Two congruent circles Γ1 and Γ2 each have radius 213,and the center of Γ1 lies on Γ2. Suppose Γ1 and Γ2intersect at A and B. The line through A perpendicular to AB meets Γ1 and Γ2 again at C and D, respectively. Find the length of CD.
Two congruent circles Γ1 and Γ2 each have radius 213,and the center of Γ1 lies on Γ2. Suppose Γ1 and Γ2intersect at A and B. The line through A perpendicular to AB meets Γ1 and Γ2 again at C and D, respectively. Find the length of CD.
Consider the Collatz function defined on the positive integers:
F (n)=
Case I n is even then n/2, case 2 n is odd then 3n+1
Find the smallest value of n such that f^(7)(n)=5.
Details and assumptions
f(7)(n) means the function f applied 7 times. I.e. f^(7)(n)=f(f(f(f(f(f(f(n))))))).Note that the function is only defined on the positive integers. Hence, your answer must be a positive integer.
S=1+2(1/5)+3(1/5)^2+4(1/5)^3…..... (goes on infinitely)
If S=a/b, where a and b are co prime positive integers, what is the value of a+b?
Find the sum of all primes p, such that p divides up, where u_p is the p-th Fibonacci number.
Suppose f(x) is a polynomial with integer coefficients of degree 100. Find the biggest possible number of pairs of integers n
ABC is a triangle with ∠BAC=60∘. It has an incircle Γ, which is tangential to BC at D. It is given that BD=3 and DC=4. What is the value of [ABC]^2?
Let
f(n)=1/(√1+√2)+1(√2+√3)+1(√3+√4)…1(√n+√n+1(whole sqrt above n+1))
For how many positive integers n, in the range 1≤n≤1000, is f(n) an integer?
ABC is an isosceles triangle with AB=BC and
∠ABC=123. D is the midpoint of AC, E is the foot of the perpendicular from D to BC and F is the midpoint of DE. The intersection of AE and BF is G. What is the measure (in degrees) of BGA?
Let S(N) denote the digit sum of the integer N. As Nranges over all 3-digit positive numbers, what value of Nwould give the minimum of M=N/S(N)?
Find the largest possible degree n≤1000 of a polynomial p(x) such that:
p(i)=i for every integer i with 1≤i≤n
p(−1)=1671
p(0) is an integer (not necessarily 0).
ABCD is a square. Γ1 is a circle that circumscribes ABCD (i.e. Γ1 passes through points A,B,C and D). Γ2is a circle that is inscribed in ABCD (i.e. Γ2 is tangential to sides AB,BC,CD and DA). If the area of Γ1 is 100, what is the area of Γ2?
Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She chooses 40 distinct pennies uniformly at random and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. After this process, what is the expected number of pennies showing heads?
Γ1 is a circle with center O1 and radius R1, Γ2 is a circle with center O2 and radius R2, and R2 is less than R1 Γ2 has O1 on its circumference. O1O2 intersect Γ2 again at A. Circles Γ1and Γ2 intersect at points B and C such that ∠CO1B=52∘. D is a point on the circumference of Γ1that is not contained within Γ2. The line DB intersects Γ2at E. What is the measure (in degrees) of the
How many positive integers less than 1000000 have the sum of their digits equal to 7?
What is the sum of all possible positive integer values of n, such that n2+19n+130 is a perfect square?
For how many positive integer values of c does the equation 2x^2+689x+c=0 have an integer solution?
wow, what kind of Thread -_-
i'll give you example..
any one get this ?
A 999-digit number starts with 9. Every 2 consecutive digits is divisible by 17 or 23. There are 2 possibilities for the last 3 digits. What is the sum of these 2 possibilities?
Details and assumptions
If you believe that the sum is over 999, enter your answer as 999.
If the number is 12345, then every 2 consecutive digits refers to the number 12, 23, 34 and 45.
The number starts with 9 if the leftmost digit is a 9.