# March 2019 (Mathcounts)

I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday. For those of you who practiced with me last year, the process is similar.

For instructions on submitting weekly answers, click on the “Logistics” tab.
For previously posted problems, click on the “Archive” tab.
For previous prizes, click on the “News” tab.

Week of March 24 – March 30.

March 24: For how many positive integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?

Answer: 4000 contains $5^3$ as a factor. Therefore, n can be 1, 2, or 3.

March 25: A unit of blood expires after 9! seconds. John donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

Answer: Jan 6th. There are 3600 seconds in an hour. 9! seconds has 9!/3600 = 100.8 hours, which is on the 5th day after Jan 1.

March 26: Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container?

Answer: Let x and y be the volumes of the two jars. We have x 3/4 = y 5/6. Therefore, x/y = 10/9.

March 27: How many positive factors does 1002001 have?

Answer: $1002001 = 1001^2 = 7^2 11^2 13^2$. It has $3 \cdot 3 \cdot 3 = 27$ factors.

March 28: The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?

Answer: $a+b = 4ab$. ${1\over a} +{1\over b} = {{a+b} \over {ab}}=4$.

March 29: At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?

Answer: The 10 people who know no one create 45 handshakes among themselves. They also shake hands with the other 20 people for a total of 200 handshakes. So the total number of handshakes is 245.

March 30: Pablo buys popsicles for his friends. The store sells single popsicles for $1$ each, $3$-popsicle boxes for $2$ each, and $5$-popsicle boxes for $3$. What is the greatest number of popsicles that Pablo can buy with $8$?

Answer: 13 popsicles. He buys 10 popsicles for $6 and 3 popsicles for$2.

Week of March 17 – March 23.

March 17: Find the sum of the digits in the product $2^5\cdot 5^8$.

Answer: The product is 12500000. The digit sum is 8.

March 18: If 1 is added to the numerator of a fraction its value becomes 1/3. If 1 is subtracted from the denominator of the same fraction, its value becomes 1/4. That is the original fraction?

Answer: (x+1)/y = 1/3.  x/(y-1)=1/4. x=2, y=9. The original fraction is 2/9.

March 19: How many 3-digit numbers consist of even digits? (Zero cannot be the leading digit of a 3-digit number.)

Answer: 4x5x5=100. (The leading digit has 4 choices: 2, 4, 6 and 8; the tens and unit digits each has 5 choices.)

March 20: Find 3 consecutive numbers such that if they are divided by 2, 3 and 4 respectively, the sum of the quotients will be the next higher number.

Answer: (x-1)/2+x/3+(x+1)/4 = x+2. x=27. These three numbers are 26, 27, 28.

March 21: There are 5 men in a car. Find the sum of the ages of the 5 men if their ages in all possible pairs are given as 96, 79, 84, 82, 95, 100, 98, 83, 81 and 86.

Answer: (96 + 79 + 84 + 82 + 95 + 100 + 98 + 83  + 81 + 86)/4 = 221

March 22: A fruit market received 540 apples packed in boxes, where each box contains the same number of apples. The number of boxes was 6 less than twice the number of apples in each box. How many boxes were used to ship the 540 apples?

Answer:  x = 2(540/x) – 6. Therefore, x = 30.

March 23: Points X and Y both have integer coordinates. Which one of the following cannot be the distance from X to Y?
A) $\sqrt 13$    B) $\sqrt 74$   C) $\sqrt 2$     D) $\sqrt 15$   E) $\sqrt 8$.

Week of March 10 – March 16.

March 10: There are 16 positive integer factors of 216 (or $6^3$). What is the product of all 16 factors?

Answer: We can pair up the 16 factors: 1 and 216, 2 and 108, 3 and 72… such that the product of each pair is 216. There are 8 such pairs. So the product is $216^8$

March 11: Find $x$ if $x = \sqrt {2 \sqrt{2 \sqrt { 2\sqrt{\dots} }} }$. (Hint: square both sides and write an equation in terms of $x$.)

Answer: $x^2 = 2x$. Therefore, x=2.

March 12: Find the value of ${1+{1\over{1+{1\over{1+{1\over {1+\dots}}} }}}}$

Answer: $x = 1+1/x$. Therefore, $x = (-1+\sqrt{5})/2$.

March 13: $B$ and 9 are two solutions to the equation $x^2-4x=A$.  What is $A+B$?

Answer: $B+9 = 4$ and $9B = -A$. So $B=-5$ and $A=45$. $A+b=40$.

March 14: If $x+{1\over x} = 3$, what is $x^2 + {1\over x^2}$? (Hint: you don’t need to solve for $x$ first.)

Answer: Square both sizes of the equation. $x^2 + 2 + {1\over x^2} = 9$. Therefore, $x^2+{1\over x^2} = 7$.

March 15: If $x+{1\over x} = 3$, what is $x^4 + {1\over x^4}$? (Hint: use your answer from the previous problem.)

Answer: Square both sides again. $x^4 +2 + {1\over x^4} = 49$. Therefore, $x^4+{1\over x^4} = 47$

March 16: If $x+{1\over x} = 3$, what is $x^3 + {1\over x^3}$?

Answer: $(x^2+{1\over x^2}) ( x+{1\over x}) = x^3 + x + {1\over x}+ {1\over x^3} = 21$. Therefore, $x^3 + {1\over x^3} = 18$.

Week of March 3 – March 9.

March 3: In quadrilateral ABCD, AB=BC=CD=DA, AC=14 and BD=48. Find the perimeter of ABCD.

Answer: This quadrilateral ABCD is a rhombus, and so the diagonals are perpendicular to each other. Each side is $\sqrt{7^2+24^2} = 25$, and the perimeter is 100.

March 4: Find the area of a rhombus with side length 6 and an interior angle with 120 degrees.

Answer: The diagonals are 6 and 6$\sqrt{3}$. The area is 18$\sqrt{3}$.

March 5: A diagonal of a rectangle has length 41, and the perimeter is 98. Find the area of the rectangle.

Answer: Let a and b two sides of the rectangle. $a+b=49$, $a^2+b^2=41^2$. We have $ab=(49^2-41^2)/2 = 360$. $a= 40, b=9$, and the area is 360.

March 6: Find the area of $\triangle ABC$ if AB=BC=12 and $\angle ABC=120^o$.

Answer: The height AH = 6, BC = 12$\sqrt 3$. So the area is 36$\sqrt 3$.

March 7: A regular hexagon has perimeter $p$ and area $A$. Compute $p^2/A$.

Answer: The side length is ${p\over 6}$. The area $A= 6\cdot {1\over 2} \cdot {p\over 6}\cdot {\sqrt{3}p\over 6} = {\sqrt{3} p^2\over 12}$. Therefore, ${p^2\over A} = {12\over \sqrt{3} } = 4\sqrt{3}$.

March 8: The number of diagonals in a certain regular polygon is equal to five times the number of sides. How many sides does this polygon have?

Answer: 5n = n(n-3)/2. The number of sides is n=13.

March 9: The interior angle measures of a pentagon form an arithmetic progression. The difference between the largest and smallest angle measures is $44^o$. Find the measure of the smallest angle, in degrees.

Answer: Let the 5 angles be a-22, a-11, a, a+11, a+22. The total angle 5a = 540. Therefore, a=108. The smallest angle is therefore 108-22=86.