MathCounts

back to index

315   
If $3x + 2 = 17$, what is the value of $x$?

316   
A mountain bike, originally priced at \$650, is on sale for 20% off. After the fixed shipping cost of \$25 is added to the new price, what is the final cost of the bike?

350   

In the table shown, $y =\frac{3x-1}{2}$. What is the value of $t$?



373   
Given the function $f(x) = 2x^2 - 3x + 7$ with domain {$-2, -1, 3, 4$}, what is the largest integer in the range of $f$?

384   
The first two terms of a sequence are 10 and 20. If each term after the second term is the average of all of the preceding terms, what is the 2015th term?

390   

This graph shows the water temperature T degrees at time t minutes for a pot of water placed on a stove and heated to 100 degrees. On average, how many degrees did the temperature increase each minute during the first $8$ minutes?


404   
A streaming video service offers its customers two plans. The first plan has a \$24 registration fee, a fixed monthly rate of \$5 and a charge of \$0.75 for each video download. The second plan has a \$15 registration fee, a fixed monthly rate of \$3 and a charge of \$1.50 for each video download. For how many downloads would the two plans cost the same for a year of service?

405   
The length of a rectangle is three times its width. A new rectangle is created by decreasing the length of the original rectangle by 9 feet and increasing its width by 4 feet. The area of the new rectangle is the same as the area of the original rectangle. What is the perimeter of the new rectangle?

406   

In equilateral triangle $ABC$, shown here, each downward pointing black triangle has its vertices at the midpoints of the sides of a larger upward pointing white triangle. What fraction of the area of $DABC$ is white? Express your answer as a common fraction.



407   
What is the greatest possible perimeter of an isosceles triangle with sides of length $5x + 20$, $3x + 76$ and $x + 196$?

408   
Consider an arithmetic sequence with $a_3 = 165$ and $a_{12} = 615$. For what value of n is $a_n = 2015$?

409   
If $\sqrt{x + 7} = 2 +\sqrt{x}$ , what is the value of $x$? Express your answer as a common fraction.

410   
The line perpendicular to $2x -2y = 2$, and with the same $y$-intercept, is graphed on the coordinate plane. What is the sum of its $x$- and $y$-intercepts?

411   
What is the units digit of the sum of the squares of the integers from $1$ to $2015$, inclusive?

412   
The doctor gave Amber ten vitamins, with instructions to take one or two each day until she runs out of vitamins. For example, Amber could take a vitamin a day for ten days, or she could take two the first day and one a day for the next eight days. A third way is to take one vitamin a day for eight days and two on the ninth day. Including the three examples given, in how many different ways can Amber take the ten vitamins?

413   
What is the radius of a circle inscribed in a triangle with sides of length 5, 12 and 13 units?

414   
A bag initially had blue, red and purple gumballs in the ratio of $2:3:4$. Five red gumballs are added to the bag. The probability of randomly drawing a red gumball is now $40%$. How many gumballs are now in the bag?

422   
The quadratic equation $x^2+ px + 2p = 0$ has solutions $x = a$ and $x = b$. If the quadratic equation $x^2+ cx + d = 0$ has solutions $x = a + 2$ and $x = b + 2$, what is the value of d?

431   
A rectangle of perimeter 22 cm is inscribed in a circle of area $16\pi$ $cm^2$. What is the area of the rectangle? Express your answer as a decimal to the nearest tenth.

433   
The units and tens digits of one two-digit integer are the tens and units digits of another two-digit integer, respectively. If the product of the two integers is $4930$, what is their sum?

435   
Octavius has eight identical blue socks, six identical red socks, four identical black socks and two identical orange socks in his drawer. If he randomly selects two socks from his drawer, what is the probability that they will be the same color? Express your answer as a common fraction.

436   
Iniki has large, medium and small metal bars. The large bars each weigh 8 kg. The medium bars each weigh 6 kg. The small bars each weigh 3 kg. Iron, nickel and lead are present in the ratio 4:1:3 in each large bar, 2:1:3 in each medium bar and 1:1:1 in each small bar. If Iniki wants to melt together a combination of bars to make an alloy that contains 40 kg of iron, 20 kg of nickel and 40 kg of lead, how many small bars will she have to use?

443   
If $x +\frac{1}{x}= 3$, what is the value of $x^4+\frac{1}{x^4}$?

446   

Points $C$ and $D$ are chosen on the sides of right triangle $ABE$, as shown, such that the four segments $AB$, $BC$, $CD$ and $DE$ each have length 1 inch. What is the measure of angle $BAE$, in degrees? Express your answer as a decimal to the nearest tenth.



451   
A restaurant sells three sizes of drinks: small for \$1.20, medium for \$1.30 and large for \$1.80. Each person at a table of ten ordered one drink, for a total cost of \$14.90, before sales tax. How many people ordered a large drink?

back to index