Practice (90/1000)

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 $a_n = 2015$?

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?

416

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $24.50$. How much would $8$ adult tickets and $6$ child tickets cost?

421

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?

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?

425

Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

427

Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?

428

A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms?

429

Let $a < b < c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$?

430

A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?

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.

434

A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?

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.

437

The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

438

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?

439

In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?

440

For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers $x$ such that $1\leq x<2014$ and $f(x)\leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?

442

The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?

443

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

444

Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?

back to index | new