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be intersection of 
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intersect circumcircle of 
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.
bisect 
,
by letting 
and 
for 
.
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such that, for some positive integer 
are all prime.
a
.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of 
such that 
for all real number 
and 
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can be obtained by concatenating a digit to the right of 
.
. Prove that
be 
for 
such that 
and 
intersect at the point 
.
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
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lies on diagonal 
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Y by aidan0626, GodGodGodGodGoose
Now unfortunately I don't know the difficulty of the problems posted here but I'll try to replicate:
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
This post has been edited 1 time. Last edited by bibidi_skibidi, Apr 15, 2025, 4:19 AM
ways to split the apples and 
ways to split the oranges, and you just multiply those two together to get the final answer.
is what I got
be the number of oranges each person gets, so 
then use stars and bars. similar process for apples
