Mathslover

What is the largest prime factor of 5^8+2^2?

12 years ago

Evaluate gcd(19!+19,20!+19). Details and assumptions The number n!, read as n factorial, is equal to the product of all positive integers less than or equal to n. For example, 7!=7×6×5×4×3×2×1.

12 years ago

Find the sum of integers c for all triples of integers (a,b,c),a≤b≤c, that satisfy the system of equations a^2−bc=91 b^2−ac=91 c^2−ab=91 Details and assumptions If a number c appears in several different triples (a,b,c), it must be counted with multiplici...

12 years ago

For how many positive integers 1≤k≤1000 is the polynomial fk(x)=x^3+x+k irreducible?

12 years ago

A graph G has 200000 edges and for any 3 vertices v,w,x, at least one of the edges vw,wx,xv is not present in G. What is the least number of vertices that G can have?

12 years ago

Let Pn be the set of all subsets of the set [n]={1,2,…,n}. If two elements of P5 are chosen at random, the expected number of elements (of [n]) that they have in common can be expressed as a/b where a and b are coprime positive integers. What is the value...

12 years ago

Samir had prepared the problem tests for Stages 1 to 5 of Geometry and Combinatorics for next week but forgot to label which test was for which stage. Since Samir didn't label them, the computer assigned them labels 1 through 5 randomly, with each label a...

12 years ago

For a set of numbers T, we say that T has distinct subset sums if all distinct subsets of T have distinct sums. How many subsets of {1,2,3,4,5,6,7,8} have distinct subset sums? Details and assumptions The empty set (the set of no elements) has sum 0 by co...

12 years ago

A national math contest consisted of 11 multiple choice questions, each having 11 possible answers. Suppose that 111 students actually wrote the exam, and no two students has more than one answer in common. The highest possible average mark for the studen...

12 years ago

Ravi wants to trisect an angle AOB, which has measure θ. From A, he drops a perpendicular to side OB, intersecting at C. He then constructs an equilateral triangle ACD on the opposite side of AC as compared to O. He claims (without any justification) that...

12 years ago

x and y are positive real numbers that satisfy log(base)x y + log(base)y x = 17/4 and xy=288√3. If x+y=a+b√c, where a, b and c are positive integers and c is not divisible by the square of any prime, what is the value of a+b+c?

12 years ago

How many pairs of positive integers (a,b), where a≤b satisfy 1/a+1/b=1/50?

12 years ago

Let A=a1,a2,…,ak and B=b1,b2,…,bj be sequences of positive integers such that a1≥a2≥⋯ak≥1, b1≥b2≥⋯bj≥1, ∑i=1k ai≤6, and ∑j i=1 bi≤6. For how many ordered pairs of sequences (A,B) satisfying the above conditions can we find a table T with {0,1} entries suc...

12 years ago

Alex and Bella play the following game. They first choose a positive integer N, and take turns writing numbers on a blackboard. Alex starts first, and writes the number 1. After that, if the number k is on the board, the next player may write down either...

12 years ago

Circle Γ with center O has diameter AB=192. C is a point outside of Γ, such that D is the foot of the perpendicular from C to AB and D lies on the line segment OB. From C, a tangent to Γ is drawn, touching Γ at E, where the foot of the perpendicular from...

12 years ago

The sequence {ak}112 (base)k=1 satisfies a1=1 and an=1337+n/an−1, for all positive integers n. Let S=⌊a10a13+a11a14+a12a15+⋯+a109a112⌋. Find the remainder when S is divided by 1000. Details and assumptions The function ⌊x⌋:R→Z refers to the greatest integ...

12 years ago

How many permutations σ of the set {1,2,…,15} are there such that σ(1)=1,∣σ(n)−σ(n−1)∣≤2 for 2≤n≤15? Details and assumptions σ(n) denotes the nth position of the permutation.

12 years ago

Calvin's River Crossing Attempt 180 points Calvin is playing a game of Dungeons and Dragons. In order to make it across the river, he needs to throw six 4-sided dice, and have their sum be a multiple of 5. How many different dice throws result in Calvin m...

12 years ago

Jack has 222 lego cubes, each of side length 1. He puts them together to form a rectangular prism. If the perimeter of the base of the prism is 10, what is the height of the prism?

12 years ago

A sphere of radius 32√ is tangent to the edges AB, AD, AA1, and the face diagonal CD1 of the cube ABCDA1B1C1D1. The volume of the cube can be written as a+bc√, where a, b are integers and c is a square-free positive integer. What is the value of a+b+c?

12 years ago

Suppose a,b, and c are positive integers such that a+b+c+ab+bc+ca+abc=1000.

12 years ago

The Fibonacci sequence a1=1,a2=1,a3=2,a4=3,a5=5,a6=8… is defined recursively using the formulas a1=a2=1 and an+2=an+an+1 for all n≥1. Find the greatest common divisor of a484 and a2013.

12 years ago

What is the sum of all integer values of x such that (x^2−17x+71)^(x^2−34x+240)=1?

12 years ago

Calvin is playing a game of Dungeons and Dragons. In order to make it across the river, he needs to throw six 4-sided dice, and have their sum be a multiple of 5. How many different dice throws result in Calvin making it across the river? Details and assu...

12 years ago

Find the smallest number N such that: The number of divisors of N is A. The number of divisors of A is B. The number of divisors of B is C. The number of divisors of C is 3. Details and assumptions The divisors include 1 and the number itself. For example...

12 years ago

Estimate the time difference between the longest day and the shortest day of a year in seconds if you lived on the Earth's equator with the assumptions below. Note: this is not the difference between solstices as we are adjusting the earth's rotation axis...

12 years ago

S=1+2*(1/5)+3*(1/5)^2+4(1/5)^3...... If S=a/b, where a and b are coprime positive integers, what is the value of a+b?

12 years ago

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?

12 years ago

If approximately 70% of the Earth's surface is covered with water, what is the order of magnitude for the number of raindrops in the world's oceans? Hint: The order of magnitude of 2478=2.478×10^3 is 3.

12 years ago

its not

12 years ago

Let ABCD be a rectangle such that AB=5 and BC=12. There exist two distinct points X1 and X2 on BC such that ∠AX1D=∠AX2D=90∘. Suppose that d is the distance from X1 to X2. What is d2?

12 years ago

meter

12 years ago

thanks

12 years ago

yes

12 years ago

20

12 years ago

its wrong

12 years ago

thanks

12 years ago

plesse tell me write answer

12 years ago

its wrong

12 years ago

please give me solution!!!!! Please

12 years ago