06/12/25

Find the sum of floor function values
For positive integers a, b, show that the sum∑m=1[a/2][bm/a] + ∑n=1[b/2][an/b] = [a/2][b/2] + [(a,b)/2].where the bracket denotes the floor function, i.e. [x] is the greatest integer not exceeding... more
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06/12/25

Counting subsets of a finite set of integers
Let S be a set of n distinct integers and A be any subset of S. Let σ(A) be the sum of the elements of A, e.g. σ(A) = 4 if A = {1,2,3} (be sure to include the empty set, for which the sum of... more
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06/09/25

Counting certain partitions
Consider the partition of an integer into the sum of some integers:n = n1 + n2 + ... + nr, with each ni ≥ 1. How many such partitions are there? An equivalent question: how many integer solutions... more
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06/09/25

Counting certain binary strings
Let b be a binary string of length r with no adjacent 0s or 1s, e.g. b can be 01010 or 1010 but not 1001 or 0010. For n > r, perform the following operation on a binary string c of length n: if... more
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06/08/25

Monotone subsequence of an infinite sequence
Does every infinite sequence have a monotone subsequce? Here by "monotone" we may either (non-strictly) increasing or decreading. More precisely, given a sequence (an)n≥1, does there exist a... more
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06/08/25

Integer square of the form 111...1?
Can an integer of the form 111...1 (there are n digits, with n ≥ 2) be the square of an integer?
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06/07/25

Perfect square of the form 111...1?
Can a number of the form 111...1 (n digits, n≥2) be a perfect square?
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06/07/25

Slope of the secant line?
Hi sorry if this seems obvious. How does one go about this and what is the formula or steps to find the secant line?If Q is the point __ , find the slope of the secant line PQ (correct to six... more
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05/28/25

ACT Math - Geometry and Optimization
A rectangular playground is being built next to a school building, which will form one of the longer sides of the playground. If the school has 80 meters of fencing available to enclose the... more
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05/28/25

integral with a derivative function
if g'(x) = f(x) with g(-11)=a express the value of the integral of -4 to 0 f(x+5)dx in terms of a
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03/28/25

Finding Slope of Lines Parallel to 9x + 4y = 7
What is the slope of any line parallel to the line 9x + 4y = 7 ?
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06/23/23

Using Power Series - Find The Solution of a Differential Equation
xy''+2y'+xy=0y(0)=1 and y'(0)=01) Find out where the first point on the positive-x-axis is located where the solution to the initial value problem is zero.2) Find the closed form of the solution... more
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05/28/23

Lesbegue Measure of a Set Given the Union of a Set is Measurable

Question:Let A ⊆R such that m∗(A) = 0.Show that if A ∪ B ∈ L, then B ∈ L.My approach:We can write B asB = (A ∪ B) ^ (A^(Bc)), where ^ denote intersection.We know that the intersection of two... more
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04/17/23

Help Linear Algebra
Find formulas for the entries of M^n, where n is a positive integer. (Your formulas should not contain complex numbers.)
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04/11/23

help Linear algebra
has 3 distinct eigen values if and only if__<k<__what are the 2 values for the blanks
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02/21/23

Prove Cassini’s identity?by induction
Prove Cassini’s identity: fn1 · fn+1 − (fn)2 = (−1)n for any n ≥ 2, there fn is the n-th Fibonacci number.
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02/16/23

Count number of binary strings that satisfy some conditions?
Let b be binary string of length r < n, b doesnt contain any adjacent 0s or 1s. Count number of binary string of length n that if we repeatedly apply these operations 1) 11 =1, 2) 00=0. b doesnt... more
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08/26/22

Consider the direction field below for a differential equation y' = y(y-3)(y-5)
1.Find the equilibrium solutions for the differential equation.where I have found y = (0,3,5)2.If the initial condition is y(0) = c, for what values of c is limt→∞ y(t) finite?Answer:(as an... more
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03/09/22

question about bid-ask spread, what do you think?
Bank Z receives U$1,000 in deposits from Tom and promises to pay him U$1,100 after one year. With this money, the bank will lend U$10 to another 100 people at a rate je, such that each of these 100... more
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02/23/22

MATHEMATICS 3 question
Prove that there is a mapping from a set to itself that is one-to-one but not onto iff there is a mapping from the set to itself that is onto but not one-to-one. [Hint: You need to distinguish... more
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02/17/22

discrete math question

prove that the sum of k C(n,k), 1<=k<=n is n x 2^(n-1)
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12/07/21

A factory manufactures three products, A, B, and C.

A factory manufactures three products, A, B, and C. Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per... more
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10/26/21

Solve the system of congruences

Solve the system of congruences ( 2x + 6y ≡ 3 (mod 15) 3x + 7y ≡ 6 (mod 15)
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04/12/21

Need help with surjective/injective
Consider the function f:C→C defined by f(x+iy)=(3x+2)+i(y2−1) where x,y are elements of R. Is the function injective? Explain your reasoning. Is the function surjective? Explain your... more
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03/03/21

Use the sandwich lemma to solve --> xn := 1/ sqrt (n^ 2 +1) + 1 /sqrt (n^ 2 +2) +...+ 1 /sqrt (n^ 2 +n) converges to 1
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