Continuation of the old thread.
These are selected problems I've done in the past few days that I've found interesting. Let's see how the ATOT crowd handles it!
1. Find the sum of all positive integers n for which n^2-19n+99 is a perfect square.
2. In trapezoid ABCD, leg BC is perpendicular to bases AB and CD, and diagonals AC and BD are perpendicular. Given that AB=sqrt(11) and AD=sqrt(1001), find BC^2.
3. The points A, B, and C lie on the surface of a sphere with center O and radius 20. It is given that AB=13, BC=14, CA=15, and that the distance from O to triangle ABC is x. Find x in exact terms.
4. The function f is defined by f(x)=(ax+b)/(cx+d), where a, b, c, and d are nonzero real numbers, has the properties f(19)=19, f(97)=97, and f(f(x))=x for all values of x except -d/c. Find the unique number that is not in the range of f.
5. Consider the polynomials P(x)=x^6-x^5-x^3-x^2-x and Q(x)=x^4-x^3-x^2-1. Given that p, q, r, and s are the roots of Q(x)=0, find P(p)+P(q)+P(r)+P(s).
6 A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects on eof the two verticies where it is not currently located, and crawls along a side of the tirangle to that vetex. Given that the probability that the bug moves to its starting vertex on its tenth move is x, find x.
7. Given that z is a complex number such that z+1/z=2 cos(3), find the least integer that is greater than z^2000+1/z^2000. (Angles are in degrees in this problem)
8. Let v and w be distinct, randomly chosen roots of the equation z^1997-1=0. Let x be the probability that sqrt(2+sqrt(3)) is lesser or equal to the absolute value of (v+w). Find x.
NOTE: Please include solutions with your answers. Tell me how you arrived at your answer.
I solved all of these problems without calculators or computers so you should too. They become too easy and not fun when you use calculators, and are designed without them.
Good Luck! :beer:
These are selected problems I've done in the past few days that I've found interesting. Let's see how the ATOT crowd handles it!
1. Find the sum of all positive integers n for which n^2-19n+99 is a perfect square.
2. In trapezoid ABCD, leg BC is perpendicular to bases AB and CD, and diagonals AC and BD are perpendicular. Given that AB=sqrt(11) and AD=sqrt(1001), find BC^2.
3. The points A, B, and C lie on the surface of a sphere with center O and radius 20. It is given that AB=13, BC=14, CA=15, and that the distance from O to triangle ABC is x. Find x in exact terms.
4. The function f is defined by f(x)=(ax+b)/(cx+d), where a, b, c, and d are nonzero real numbers, has the properties f(19)=19, f(97)=97, and f(f(x))=x for all values of x except -d/c. Find the unique number that is not in the range of f.
5. Consider the polynomials P(x)=x^6-x^5-x^3-x^2-x and Q(x)=x^4-x^3-x^2-1. Given that p, q, r, and s are the roots of Q(x)=0, find P(p)+P(q)+P(r)+P(s).
6 A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects on eof the two verticies where it is not currently located, and crawls along a side of the tirangle to that vetex. Given that the probability that the bug moves to its starting vertex on its tenth move is x, find x.
7. Given that z is a complex number such that z+1/z=2 cos(3), find the least integer that is greater than z^2000+1/z^2000. (Angles are in degrees in this problem)
8. Let v and w be distinct, randomly chosen roots of the equation z^1997-1=0. Let x be the probability that sqrt(2+sqrt(3)) is lesser or equal to the absolute value of (v+w). Find x.
NOTE: Please include solutions with your answers. Tell me how you arrived at your answer.
I solved all of these problems without calculators or computers so you should too. They become too easy and not fun when you use calculators, and are designed without them.
Good Luck! :beer:
