Feb 2018 (Mathcounts)

SMC is now finished for the 2017-2018 season. I’ll be back in Sept!  Meanwhile, you can practice problems from the SMC Archive

This is the last week for the 2017-2018 season. I’ve picked out seven problems that had the most number of wrong answers in the submissions.  It looks like most of these are counting and probability problems!

Click GoogleForm to submit answers by Sat, March 3.

Week of Feb 25 – Mar 3.

Feb 25: How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?

Answer: 84.   If the unit digit is 5, the thousands digit has 4 options (1,2,3,4), the hundreds digit has 4 options and tens has 3 options. If the unit digit is 0, then 5 can appear in tens, hundreds or thousands digit. The remaining 2 digits have 4*3 options.  So in total 4*4*3 + 3*4*3.

Feb 26: Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$ $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?

Answer: 24.  I post a different way to solve this problem from what I posted in Oct, 2017. Let f(k) be the number of ways to climb k steps in total, and we want f(6).
f(1)=1
f(2)=1+f(1)=2 (2 steps at a time; or do f(1) followed by 1 step)
f(3)=1+ f(2) + f(1) = 4 (3 steps at a time; or do f(2) followed by 1 step;  or do f(1) followed by 2 steps at a time)
f(4)=f(1)+f(2)+f(3) = 7 (do f(1) first followed by 3 steps at a time; or do f(2) first followed by 2 steps at time; do f(3) first followed by 1 step)
f(5)=f(2)+f(3)+f(4) = 13
f(6)=f(3)+f(4)+f(5) = 24

Feb 27: Let $R$ be a set of nine distinct integers. Six of the elements are 2, 3, 4, 6, 9, and 14. What is the number of possible values of the median of $R$ ?

Answer: 7. Let x<y<z be the other 3 numbers. If we put x, y, z in front of 2, 3, 4, 6, 9, 14, then 3 can be the median. If we put x, y, z after 14, then 9 is the median. Therefore, 3, 4, 6, 9, x, y and z can all be the medians. So the answer is 7.

Feb 28: Two cards are dealt from a deck of four red cards labeled $A, B, C, D$ and four green cards labeled $A, B, C, D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?

Answer: 4/7.  Here are the number of ways to choose a winning pair: 4 matching letters, 6+6=12 matching colors.  The number of ways to choose all possible pairs is 8*7/2 = 28.  Therefore, the probability is 16/28 = 4/7.

Mar 1: There are 64 identical-looking coins, one of which is slightly heavier than the others. A balance scale can be used to show which one of two groups of coins is heavier or that the two groups weigh the same. What is the minimum number of uses of the balance scale that is guaranteed to determine which of the coins is the heavier one?

Answer: 4 times. First weighing: divide 64 coins into 3 groups, 21-21-22. Weigh group 1 (21 coins) against group 2 (21 coins).  If either one is heavier, it contains the heavy coin; else group 3 (22 coins) has the heavy one.   Second weighing: assume the 22-coin group contains the heavy one.  divide it into 3 groups 7-7-8. Weigh group 1 (7 coins) against group 2 (7 coins). Again we know which group has the heavy coin. Third weighing assume the 8-coin group contains the heavy one.  Divide into 3-3-2. Weigh group 1 (3 coins) against group 2 (3 coins).  Fourth weighing finds the coin.

Mar 2: What is the value of $32 _4+ 43_5 + 54_6$ when written in base 7? ( $32_4$ is 32 written in base 4)

Answer: $32_4=14$ $43_5=23$ $54_6=34$ $14+23+34=71=131_7$  therefore $131_7$

Mar 3: When the circuit containing blinking lights A and B is turned on, lights A and B blink together. Then A blinks once every 5 seconds and B blinks once every 11 seconds. Lindsey looks at the two lights just in time to see A blink alone. What is the percent probability that the next light to blink will be A blinking alone?

Answer:  50%.   Consider a 55-second period:  0(AB), 5(A), 10(A), 11(B), 15(A), 20(A), 22(B), 25(A), 30(A), 33(B), 35(A), 40(A), 44(B), 45(A) and 50(A). When Lindsey sees A blink alone, it must be one of the 10 green times. Among them, exactly 5 will be followed by a green time. So the prob is 1/2.

