2014 SDMO (Middle School)

Part of SDMO (Middle School)

Subcontests

(5)

1

Triangular cupola volume

Below is a net consisting of

3

4

equilateral triangles, and

1

regular hexagon. Each polygon has side length

1

. When we fold this net to form a polyhedron, what is the volume of the polyhedron? (This figure is called a "triangular cupola".)
Net:
[asy] pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P; A = origin; B = (0,3); C = 3*dir(150); D = (0,1); E = (0,2); F = C+2*dir(30); G = C+dir(30); H = 2*dir(150); I = dir(150); J = (1,1); K = J+dir(30); L = (1,2); M = F+dir(120); N = G+dir(120); O = H+dir(240); P = I+dir(240); draw(A--B--C--cycle); draw(D--E--F--G--H--I--cycle); draw(D--E--L--J--cycle); draw(F--G--N--M--cycle); draw(H--I--P--O--cycle); draw(J--K--L--cycle); [/asy]
Resulting polyhedron:
https://upload.wikimedia.org/wikipedia/commons/9/93/Triangular_cupola.png

1

Sum of numbers and their reciprocals not both being zero

a

b

c

be nonzero real numbers. Prove that

a+b+c

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}

0

1

Restrictions on a,b with a+b=999

How many ordered pairs of nonnegative integers

\left(a,b\right)

a+b=999

such that each of

a

b

consists of at most two different digits? (These distinct digits need not be the same digits in both

a

b

. For example, we might have

a=622

b=377

1

Three trainers of a dog

A dog has three trainers:
[*]The first trainer gives him a treat right away. [*]The second trainer makes him jump five times, then gives him a treat. [*]The third trainer makes him jump three times, then gives him no treat.
The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?

1

Mix of Pythagorean Triples and Fermat's Last Theorem

Find the smallest positive three-digit integer

n

3^n+4^n

is divisible by

5