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Questions (43)

If approximately 70% of the Earth's surface is covered with water, what is the order of magnitude for the number of raindrops in
1. 3 answers
2. 525 views
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
1. 4 answers
2. 597 views
The real numbers x and y satisfy the nonlinear system of equations2x^2−6xy+2y^2+43x+43y=174, x^2+y^2+5x+5y=30. Find the
1. 1 answer
2. 465 views
Integers a, b, c, d and e satisfy 50<a<b<c<d<e<500, and a,b,c,d,e form a geometric sequence. What is the sum of all possible
1. 1 answer
2. 397 views
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?
1. 1 answer
2. 418 views
Let α and β be the roots of 3x^2+4x+9=0. Then (1+α)(1+β) can be expressed in the form a/b, where a and b are coprime
1. 1 answer
2. 409 views
Estimate the time difference between the longest day and the shortest day of a year in seconds if you lived on the Earth's
1. 4 answers
2. 765 views
The game Slice is played using a m×n rectangular piece of paper as a board. Players alternate turns, on each turn they choose a
1. 5 answers
2. 644 views
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
1. 4 answers
2. 523 views
A tunable capacitor (with variable capacitance) is charged by a U0=12 V battery and then is connected in parallel to a R=3 Ω
1. 1 answer
2. 496 views
Many of you may have noticed the phenomenon that basketballs get flat if the weather is cold. If a basketball was inflated to a
1. 2 answers
2. 676 views
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
1. 2 answers
2. 595 views
Let x,y,z be non-negative real numbers satisfying the condition x+y+z=1. The maximum possible value ofx^3*y^3+y^3*z^3+z^3*x^3
1. 1 answer
2. 412 views
What is the sum of all integer values of x such that(x^2−17x+71)^(x^2−34x+240)=1?
1. 3 answers
2. 532 views
The Fibonacci sequencea1=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
1. 2 answers
2. 1,193 views
Suppose a,b, and c are positive integers such thata+b+c+ab+bc+ca+abc=1000.
1. 1 answer
2. 386 views
There are four complex fourth roots to the number 4−43√i. These can be expressed in polar form asz1=r1(cosθ1+isinθ1)
1. 3 answers
2. 1,178 views
Let S be a set of 31 equally spaced points on a circle centered at O, and consider a uniformly random pair of distinct points A
1. 4 answers
2. 965 views
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
1. 10 answers
2. 460 views
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
1. 6 answers
2. 561 views
Calvin's River Crossing Attempt180 points Calvin is playing a game of Dungeons and Dragons. In order to make it across the
1. 5 answers
2. 524 views
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
1. 1 answer
2. 374 views
Let ABCD be a rectangle such that AB=5 and BC=12. There exist two distinct points X1 and X2 on BC such that
1. 1 answer
2. 345 views
Equilateral triangle ABC has a circumcircle Γ with center O and circumradius 10. Another circle Γ1 is drawn inside Γ such
1. 1 answer
2. 545 views
The sequence {ak}112 (base)k=1 satisfies a1=1 and an=1337+n/an−1, for all positive integers n. LetS=⌊a10a13+a11a14+a12a15+�
1. 1 answer
2. 478 views
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
1. 6 answers
2. 1,084 views
Alex and Bella play the following game. They first choose a positive integer N, and take turns writing numbers on a blackboard.
1. 2 answers
2. 559 views
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,
1. 3 answers
2. 517 views
How many pairs of positive integers (a,b), where a≤b satisfy 1/a+1/b=1/50?
1. 4 answers
2. 838 views
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
1. 1 answer
2. 470 views
A point charge of charge 1 mC and mass 100 g is attached to a non-conducting massless rod of length 10 cm. The other end of the
1. 2 answers
2. 672 views
Ravi wants to trisect an angle AOB, which has measure θ. From A, he drops a perpendicular to side OB, intersecting at C. He
1. 1 answer
2. 457 views
A national math contest consisted of 11 multiple choice questions, each having 11 possible answers. Suppose that 111 students
1. 1 answer
2. 446 views
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
1. 1 answer
2. 520 views
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
1. 1 answer
2. 480 views
Samir had prepared the problem tests for Stages 1 to 5 of Geometry and Combinatorics for next week but forgot to label which
1. 1 answer
2. 1,277 views
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
1. 1 answer
2. 479 views
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
1. 5 answers
2. 449 views
For how many positive integers 1≤k≤1000 is the polynomial fk(x)=x^3+x+k irreducible?
1. 1 answer
2. 324 views
Find the sum of integers c for all triples of integers (a,b,c),a≤b≤c, that satisfy the system of equationsa^2−bc=91
1. 1 answer
2. 471 views
Evaluate gcd(19!+19,20!+19).Details and assumptions The number n!, read as n factorial, is equal to the product of all positive
1. 4 answers
2. 1,257 views
Determine the last three digitsof 2^5+3^5+4^5+......+10,000,000^5 + 2^7+3^7+4^7+......+10,000,000^7
1. 1 answer
2. 296 views
What is the largest prime factor of 5^8+2^2?
1. 1 answer
2. 344 views

Answers (11)

please give me solution!!!!! Please
its wrong
plesse tell me write answer
thanks
its wrong
20
yes
thanks
meter
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?
its not

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