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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

The DeciMate encryption scheme requires a sequence of true random decimal digits.

The sequence will be hand-generated by throwing a set of dice to determine each digit.
The dice are perfect cubes with 6 blank faces.
Any number or symbol can be written on each face.

Design such a set of dice so that each sum 0 through 9 has exactly the same probability.

** 6 cannot be used as a 9 by turning it upside down.

While attending W.G.’s birthday party I reminded him that exactly one year ago I presented (double entendre) him a 15 digit number and challenged him to examine its structure and using any method to keep it in memory for another year.

“Yes, I sure remember it’s structure”, quipped he. “ Your number was composed of three 5-digit parts, each containing the same 5 distinct digits, the 1st had 4 successive digits like 1234 and additional digit, the second was like the 1st reversed and the 3rd used the same digits in a higgledy-piggledy arrangement. There were no leading zeroes, but to reconstruct the whole number I need some additional input….Otherwise there are like (paused for a minute) 1180 possible valid solutions…” “OK, I do not check this number, and I am more than happy to inform you that adding the 3 components of the 15-digit chain I got 175497.”

a. Could WG decipher my number?
b. How about you?
c. Is the 1180 correct number?

ROUTE TAPER PASTE MITRE BRUSH TRADE YEAST PRIME TRAIN INURE GOUTY LINER CIDER

To each of the above 5-letter words add two additional two letters, not existing in the word, then permute the 7 letters to form a new non-esoteric English word, e.g. TRAIN+HS=TARNISH or TRAIN+CG=TRACING.
Never add 2 letters that were used before, thus utilizing all 26 letters to fully perform the task.

Rem: Posting partial results is counterproductive…

Some time has passed since Lighthouse Crossed Three Lights Muse and each lighthouse has developed a glitch. Now, instead of switching off when it should, the light will (50% of the time) stay on for an extra time period.
The extra time is exactly 1 second for the 1st lighthouse, exactly 2 seconds for the 2nd, and exactly 3 seconds for the 3rd.

In summary:
- The first light shines for 3 or 4 seconds, then is off for 3 seconds.
- The second light shines for 4 or 6 seconds, and then is off for 4 seconds.
- The third light shines for 5 or 8 seconds, then is off for 5 seconds.

What is the expected value of the first time all the lights will be off at the same time?

In triangle ABC, a=9 and a(cos A)+b(cos C)+c(cos B)=450/29. Find the value of sin A.

Part of me is in terrible, but not in awful,
Part of me is in I, but not in me,
Part of me is in copper, but not in beryllium,
Part of me is in hello, but not in goodbye,

What am I?

A semicircle with radius 1 is centered at O, and has diameter AB. Point P lies on the semicircle so that angle ABP=60 degrees. Compute the radius of the circle tangent to diameter AB, segment BP, and minor arc AP.

You will be given a list of 9 words and your job is to search each word and find the airport codes hidden in each word. For some airport codes, a business name or person may be used in place of a word when one does not exist. For example, Concord corresponds to ORD, or Chicago O'Hare international Airport.

Relax
Mlive News Michigan
Diarrhea
Atlas
Tulsi Gabbard
Mspy
Sforzando
Santa
Algae

Each pair of definitions is for two words, where the second word is the first word with a letter deleted (example: brand & band). The length of the first word in each pair is provided, along with the position of the deleted letter to obtain the second word.

1) something absurd or ridiculous (5 letters) & (delete 4th letter) money paid for transportation.
2) a type of fruit (6 letters) & (delete 1st letter) a place for shooting practice.
3) a dead body (6 letters) & (delete 3rd letter) a thicket of small trees or shrubs.
4) to chew loudly (5 letters) & (delete 3rd letter) a great amount or quantity.

Consider the individual letters that make up the word: ANTHEM

There is a certain case-sensitive rule by which the letters of the alphabet can be separated into 3 distinct groups.
Each of letters in "ANTHEM" fall into the same group.

Name another letter that belongs in this group.

Participants: Book, Candy, Flowers, Money and Scarf.
Gifts: book, candy, flowers, money and scarf.
The participants are unmarried, grown-up people of mixed genders, each being both a giver (donor) and a receiver, in a way that no one gave or received an item bearing his or her name and none gave a present to the person from whom he or she received one.

The present received by Flowers was the name of the donor of the scarf, Scarf sent flowers to the book-giver, Candy gave Scarf a present and Book received candy.

a. Find out, who gave what to whom.
b. You may deduce (with no certainty) from the nature of the gifts the probable gender of each of the people mentioned,

Source : (Slightly abridged) problem #98 from Penguin’s Problems book by William and Savage 1940.

(i) BP in a MB.
(ii) 17P x 2 + 3R = 37 P in a R L M
(iii) 1= P of PP that PPP.
(iv) 120=N of B B on the F A

A bug is placed at one corner of a wire frame in the shape of a cube. At the diagonally opposite corner is a piece of sugar.

The bug crawls along the 12 wires of the frame searching for the sugar. At each of the 8 corners the bug randomly chooses one a wire to follow next with the additional rule that it can never cross the same wire twice.

What is the probability that it will dead-end by reaching a corner with no available wires? In the case where it does reach the sugar, what is the expected number of edges the bug traverses?

On the coast there are 3 lighthouses. All 3 turn on at time zero.
- The first light shines for 3 seconds, then is off for 3 seconds.
- The second light shines for 4 seconds, then is off for 4 seconds.
- The third light shines for 5 seconds, then is off for 5 seconds.

A distant ship has an optical sensor aimed at the set of lighthouses, but it is far enough away that the sensor only activates when two of the lights are on at the same time.
Over the course of one minute, how often is the sensor active?

Original problem: Lighthouse Crossed Three Lights Muse