We are going to have a Regular Polygon stretch.  In a regular n-gon, all sides are the same length and all angles are the same size.

Click GoogleForm to submit answers by Sat, Feb 24.

Week of Feb 18 – Feb 24

Feb 18:  A regular 3-gon is an equilateral triangle.  What is the area of an equilateral triangle if its side length is three?

Answer: ${1\over 2} \cdot 3 \cdot {3\sqrt{3}\over 2} = {9\sqrt{3}\over 4}$

Feb 19: A regular 4-gon is a square. If the diagonal of a square is $\sqrt{6}$, what is the side length of the square?

Answer: $\sqrt{6}/ \sqrt{2} =\sqrt{3}$

Feb 20: A regular 5-gon is a pentagon. How big is an interior angle of a regular pentagon?

Answer: 180 * 3 / 5 = 108

Feb 21: A regular 6-gon is a hexagon.  If a circle of radius 3 is inscribed in a regular hexagon, what is the perimeter of this hexagon?

Answer: $12\sqrt{3}$

Feb 22: A regular 7-gon is a heptagon.  How many diagonals does a heptagon have?

Answer:  4*7/2 = 14

Feb 23: A regular 8-gon is an octagon. It can be constructed from a square by cutting off an isosceles right triangle from each corner of the square. If the square has side length 4, what is the area of the octagon?

Answer: Let $x$ be the side length of the octagon. $2x/\sqrt{2}+ x = 4$, therefore $x= 4/(\sqrt{2}+1)= 4(\sqrt{2}-1)$. Area of octagon is $16-x^2=16-16(3-2\sqrt{2}) =32\sqrt{2}-32$

Feb 24: Do you know the name for a regular 9-gon?

We are going to have a Quadratic Equation Stretch this week.  For $ax^2 +bx+c=0$, there are multiple ways to solve the equation. You can factor if the numbers are nice.  If not, the quadratic formula always works.  If you don’t remember the formula, you can complete the square and go from there.  Do you remember when the equation has 2 real roots, 1 root and no root?  If the formula has 2 roots, the sum of the roots is $-b/a$ and the product of the roots is $c/a$.

Click GoogleForm to submit answers by Sat, Feb 17.

Week of Feb 11 – Feb 17

Use any method you like to solve the equations for Feb 11-14.

Feb 11: $r^2-7r+10=0$.

Answer: r=2,5   (r-2)(r-5) = 0

Feb 12: $x^2 +9 = 10x$.

Answer: x = 1,9  (x-1)(x-9) = 0

Feb 13: ${{2x} \over {x-2}} + {{3}\over {x-3}} =1$.

Answer: x = -1+root(13), -1-root(13)

Feb 14: ${2\over {3x+{5\over x}}} = 1$.

Answer: x has no real root.

Feb 15:  What is the sum of the solutions of $x^2 + 5x + 6 = 0$?

Feb 16: For what values of $b$ does the equation $3x^2+bx+27=0$ have one root?

Answer: b = 18, -18.  (b^2-4*3*27=0)

Feb 17: The product of the roots of $6x^2+cx+4=0$ is 2 greater than the sum of the roots.  What is c?

Answer:  c= 8  (4/6 = 2-c/6)

We are going to have a Fraction Stretch this week.

Click GoogleForm to submit answers by Sat, Feb 10.

Week of Feb 4 – Feb 10

Feb 4: On a number line, what common fraction is $3\over 4$ of the way from $-{11\over 2}$ to 4?

Answer:   13/8 = 4-(4+11/2)/4

Feb 5: What is the reciprocal of $1\over{2+{1\over 3}}$?

Feb 6: What common fraction is equal to $20.1\bar 8$?

Answer: 20+(1/10)+(8/90) = 1817/90

Feb 7: If ${1\over { {1\over{{1\over n} +{1\over 3} }}+{1\over{{1\over n} +{1\over 3} }}} } = {5 \over 12}$, what is the value of n?

Feb 8: If ${ {2x} \over {x-3}} - 2 = {4\over{x+2}}$, what is the value of x?
Feb 9: What fraction of $3\over 8$ is $9\over 16$?
Feb 10:  What is coordinate of the point that is $1\over 3$ distance from $(3,0)$ to $(0,-6)$?