Question 1

What is the value of when ?

Solution

Question 2

If , what is ?

Solution

Question 3

Let . What is the value of

Solution

Question 4

Zoey read books, one at a time. The first book took her day to read, the second book took her days to read, the third book took her days to read, and so on, with each book taking her more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her th book?

Solution

Question 5

The mean age of Amanda's cousins is , and their median age is . What is the sum of the ages of Amanda's youngest and oldest cousins?

Solution

Question 6

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number . What is the smallest possible value for the sum of the digits of ?

Solution

Question 7

The ratio of the measures of two acute angles is , and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?

Solution

Question 8

What is the tens digit of

Solution

Question 9

All three vertices of are lying on the parabola defined by , with at the origin and parallel to the -axis. The area of the triangle is . What is the length of ?

Solution

Question 10

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length inches weighs ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of inches. Which of the following is closest to the weight, in ounces, of the second piece?

Solution

Question 11

Carl decided to fence in his rectangular garden. He bought fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl???s garden?

Solution

We can use 4 yards as the unit for the dimensions. And let the dimensions be a * b, then we have one side will have a+1 posts (including corners) and the other b+1 (see example diagram below with a=4 and b=3).

The total number of posts is 2(a+b)=20.

Solve the system b+1=2(a+1) and 2(a+b)=20,

We get: a=3 and b=7. Then the area is 4yard * a * 4yard * b=336 square yards.

Question 12

Two different numbers are selected at random from and multiplied together. What is the probability that the product is even?

Solution

Question 13

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these babies were in sets of quadruplets?

Solution

Question 14

How many squares whose sides are parallel to the axis and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line

Solution

Question 15

All the numbers are written in a array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to . What is the number in the center?

Solution

Question 16

The sum of an infinite geometric series is a positive number , and the second term in the series is . What is the smallest possible value of

Solution

Question 17

All the numbers are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?

Solution

Question 18

In how many ways can be written as the sum of an increasing sequence of two or more consecutive positive integers?

Solution

Question 19

Rectangle has and . Point lies on so that , point lies on so that . and point lies on so that . Segments and intersect at and , respectively. What is the value of ?

Solution

Question 20

A dilation of the plane???that is, a size transformation with a positive scale factor???sends the circle of radius centered at to the circle of radius centered at . What distance does the origin , move under this transformation?

Solution

Question 21

What is the area of the region enclosed by the graph of the equation

Solution

With |x| and |y|, x and y can be positive or negative. Suppose they are both positive

x^2 + y^2 = x + y

(x-1/2)^2 + (y-1/2)^2 = 1/2

Review equation for circle: https://www.homesweetlearning.com/resources/math/math910/coordinate_graphs/circles_in_coordinate_plane.html

This is a circle with center at point (1/2, 1/2) and radius to be 1/sqrt(2).

Since x an y can take positive and negative values, so the original equation is something like below.

Question 22

A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won games and lost games; there were no ties. How many sets of three teams were there in which beat , beat , and beat

Solution

Question 23

In regular hexagon , points , , , and are chosen on sides , , , and respectively, so lines , , , and are parallel and equally spaced. What is the ratio of the area of hexagon to the area of hexagon ?

Solution

Question 24

How many four-digit integers , with , have the property that the three two-digit integers form an increasing arithmetic sequence? One such number is , where , , , and .

Solution

Question 25

Let , where denotes the greatest integer less than or equal to . How many distinct values does assume for ?

Solution

Answer Keys