3 minutes individual, rest team

algebra

100

Simplify into an improper fraction. 

121/84

200

Find integers m and n, with m < n, for which the least common multiple of m and n is 105 and 50 < m + n < 70

105 = 3x5x7 . The divisors of 105 are: 1, 3, 5, 7, 15, 21, 35, 105. Of these, 21 and 35 are the only pair whose sum is between 50 and 70. 

300

If n + 1/n=5, find the value of n²+1/n²

23 

400

The roots of a² + bx + 1 = 0 are -2 and 3. What are the roots of bx^{2 +} ax - 1 =0?

-2, 3 

geometry

100

∆ABC is an isosceles triangle with AB = AC. Find m<DAC +m<BCF + m<CBE.

360^o

These are the exterior angles of the triangle! By the Exterior Angle Theorem, their sum is 360º

200

On hypotenuse PR of right triangle PQR (Figure 2), X is the point for which PX = PQ. If m<P = 40^o , find m<RQX .

20^o 

40 + n+n =180 

90-n =20 

300

 A rectangle is “hinged” at its vertices so that its side lengths remain constant, but its angle measurements become variable. A new quadrilateral is formed in this way, with half the area of the original rectangle. What is the measure of the obtuse angle(s) in this new quadrilateral?

Let h = the height of the original rectangle. The new quadrilateral (a parallelogram) shares the same base, so in order to have half the area, it must have half the height, as shown in the figure. The small right triangle, therefore, has the side length ratios of a 30-60-90 triangle. So the acute angles are 30º each, making the obtuse angles 150^o

400

Two of the side lengths in an isosceles triangle are 7 and 16. Compute the triangle’s perimeter. 

16 + 16 +7 = 39

word problem

100

Kiafa has two younger twin sisters. The product of their three ages is 18. How old is Kiafa?

Don’t forget about the possibility of 1 as a factor! If each twin is 1 year old, then Kiafa is 18.

200

Express 8^-1/3 + 9^-1/2 as the quotient of two relatively prime integers. 

hint! as a fraction 

1/2 +1/3 = 5/6

300

Alec is hosting a large party and needs to buy 100 bottles of Sierra Dew. Luckily, CostCorp sells them in bulk! He can buy 36 bottles for $9.00, 15 bottles for $4.50, or 4 bottles for $1.60. What’s the smallest amount Alec could pay, if he were to buy exactly 100 bottles?

$28.60

36 bottles cost $0.25 each, 15 bottles cost $0.30 each, and 4 bottles cost $0.40 each. Working based on the number of 36-bottle packs: Two 36’s can only reach 100 with seven 4’s: 2($9) + 7($1.60) = $29.20; One 36 reaches 100 with four 15’s and one 4: 1($9) + 4($4.50) + 1($1.60) =$28.60; With no 36’s, we maximize 15’s: 4($4.50) + 10($1.60) = $34.

400

While hosting some European guests this past summer, I attempted to mentally convert the day’s high temperature into degrees Celsius. However, I quickly realized I had incorrectly substituted the Fahrenheit temperature for C in the formula F = 9/5 C +32,  rather than for F. If my incorrect conversion yielded 186.8^o C , what would the correct conversion have yielded?

original high temperature 86ºF

C= 30