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ABCDEF is a regular hexagon with sides of length h units. A square PQRS is drawn inside the hexagon, with PQ parallel to AB, SR parallel to ED, and vertices PQRS lying on FA, BC, CD and EF respectively.
The area of PQRS is 4 square units.
What is the exact value of h?
What is the sum of all the exact divisors of the number N=19^{88}1 that are of the form 2^{a}*3^{b}, where a and b are greater than zero?
Mo and Stef have challenged each other to race twice the length of the school hall.
The hall is rectangular with a platform at one end and doors at the opposite end.
Mo starts from the platform end and Stef from the end with the doors. They each run at a constant speed.
They first pass each other 30m from the platform end and then pass each other again 18m from the door end, as they return to their starting points.
What is the length of the school hall?
What is the ratio of Mo and Stef’s speeds?
You are given 99 thin rigid rods with lengths 1, 2, 3, ..., 99. You are asked to assemble these into as many right triangles as you wish. What is the largest total area that can be obtained? (Each side of a triangle must be one entire rod.)
Suppose that one bowl costs more than two plates, three plates cost more than four candlesticks, and three candlesticks cost more than one bowl.
If it costs precicely $100 to purchase a plate, bowl, and candlestick, how much does each item cost?
In how many ways can 8 different numbers can be chosen from the first 49 positive integers such that the product of these numbers is divisible by 8?
Does there exist any triplet (A,B, N) of integers that satisfy this equation:
1999^{A} + 2000^{B} = N^{3}?
If so, provide an example.
If not, then prove that any solution to the given equation is nonexistent.
A triangle ABC has sides of length a, b and c.
A solid composed from two cones is produced by rotating the triangle by 360° about the side of length a. This process is then repeated for sides b and c to produce two more solids, both formed from pairs of cones.
Find the ratio of the volumes of the resulting solids.
Consider a regular 8x8 chessboard. Precisely 6 distinct squares are chosen randomly on the chessboard.
Determine the probability that they lie in the same diagonal.
You have been hired to build the Temple of Heterodoxy for a fixed fee. The Temple must be rectangular, divided into two or more rectangular interior rooms, with each side an integral number of bozols. The outer dimensions and the dimensions of each room must all be different. For example, you could not have both a 5x9 room and a 9x11 room. The thickness of the walls is negligible.
What is the smallest possible area for the floor of the temple?
The parallel sides of a trapezium are a cm and b cm long, where a and b are integers, a < b.
The trapezium is split into two smaller trapezia of equal area by a line of length c cm which is parallel to the sides of length a cm and b cm.
Given that c is also an integer, what is the smallest possible value of c?
The census taker comes to the home of Mr. and Mrs. Lobotomy. He jots down the 4digit house number and the ages of the wife and her somewhat older husband. Then he asks about their children.
Mrs. Lobotomy tells him that they have 3 daughters and 3 sons, that the product of the 3 daughters' ages is the same as their house number, the sum of the 3 daughters' ages is the same as her own age, the product of the 3 sons' ages is also the same as their house number, and the sum of the 3 sons' ages is the same as her husband's age.
The census taker is an accomplished mathematician, but after some time he determines that it is not possible to figure the ages of the sons or the daughters.
Mr. Lobotomy then tells the census taker that the difference in age between the youngest daughter and youngest son is the same as the age of their cat. The census taker does not see a cat, and they have not mentioned its age, but he knows that it still would be impossible to determine the ages of the 6 children.
The cat is 3 years old, and the sum of the ages of the oldest son and the oldest daughter is not 60. What are the ages of the 8 people in the Lobotomy family?
Show that the equation: 987a + 789b + 12321c = (a + b + c)^{2} has infinitely many integer solutions.
*** Do not include solutions where a+b+c=0.
Prove that the expression 2^{n} + n^{222} is never prime for any positive integer values of n, except for n=1.
One year, on Sue's birthday, Sue started a collection of thimbles. The following birthday she added to her collection, which went from strength to strength. In all subsequent years when she counted the thimbles on her birthday the total had increased from the previous year’s total by a number equal to the total she had on her birthday the year before that. (So, for example, her 1983 total equalled her 1982 total added to her 1981 total).
