AoPS Academy
. What are the last three digits (from left to right) of the 
th term?
has the property that the sum of its first 
terms is divisible by for every positive integer 
. Let 
be the number of such sequences satisfying 
. Compute the remainder when is divided by 
.
has side length 
. Point 
lies on segment 
such that line 
is perpendicular to line 
. Given that 
, the area of can be expressed as 
, where 
and are relatively prime positive integers. Compute 
.
and 
satisfy 
, 
, and 
. If the number of possible ordered pairs 
is equal to 
, compute the remainder when is divided by .
be a triangle. Point 
lies on side 
, point 
lies on side 
, and point 
lies on side 
such that 
, 
, and 
. Let 
be the foot of the altitude from 
to . Given that 
, 
, and ![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
, the value of 
can be expressed as , where and are relatively prime positive integers. Compute .
and tells Bob that the three resulting remainders are 
, 
, and , in some order. For how many values of is it possible for Bob to uniquely determine ?
satisfying the system of equations 
The value of 
can be expressed as 
, where 
and 
are positive integers such that is square-free. Compute 
. be the set of all monotonically increasing six-term sequences whose terms are all integers between 
and 
inclusive. We say a sequence 
in is symmetric if for every integer 
, the number of terms of 
that are at least 
is 
. The probability that a randomly chosen element of is symmetric is 
, where and are relatively prime positive integers. Compute ., let 
denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer 
such that the equation 
has at least ten positive integer solutions . be a quadratic polynomial with a positive leading coefficient. There exists a positive real number 
such that 
and 
for 
. Compute 
.
such that 
and 
is a multiple of 
.
are such that 
and ![\[\frac{x}{26-x^2}=\frac{y}{26-y^2}=\frac{xy}{26-(xy)^2}.\]](http://latex.artofproblemsolving.com/d/7/4/d744d40b3d9e719d30777cc47035fe9e9f6fc44d.png)
What is 
?
. [/center]
, and 
, 
. This is proven in solution 1a-1. We will refer to this as Trick 1.
be the set of permutations of n-degree and 
. Prove that if 
, then 
(where 
is the identical permutation).
be subgroups of the group 
and 
. Prove that exists 
such as 
.
, which is equivalent to 
for any integer .
and be positive integers such that 
is coprime with for 
.
for 
.
for some 
where 
.
is said to be good if there exists a partition of 
into disjoint proper subsets such that the sum of the numbers in each subset of the partition is . How many good numbers are there?
the product of whose digits equals 
.
, where there are hundred 
in the last term in the sum. How many times does the digit 
occur in the number
wants the 5th derivative, raise the function to the 5th power, or plug the thing back in the function five times?
on any given shot. During the hunt, Ali shoots 
times, and Veli shoots 
times. What is the probability that Veli hits more ducks than Ali?
ducks
Now, we find the probability that Veli shoots at least 
duck. The probability of this happening is
Therefore, the probability that Veli hits more ducks than Ali when Ali hits ducks is:
Full answer:
and 
be the number of ducks Ali and Veli have hit after both have taken 12 shots, respectively. Let 
be the probability Veli wins overall. We see that: 
We know all of the conditional probabilities already. If Ali is ahead on Veli's 13th shot, Veli cannot beat Ali, so 
. If they are tied going into the 13th shot, Veli has a 
chance of winning, so 
. If Veli is already ahead going into the 13th shot, she wins no matter what, so 
. If we then let 
, then 
. Therefore, our final answer is 
chance of hitting a duck on the 13th shot