A positive integer $N$ has a remainder of 4 when divided by 5, a remainder of 3 when divided by 4, and a remainder of 2 when divided by 3. Which of the following could be the value of $N$?
49
39
59
69
29
A school is selling tickets for a play. Student tickets cost 8 dollars each and adult tickets cost 15 dollars each. The school sold 320 tickets in total and collected 3610 dollars. How many more student tickets than adult tickets were sold?
90
70
60
100
80
A visual pattern is constructed with dots. The first figure has 1 dot. The second figure has 5 dots arranged in a cross shape. The third figure has 13 dots, formed by a central dot and squares of dots around it. The number of dots in the $n$-th figure is given by $2n(n-1)+1$. How many dots are in the 12th figure?
289
313
265
221
241
A rectangular garden is to be fenced on three sides, with a long stone wall forming the fourth side. If 120 meters of fencing material are available, what is the maximum possible area of the garden in square meters?
1800
1200
3600
1500
2400
Solution A is 20% acid and Solution B is 50% acid. A chemist wants to create a 120-litre mixture that is 30% acid. How many more litres of Solution A than Solution B must be used?
20
30
40
50
In a right-angled triangle, the hypotenuse is 2 cm longer than one leg and 9 cm longer than the other leg. What is the perimeter of the triangle?
45 cm
30 cm
40 cm
24 cm
36 cm
How many non-congruent triangles can be formed whose vertices are three of the vertices of a regular decagon (a 10-sided polygon)?
8
12
10
15
A box contains 4 red balls, 5 blue balls, and 6 green balls. Two balls are drawn in succession without replacement. What is the probability that both balls are the same color?
$41/105$
$41/210$
$47/210$
$47/105$
$1/3$
A composite solid is formed by a cone placed on top of a cylinder with the same radius. The radius of the base is 5 cm, the height of the cylinder is 10 cm, and the slant height of the cone is 13 cm. What is the total volume of the composite solid?
$250\pi$ cm$^3$
$300\pi$ cm$^3$
$400\pi$ cm$^3$
$350\pi$ cm$^3$
$375\pi$ cm$^3$
A tank is filled by two pipes, A and B, and emptied by a third pipe, C. Pipe A can fill the tank in 6 hours, pipe B can fill it in 8 hours. When all three pipes are open, the tank fills in 12 hours. How long would it take for pipe C alone to empty the full tank?
4.8 hours
4.2 hours
5.0 hours
4.4 hours
4.6 hours
In triangle ABC, a point D is on AC such that $AB = AD$. Given that $\angle ABC - \angle ACB = 30^\circ$, find $\angle CBD$.
$10^\circ$
$15^\circ$
$25^\circ$
$20^\circ$
$30^\circ$
The number $N = 2^4 \times 3^3 \times 5^2 \times 7^1$. How many positive divisors of $N^2$ are perfect cubes but not perfect squares?
2
6
4
A factory has two machines, A and B, that produce widgets. Machine A produces 60% of the daily output, and 5% of its widgets are defective. Machine B produces the remaining 40%, and 8% of its widgets are defective. If a widget selected at random is found to be defective, what is the probability that it was produced by Machine A?
$16/31$
$1/2$
$32/62$
$30/62$
$15/31$
A large cube with a side length of 4 cm is painted red on all its faces. It is then cut into 64 smaller cubes of side length 1 cm. What is the ratio of the number of small cubes with exactly two red faces to the number of small cubes with exactly one red face?
2:1
1:1
1:2
2:3
3:2
What is the sum of the integer roots of the polynomial $P(x) = x^3 - 7x - 6$?
7
1
-1
0
A grid of points is arranged in 5 rows and 6 columns. How many distinct rectangles can be drawn whose vertices are points on this grid and whose sides are parallel to the grid lines?
210
150
180
120
A hollow cylindrical metal pipe is 1 meter long. Its inner radius is 3 cm and its outer radius is 4 cm. What is the volume of the metal used to make the pipe, in cubic centimeters?
$300\pi$
$400\pi$
$100\pi$
$900\pi$
$700\pi$
In $\triangle ABC$, point $D$ lies on side $AC$ such that $AB = AD$. If $\angle ABC - \angle ACB = 30^\circ$, what is the measure of $\angle CBD$?
$22.5^\circ$
What is the smallest three-digit positive integer which leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5?
101
119
121
179
239
A metallurgist needs to make 100 kg of an alloy that is 50% silver. She has two alloys available: Alloy A is 40% silver and Alloy B is 65% silver. How many kilograms of Alloy A must she use?
75 kg
30 kg
50 kg
40 kg
60 kg
In the following cryptarithmetic puzzle, each letter represents a unique digit from 0 to 9.
