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A natural number n is given. Let f(x,y) be a polynomial of degree less than n such that for any positive integers x,y≤n, x+y≤n+1 the equality f(x,y)=x/y holds. Find f(0,0).
Let f(z) be the degree n polynomial z^n + 2*z^(n-1) + 3*z^(n-2) + ... + (n-1)*z^2 + n*z + n+1.
Prove all n roots of f(z) have a magnitude greater than 1; i.e. if f(z)=0 then |z|>1.
A pentagram is typically constructed by taking the diagonals of a regular pentagon.
This common pentagram has five angles measured at the vertices, each of which equals 36 degrees. In total all five angles sum to 180 degrees.
Generalize to make an irregular pentagram by taking the diagonals of a convex irregular pentagon.
This irregular pentagram also has five angles measured at the five vertices. Show that the sum of these five angles equals 180 degrees.
A certain road has a path of a perfect circle with a single entrance/exit. A woman enters the road and walks the full circumference at a constant speed without stopping or changing direction. During her time on the road, N cars, each at its own random time during the duration of the walk, enter the circle. Each car proceeds, on the shortest path, to its own randomly selected stopping point on the circle. If cars travel 10 times as fast as the woman walks, answer the following:
1) For N=1, what is the probability that the woman “encounters” a car? Definition: An “encounter” is when a moving car either overtakes the woman in the same direction or passes her while going in the opposite direction. If a car is stationary, there can be no encounter.
2) What is N such that there is at least a 75% chance of encountering a car?
3) For N=20, what is the expected number of encounters?
On my jogging days I start at 6:00 sharp and follow a paved road, heading strictly North. At some point this same-level road turns West, but I go on on a path heading North going uphill till I reach an antenna site located on top of the hill.
I rest there for 10 minutes exactly then return following the same route in opposite direction.
I arrive home at 08:10. My average speeds are: on the paved road 6 mph, uphill 4.8 and downhill 8,
What is the distance between my home and the point of return?
The radius of the circumscribed circle of an acute-angled triangle is 23 and the radius of its Inscribed circle is 9. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.
The following fractions are written on the board 1/n, 2/(n-1), 3/(n-2), ... , n/1 where n is a natural number. Alice calculated the differences of the neighboring fractions in this row and found among them 10000 fractions of type 1/k (with natural k). Prove that he can find even 5000 more of such these differences.
a) Prove that there exists a differentiable function f:(0, ∞)->(0, ∞) such that f(f'(x))=x, for all x>0.
b) Prove that there is no differentiable function f:R->R such that f(f'(x))=x, for all x∈R.
Find the number of trapeziums that it can be formed with the vertices of a regular polygon of n sides.
Determine a cuboid with minimal surface area, if its volume is strictly greater than 1000, and the lengths of its sides are integer numbers.
In the Project Euler problem Tom and Jerry, it defines Tom graphs. A cat named Tom and a mouse named Jerry play a game on a graph G. Each vertex of G is a mouse hole. Jerry hides in one of the holes. Then, Tom tries to catch Jerry by picking a hole. Then, Jerry moves to an adjacent hole. Then, Tom picks another hole. They keep doing this, where Jerry moves to an adjacent hole and Tom picks a hole. A Tom graph is a graph that Tom can always catch Jerry by following a sequence of holes.
Example: This is a Tom graph.
1---2---3
Tom can do the sequence 2, 2 to guarantee catching Jerry.
1. Prove that all Tom graphs are cycle-free.
2. Find the smallest cycle-free graph that is not a Tom graph.
A well-known phrase:
6 2 2/3
→ 7 - E & 7/8 ←
8 15/7
Pick up at random a 3-digit number and decrease it by the sum of its digits.
Clearly,applying this procedure again and again you will hit zero no matter how big your initial number was .
How many iterations at most are needed and how many numbers cause it?
Three students write on the blackboard next to each other three two-digit squares. In the end, they observe that the 6-digit number thus obtained is also a square. Find this number!
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