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Point D is lies on side BC of triangle ABC such that CD=2BD. It turned out that angle ADC=60 and angle ABC=45. Find angle BAC.

Let P(x) be a polynomial of degree 3 and x₁,x₂,x₃ are the solutions of P(x)=0.

Let (P(1/3)-P(-1/3))/P(0)=8, (P(1/4)-P(-1/4))/P(0)=9 and x₁+x₂+x₃=35.

Find the value of (x₂+x₃)/x₁+(x₁+x₃)/x₂+(x₁+x₂)/x₃.

A bug is crawling North along a wire when it comes to a wire frame in the shape of a cube. The top and bottom faces of this cube are perfectly level with the bug arriving at one of the lower level corners labelled A. Corners B, C, and D are also on the lower level naming counterclockwise. Upper level corners labelled E, F, G, and H are directly above A, B, C, and D respectively. The only connections to the outside world are at A and C. The wire exiting the cube at C is headed North.

Thus the cube represents a type of temporary trap for the poor bug. The bug travels along the wires taking 1 second to go from one corner to another. When it gets to either A or C there is a chance it will exit, and continue walking North from C or South from A. Once the bug decides to go in the exit direction, it is instantly free (do not count any time for the exiting move)

Bug Logic:
1. The bug never travels back along the same wire (and cannot reverse direction on a wire).
2. When encountering multiple choices which are all level, each path has an equal chance of being chosen.
3. The bug prefers going up to going level; and prefers level to down. Each is a 2:1 preference.
4. At a corner, the bug first decides between up/level or level/down; then if the choice was "level", it applies Rule 2.

What is the probability the bug will exit North?
What is the expected value of the number of seconds spent on the cube?

Same two questions if there were no up/level and level/down preference, and all directions at a corner (except returning back the same way) were equally likely?

Let ABC be a triangle with integral side lengths such that angle A=3 * angle B. Find the minimum value of its perimeter.

Positive integers a,b are such that 137 divides a+139b and 139 divides a+137b. Find the minimum possible value of a+b.

Let X₁, X₂, X₃, ..., X_n be a permutation of the integers 1,2,3,...,n.

Consider the sum:
abs(X₁-X₃) + abs(X₂-X₄) + abs(X₃-X₅) + ... + abs(X_n-2-X_n).

What is the mean value of this sum taken over all possible permutations?

Find the smallest three distinct whole numbers A, B and C such that you can rearrange the digits of A and B to get C^2, the digits of A and C to get B^2, and the digits of B and C to get A^2.

**** Leading zeroes are not allowed.

Seven horses in a farm in Kentucky,
Of three blacks, the oldest is a daddy;
Of two browns, the 3 year old male is called BILLY,

And the horse SNOWY is white as a lily.

Of two half-sister mares, the black won in the Derby
She's a shade lighter than her daughter, BEAUTY
BEAUTY and her cousin LUCKY, both half the age of their auntie,

But faster than either ROCKY or SUNNY.

Three of the horses are ready for the Derby,

A horse must be three years to be an entry;

Maybe this year the farm gets lucky, Their stallion's loss last year was a pity.

CHALLENGE: Describe the horse that lost in last year's derby.

Clues:

a. Famous fiddler
b. “The brave of the bravest”
c. Dazzling light
d. European city

By adding 2 letters to each of the words (not the same letters) defined by the above clues and scrambling all the letters create 4 new words matching the definitions below:

a. fruit
b. coin
c. country
d. drug

Permuting the 8 added letters one gets the name of another famous violinist.

Based on a concept of puzzle #177 of “Penguins problems book” ed. 1940.

A bug is placed at one corner of a wire frame in the shape of a cube. This cube currently does not have any sugar for the hungry bug.

The bug crawls along the 12 wires of the frame searching for something to eat. At each of the 8 corners the bug randomly chooses one of the 3 wires to follow next (including the one it just traveled).

The moment the bug crosses the first wire to the next corner, a piece of sugar is placed on the original starting corner. At some point, the bug will return to the starting corner and reach the sugar.

What is the probability that the bug will have visited all 8 corners by the time it returns to the starting corner?

Zachary has a private zoo. He has five groups of animals in his zoo: snakes, birds, mammals, insects, and spiders. Assume that, typically: animals have 1 head, snakes have 0 legs, birds have 2 legs, mammals have 4 legs, insects have 6 legs, and spiders have 8 legs. Zachary has some unusual animals in his zoo. He has: a snake with 3 heads, a bird with 2 heads, a mammal with 3 legs, an insect with 4 legs, and a spider with 7 legs.

1) There are a total of 100 heads and 376 legs.
2) Each group has a different quantity of animals.
3) The most populous group has 10 more members than the least populous group.
4) There are twice as many insect legs as there are bird legs.
5) There are as many snake heads as there are spider heads.

From the following information, determine how many of each group of animals that Zachary has in his menagerie.

Solve the following radical equation, with the domain x>=1:
x + sqrt(x-1) + sqrt(x+1) + sqrt(x^2-1) = 4