#### NEW 2020 Problems of the Week

##### Week of November 30th:

**Problem #1:** How many two digit numbers are there (i.e. between 10 and 99) where the sum of that number’s digits is also a two digit number? (for example, 87 would be 8+7=15)

**Problem #2:** (hard) How many *three*-digit numbers are there where the sum of that number’s digits is a two-digit number?

#### Intermediate Problems:

**Problem #1:** What is the largest possible prime number where all its digits are consecutively descending prime digits?

**Problem #2:** X and Y are real nonzero numbers. The sum of x and y is equal to x cubed. If (x*y)/(y/x) = 16 what is y/x?

**Problem #3:** Shlomo bought a bag of Rugalach for $10. The next day he took $10 total, from 46 coins consisting of only dimes and quarters to the shop and went to buy another bag. This time the Rugalach were 50% off and after paying he had 16 quarters remaining and x dimes. How many dimes were left?

**Problem #4:** Make 0 0 0 0 = 7 with only those digits, but all other common operands (!, ln, log, √, () included). Example: ( 0+0-0*0=0)

**Problem #5:** Find the minimum possible value for x, if x and y are positive whole numbers and 500+x=2y.

**Problem #6:** Joe flips a coin 4 times, what is the chance he gets 2 or more heads?

#### Hard Problems:

**Problem #1:** Given a tetrahedron (triangular pyramid) of side length r, what is the closest that two of the edges which are skew (do not meet) approach each other?

**Problem #2:** If infinite circles are packed on a plane in a hexagonal tiling (such that every circle is touching six other circles), what fraction of the plane is left uncovered? (0 is none, 1 is all)

**Problem #3:** For the function f(x)=nxx!, where n is constant, between which two consecutive whole numbers (expressed in terms of n) does the slope of f(x) equal zero? (i.e. between which two numbers does f(x) begin to decrease)

**Problem #4:** If a light beam with a circular cross-section is shone (parallel to the ground) at the corner of a room (where two walls meet at a right angle) such that half of the beam shines on each wall, and the surface area of the portion of wall #1 in light covers 3times the surface area of the light on wall #2, what angle does the light beam make with wall #1?

#### 2018 Problems of the Week Favorites:

**Problem #1:** What is the largest possible prime divisor of every 3-digit number with three identical non-zero digits?

**Problem #2:** What is the greatest possible length of the longest side of an isosceles triangle whose sides each have an integral length whose perimeter is 5779?

**Problem #3:** Yossele’s coins consist of only pennies, nickels, dimes, and quarters. Yossele has less than 100 pennies and the total value of all of Yossele’s coins is $6.13. At most, how many pennies does he have?

**Problem #4:** If x and y are real numbers whose numbers whose sum is three times their product, and whose product is not 0. What is the value of x + y?

**Problem #5:** For all real numbers x, the function f is defined by f(x) = 5779. What is the value of f(x + 5779)?

**Problem #6:** Moshe runs three times as fast as he walks. It takes Moshe 21 minutes to get to work from home if he walks for twice the amount of the time he runs. How many minutes does it take Moshe to get to work from home if he runs for twice the amount of time that he walks?

**Problem #7:** Line J is perpendicular to lines m and n. If the product of the slopes of all 3 lines is -8, what is the slope of the line J?

#### 2017 Problems of the Week Favorites:

**Problem #1:** Steve and Josh played a game. Steve wrote on a blackboard all integers from 1 to 18 and challenged Josh to choose 8 different integers from this list. To win the game Josh had to choose 8 integers so that among them the difference between any two of them is either less than 7 or greater than 11. Can Josh win the game? Justify your answer.

**Problem #2:** What is the smallest positive number k such that there are real numbers a and b satisfying a + b = k and ab = k?

**Problem #3:** Patty is picking peppermints off a tree. They come in two colors, red and white. She picks fewer than 100 total peppermints but at least one of each color. The white flavor is stronger, so she prefers red to white. Thus, she always picks fewer white peppermints than ten times the number of reds. How many different combinations of peppermints can she go home with?

**Problem #4:** A class average mark in an exam is 70. The average of students who scored below 60 is 50. The average of students who scored 60 or more is 75. If the total number of students in this class is 20, how many students scored below 60?

**Problem #5:** The sum of 5 real numbers is 8 and the sum of their squares is 16. What is the largest possible value for one of the numbers?

**Problem #6:** My grandson is about as many days as my son in weeks, and my grandson is as many months as I am in years. My grandson, my son, and I together are 120 years. Can you tell me my age in years?

**Problem #7:** David, Steven, and Harry are going to the mall to buy clothes. 1/4 of David’s money plus 1/4 of Steven’s money is equal to 3 dollars. 1/5 of David’s money plus 1/5 of Harry’s money is equal to 2 dollars. If the sum of all their money is a multiple of 7 and they individually have whole number amounts of dollars, what are the possible values for David’s money? (there may be more than one)