Now, by coincidence, her daughter was born on her birthday. And, with her collection growing following the described pattern, on their birthday in 1983 the number of thimbles Sue owned had reached exactly four times her daughter’s age on that day. On her birthday this year the total of thimbles was four times her age. On only one other occasion has the total been divisible by four, and that was in the year Sue's son was born.
How many thimbles were there in Sue's collection on her birthday this year? How many (if any) did she have the day her daughter was born?
Antoine has been given 3 Hourglasses H _{1}, H _{2}, and H _{3} which are respectably able to measure out precisely 11 minutes, 15 minutes, and 17 minutes.
• Antoine is asked to measure out precisely 23 minutes, and he is permitted to use any 2 of the 3 hourglasses to perform this task.
What choice of 2 hourglasses will enable Antoine to achieve the said objective in a minimum number of steps?
Provide valid reasoning for your answer.
Arrange 25 fiveletter words in a 5by5 grid. The five words in each row share a common letter. The five words in each column share a common letter. The five words in each diagonal share a common letter. There are 12 different common letters.
The words are: amuse, charm, clasp, cough, crypt, drive, dunce, first, fresh, graft, grime, heavy, money, pixel, pluck, power, quiet, shark, smirk, stein, syrup, tidal, twice, weigh, yearn.
One letter of each word is provided in the grid.
Row 1: k a s g h
Row 2: d n e c u
Row 3: r y a l k
Row 4: t p f d e
Row 5: a m r i s
Given that:
A+B+C=2023, and:
1/A + 1/B + 1/C =1/2024
Find the value of:
1/A^{2024} + 1/B^{2024} +1/C^{2024}
Find the smallest convex pentagon such that each of its 5 sides and each of its 5 diagonals have distinct integer lengths.
A positive integer is called vaivém when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, 2021 is vaivém, as 2 > 0 and 0 < 2 and 2 > 1. The number 2023 is not vaivém, as 2 > 0 and 0 < 2, but 2 is not greater than 3.
a) How many vaivém positive integers are there from 2000 to 2100?
b) What is the largest vaivém number without repeating digits?
c) How many distinct 7digit numbers formed by all the digits 1, 2, 3, 4, 5, 6 and 7 are vaivém?
If x and y are real numbers, then solve this system of equations:
√x + √y =3, √(x+5) + √(y+3) = 5
A 11digit number is such that it contains each of the digits from 1 to 9 at least once.
What is the percentage of prime numbers in such an occurrence?
Show that for any prime p there exists a nonnegative integer N such that:
2^N+3^N+6^N1 is a multiple of p.
A company receives a gift of $ 155,000. The money is invested in stocks, bonds, and CDs.
CDs pay 4.5 % interest, bonds pay 2.8 % interest, and stocks pay 8.4 % interest.
The company invests $ 40,000 more in bonds than in CDs.
If the annual income from the investments is $ 7,930, how much was invested in each account?
Without direct evaluation, or using a calculator, determine which of these is larger: (1+√5)/2 vs log_{e}4
The sequence a_{n} is defined by the recurrence relation a_{n+4} = a_{n+3}  a_{n+2} + a_{n+1}  a_{n} with initial values a_{0}=1607, a_{1}=1707, a_{2}=1814 and a_{3}=1914. Find a_{100}.
Given:
f(x) = √(x^210x+314) + √(x^2+20x+325)
Determine the minimum value of f(x) for a real number x.
*** Adapted from a problem appearing in 2017 Singapore M.O. open.
If the quadratic equation ax^{2}bx+12=0 where a and b are positive integers not exceeding 10, has roots both greater than 2. Then the number of possible ordered pair (a,b) is?
Rationalize the denominator: √2 + √3 + √6

√2 + √3 + √6 + √8 + √16
Find at least two common words with seven consonants in a row.
If, x  6/√x = 37
Then, find the value of x6√x
A logician named John went to the Land of Knights and Liars. He met two inhabitants, A and B. He asked A two questions.
John:Has B ever said that you are a liar?
A answered, either "Yes" or "No."
John:Is B a liar?
A answered, either "Yes" or "No."
The next day, another logician named Jack went to the Land of Knights and Liars. He met the same two inhabitants, A and B. He also asked A two questions.
Jack:Has B ever said that you are both liars?
A answered, either "Yes" or "No."
Jack:Is B a liar?
A answered, either "Yes" or "No."
One of the two logicians, John and Jack, could figure out what types A and B were, but the other logician could not figure out their types. What are A and B?
