April 29th:
1. [easy] Find a 10-digit number so that the first digit is divisible by 1, the first two digits are divisible by 2, the first three digits are divisible by 3, and so on. Solved by JujuFish
2. [medium] Find the smallest integer k such that 1^2+2^2+3^2+......+k^2 is divisible by 200. Solved by chuckywang
3. [hard] Two positive integers differ by 60. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? Solved by chuckywang
4. [bonus] The equation 26+62=126 is FALSE. However, if one moves one of the digits to another place, the equation becomes true. Find the move. (Note: Rotation, Translating, is allowed. Reflection, i.e. flipping, is NOT allowed).Solved by dullard
5. [easy] Let N be the greatest integer multiple of 8, no two of whose digits are the same. Find N. Solved by JujuFish
*6. [medium] Circles A, B, and C are externally tangent to each other and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?
7. [hard] Given that log(sin x)+log(cos x)=-1 and that log(sin x+cos x)=1/2*(log(n) -1), find n. (We are assuming base 10.) Solved by dullard
8. [easy] Let N be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of N forms a perfect square. What are the leftmost three digits of N? Solved by chuckywang
9. [medium] Two distinct, real, infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is 1/8. Find the second term of both series. Solved by chuckywang
*10. [hard] How many positive integer divisors of 2004^(2004) are divisible by exactly 2004 positive integers?
11. [hard] How many positive integers less than 10,000 have at most two different digits? Solved by sao123
*12. [easy] Find the sum of all positive two-digit integers that are divisible by each of their digits.
*13. [medium] Jane is 25 years old. Dick is older than Jane. In n years, where n is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let d be Dick's present age. How many ordered pairs of positive integers (d,n) are possible?
*14. [hard] The perimeter of triangle APM is 152, and angle PAM is a right angle. A circle of radius 19 with center O on AP is drawn so that it is tangent to AM and PM. Find the length of OP.
*15. [hard] Given that 1/(2!17!)+1/(3!16!)+1/(4!15!)+1/(5!14!)+1/(6!13!)+1/(7!12!)+1/(8!11!)+1/(9!10!) =N/(18!), find N.
*16. [hard x3] A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of hte paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of the strip is folded over to coincide with and lie on top of hte left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. How many of these squares lie below the square that was originally the 942nd square counting from the left?
*17. [hard x10] Find the least positive integer n such that 1/(sin45*sin46) + 1/(sin47*sin48) + .... + 1/(sin133*sin134)=1/sin(n) (Note: all the angles are in degrees.)
*=unclaimed problem
The first person to provide the answer to any of these problems will have their s/n edited in.
If you need clairification with a question, please speak up.
Good Luck!
:beer:
Edit: From now on, any solution must have a short description along with it. I dont' need a proof, but a short explanation would work. Of course, hard problems should have longer explanations than easy problems.
Edit2: All these problems can be done without electronics, and should be done without electronics.. So, in your explanation, I shouldn't see methods that require massive amounts of computation.
1. [easy] Find a 10-digit number so that the first digit is divisible by 1, the first two digits are divisible by 2, the first three digits are divisible by 3, and so on. Solved by JujuFish
2. [medium] Find the smallest integer k such that 1^2+2^2+3^2+......+k^2 is divisible by 200. Solved by chuckywang
3. [hard] Two positive integers differ by 60. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? Solved by chuckywang
4. [bonus] The equation 26+62=126 is FALSE. However, if one moves one of the digits to another place, the equation becomes true. Find the move. (Note: Rotation, Translating, is allowed. Reflection, i.e. flipping, is NOT allowed).Solved by dullard
5. [easy] Let N be the greatest integer multiple of 8, no two of whose digits are the same. Find N. Solved by JujuFish
*6. [medium] Circles A, B, and C are externally tangent to each other and internally tangent to circle D. Circles B and C are congruent. Circle A has radius 1 and passes through the center of D. What is the radius of circle B?
7. [hard] Given that log(sin x)+log(cos x)=-1 and that log(sin x+cos x)=1/2*(log(n) -1), find n. (We are assuming base 10.) Solved by dullard
8. [easy] Let N be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of N forms a perfect square. What are the leftmost three digits of N? Solved by chuckywang
9. [medium] Two distinct, real, infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is 1/8. Find the second term of both series. Solved by chuckywang
*10. [hard] How many positive integer divisors of 2004^(2004) are divisible by exactly 2004 positive integers?
11. [hard] How many positive integers less than 10,000 have at most two different digits? Solved by sao123
*12. [easy] Find the sum of all positive two-digit integers that are divisible by each of their digits.
*13. [medium] Jane is 25 years old. Dick is older than Jane. In n years, where n is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let d be Dick's present age. How many ordered pairs of positive integers (d,n) are possible?
*14. [hard] The perimeter of triangle APM is 152, and angle PAM is a right angle. A circle of radius 19 with center O on AP is drawn so that it is tangent to AM and PM. Find the length of OP.
*15. [hard] Given that 1/(2!17!)+1/(3!16!)+1/(4!15!)+1/(5!14!)+1/(6!13!)+1/(7!12!)+1/(8!11!)+1/(9!10!) =N/(18!), find N.
*16. [hard x3] A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of hte paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of the strip is folded over to coincide with and lie on top of hte left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. How many of these squares lie below the square that was originally the 942nd square counting from the left?
*17. [hard x10] Find the least positive integer n such that 1/(sin45*sin46) + 1/(sin47*sin48) + .... + 1/(sin133*sin134)=1/sin(n) (Note: all the angles are in degrees.)
*=unclaimed problem
The first person to provide the answer to any of these problems will have their s/n edited in.
If you need clairification with a question, please speak up.
Good Luck!
:beer:Edit: From now on, any solution must have a short description along with it. I dont' need a proof, but a short explanation would work. Of course, hard problems should have longer explanations than easy problems.
Edit2: All these problems can be done without electronics, and should be done without electronics.. So, in your explanation, I shouldn't see methods that require massive amounts of computation.
Facebook
Twitter