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what is a diophantine equation?
1.We have 8 pieces of strawberry candy and 7 pieces of pineapple candy. In how many ways can we distribute this candy to 4 kids?
2. In how many ways can we distribute 13 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on receiving an equal number of pieces?
3. 4 students are running for club president in a club with 50 members. How many different vote counts are possible, if everyone is required to vote?
I don't really know what is an absorbing Markov chain. Do you have a way to solve it using only algebra and combinations?
1. A point is always at 0, 1, or 2. At each step, the point moves 1 unit, either up or down. If the point is at 1, it moves to either 0 or 2 with equal probability. However, if the point is at 2, it must move to 1 with its next step.
For a point currently at 1, let t_1 be the expected number of steps the point will take before it reaches 0 for the first time. Similarly, let t_2 be the expected number of steps for the point to first reach 0 if the point instead starts at 2.
(a) Consider the next step for a point that is currently at 2. Express t_2 in terms of t_1.
(b) Now consider the next step for a point that is currently at 1. Give an expression for t_1 in terms of t_2 that is not equivalent to (a).
(c) Use your answers to (a) and (b) to find t_1 and t_2.
2. Prove that (2n combination n) is always even for n>0.
In this problem we will discover and prove another identity.
(a) Evaluate:
(i) {1\choose 0}^2 + {1\choose 1}^2.
(ii) {2\choose 0}^2 + {2\choose 1}^2 + {2\choose 2}^2
(iii) {3\choose 0}^2 + {3\choose 1}^2 + {3\choose 2}^2 + {3\choose 3}^2.
(iv) {4\choose 0}^2 + {4\choose 1}^2 + {4\choose 2}^2 + {4\choose 3}^2 + {4\choose 4}^2.
(b) Do your answers in part (a) appear on Pascal's Triangle? If so, where?
(c) Guess an identity based on your observations in parts (a) and (b).
(d) Test your identity by putting in n=1, 2, 3, 4 and making sure you get the relationships you found in part (a).
(e) Prove your relationship using a block walking argument.
(f) Prove your relationship using a committee-forming argument.
1. Two lines \ell and m intersect at O at an angle of 28^\circ. Let A be a point inside the acute angle formed by \ell and m. Let B and C be the reflections of A in lines \ell and m, respectively. Find the number of degrees in \angle BAC.
2.Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC].
3.A sphere with radius 3 is inscribed in a conical frustum of slant height 10. (The sphere is tangent to both bases and the side of the frustum.) Find the volume of the frustum.
4.In tetrahedron ABCD, \angle ADB = \angle ADC = \angle BDC = 90^\circ. Let a = AD, b = BD, and c = CD.
(a) Find the circumradius of tetrahedron ABCD in terms of a, b, and c. (The circumradius of a tetrahedron is the radius of the sphere that passes through all 4 vertices of the tetrahedron.)
(b) Let O be the circumcenter of tetrahedron ABCD. Prove that \overline{OD} passes through the centroid of triangle ABC.
(The circumradius of a tetrahedron is the radius of the sphere that passes through all four vertices, and the circumcenter is the center of this sphere.)
so its 72?
but its wrong
thanks, its correct
so i guess it is computer marked
ok so is \theta 144 degrees?
yeah its for my hw on aops
so is the answer 10?
is it still five
so how many different ways are there
what do you mean by computer marked?
OPQRSTUVWX is a regular decagon. A rotation of \theta degrees about U maps X to R. Given \theta < 180 degrees, find \theta
i don't get about how many ways you can get when you can also reflect
for no. 2 i got 5 and for no. 1 i got 6 but i don't know about no.3
i got 0 since 3/2+-3/2 is 0
what do you mean by just say minus the same for the second one
i got one value for k (3/2) but i don't know how to get the other one
When a square of area 4 is dilated by a scale factor of k, we obtain a square of area 9. Find the sum of all possible values of k.
but i don't really understand the first problem
1. Point Q is the image of point P under a dilation with center O and scale factor 4. If PQ = 18, then what is OP?
2.We cut a regular pentagon out of a piece of cardboard, and then place the pentagon back in the cardboard.
How many different ways can we place the pentagon back in the cardboard, if we are allowed to rotate but not reflect the pentagon?
3.How many different ways can we place the pentagon back in the cardboard, if we are allowed to rotate and reflect the pentagon?
thanks