# Search results

1. ### Challenge Determine the minimum value

Determine the minimum value of the function f(x)=x^4-6x^2+8x-3 .
2. ### Challenge Evaluate the sum

Given that abx^2=(a-b)^2(x+1). Find 1+\dfrac{4}{x}+\dfrac{4}{x^2}.
3. ### Challenge Quadratic equation that has two roots

Find all integer valuse of k such that the quadratic expression (x+k)(x+1991)+1 can be factored as a product of (x+p)(x+q), where p and q are integers.
4. ### Challenge Simplify a sum

Simplify \tan x \cos 1 ^\circ + \sin 1^\circ.
5. ### Challenge Find the sum of squares

If x^3=12x+7y and y^3=7x+12y, evaluate x^2+y^2.
6. ### Challenge Solve for x

Solve the equation x^{\log x}=\dfrac{x^3}{100}.
7. ### Challenge Find the sum to infinity of a sequence

If \dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots=\dfrac{\pi^2}{6}, evaluate the exact value of \dfrac{1}{1^2}+\dfrac{1}{3^2}+\dfrac{1}{5^2}+\cdots
8. ### Practice Find derivative of the sum of two quotients

Find the derivative of \dfrac{\sin^2 x}{1+\cot x}+\dfrac{\cos^2 x}{1+\tan x}.
9. ### Challenge Logarithmic Challenge

Given that \log_{4n} 40\sqrt{3}=\log_{3n} 45, find n^3.
10. ### Challenge Find the area of the shaded region

Points P and Q are centers of the circles as shown below. Chord AB is tangent to the circle with center P. Given that the line PQ is parallel to chord AB and AB=x units, find the area of the shaded region.
11. ### Challenge Simplifying a quotient

Simplify \dfrac{2020^3-2002^3-18^3}{2020\cdot 2002 \cdot 18}.
12. ### Challenge Evaluation of a sum

Evaluate a^{a^8}+a^{a^2} if a^{a^4}=4 and a is real.

There is huge difference between good practice questions and bad practice questions. Bad practice questions, especially the one made up by anyone who knows nothing or just a little of mathematics can make a forum looks in a very bad taste. :confused: On the other hand, good practice problems...
14. ### Optimization problem

Suppose a and b are the values of x that satisfy the equation x^2-2px+p^2+2p+3=0 for some real number p. Minimize the expression of a^2+b^2.
15. ### Challenge Prove a sum identity

Prove that 1^2-2^2+3^2-4^2+\cdots+(-1)^{n-1}n^2=(-1)^{n-1}\dfrac{n(n+1)}{2}.

17. ### Product over Sum Problem

Evaluate \dfrac{1\cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}.
18. ### Challenge Solve for a side of triangle

In a triangle ABC, it is given that \sin(A+B)+\cos(2A-B)=2 and AB=4. Find BC.
19. ### Challenge Evaluate P(3)+P(9)

Evaluate P(3)+P(9) if P(x)=\dfrac{(x-\sqrt{2})(x-\sqrt{3})}{(1-\sqrt{2})(1-\sqrt{3})}+2\left(\dfrac{(x-1)(x-\sqrt{3})}{(\sqrt{2}-1)(\sqrt{2}-\sqrt{3})}\right)+3\left(\dfrac{(x-1)(x-\sqrt{2})}{(\sqrt{3}-1)(\sqrt{3}-\sqrt{2})}\right).
20. ### Challenge Evaluation of an expression

What is the value of 2(2(2(2(2(2+1)+1)+1)+1)+1)+1?