$L \ O \ G \ I \ C$
$+ \ L \ O \ G \ I \ C$
$-----------$
$P \ U \ Z \ Z \ L \ E$
What is the digit represented by the letter P?
9
A sequence of figures is constructed from dots. Figure 1 is a single dot. To create the next figure, a layer of dots is added around the previous figure to form a perfect square. Figure 1 is a $1 \times 1$ square, Figure 2 is a $3 \times 3$ square, Figure 3 is a $5 \times 5$ square, and so on. How many dots are in the layer added to create Figure 10?
36
361
72
A rectangular garden is to be enclosed on three sides by a fence, with the fourth side being a long, straight rock wall. If 100 metres of fencing are available, what is the maximum possible area of the garden in square metres?
$1000 \ m^2$
$5000 \ m^2$
$2500 \ m^2$
$1250 \ m^2$
$625 \ m^2$
The polynomial $P(x) = x^3 - 6x^2 + 11x - 6$ has three distinct integer roots. What is the sum of the squares of these roots?
21
14
11
Pipe A can fill a tank in 4 hours. Pipe B can fill the same tank in 6 hours. Pipe C can empty the full tank in 3 hours. If the tank is empty and all three pipes are opened at the same time, how long will it take for the tank to be filled?
7 hours
10 hours
12 hours
The tank will never be filled.
5 hours
A circle with radius $R$ is inscribed in a square. A smaller circle with radius $r$ is placed in one corner, tangent to the larger circle and to two adjacent sides of the square. What is the exact value of the ratio $R/r$?
$4$
$1 + \sqrt{2}$
$2 + \sqrt{3}$
$2\sqrt{2}$
$3 + 2\sqrt{2}$
How many distinct diagonals can be drawn in a regular decagon (a 10-sided polygon)?
45
35
27
From a standard 52-card deck, two cards are drawn in succession without replacement. What is the probability that the second card is a Queen, given that the first card was also a Queen?
$\frac{4}{51}$
$\frac{1}{17}$
$\frac{3}{52}$
$\frac{1}{13}$
$\frac{1}{26}$
A solid toy consists of a hemisphere mounted on a cylinder of the same radius. The radius of the base is 7 cm and the total height of the toy is 17 cm. What is the total surface area of the toy? (Use $\pi \approx \frac{22}{7}$)
$902 \ cm^2$
$1056 \ cm^2$
$748 \ cm^2$
$858 \ cm^2$
$1078 \ cm^2$
In a square $ABCD$, point $E$ lies on side $CD$ and point $F$ lies on side $BC$ such that $\angle EAF = 45^\circ$. Which of the following statements must be true?
$EF = CE + CF$
$EF = DE + BF$
Area($\triangle AEF$) = Area($\triangle ADE$) + Area($\triangle ABF$)
$AC$ bisects $EF$
$AE = AF$
Find the sum of all positive integers $n$ for which the expression $\frac{n^2 + n + 10}{n+2}$ is an integer.
19
17
28
A boat travels 36 km downstream in 3 hours and 24 km upstream in 4 hours. What is the speed of the boat in still water?
10.5 km/h
6 km/h
7.5 km/h
12 km/h
9 km/h
In the cryptarithm $WRONG + WRONG = RIGHT$, each letter represents a distinct digit from 0-9. Given that W=1 and the solution is unique, what is the value of the digit represented by R?
A structure is built by stacking $n$ identical cubes one directly on top of another. The structure rests on the ground. If the total number of exposed square faces (including the top and sides, but not the bottom face touching the ground) is 85, how many cubes are in the stack?
42
84
22
A piece of wire is cut into two parts. One part is bent to form a square, and the other to form a circle. To make the sum of the two areas a minimum, what must be the ratio of the length of the wire used for the circle to the length of the wire used for the square?
$\frac{\pi}{4}$
$\sqrt{\pi}$
$\frac{\pi}{2}$
$\frac{4}{\pi}$
A box contains 4 red balls and 6 blue balls. Two balls are drawn from the box one after the other, without replacement. What is the probability that both balls are of the same color?
$\frac{8}{15}$
$\frac{1}{2}$
$\frac{7}{15}$
$\frac{1}{3}$
$\frac{13}{25}$
A positive integer $N$ leaves a remainder of 3 when divided by 5, a remainder of 4 when divided by 7, and a remainder of 5 when divided by 9. What is the smallest possible sum of the digits of $N$?
$13$
$12$
$14$
$16$
$15$
At a funfair, an adult's ride pass costs 15 dollars more than a child's pass. A group of 20 people, consisting of adults and children, paid a total of 380 dollars for their passes. A second group, with the same number of adults but twice the number of children as the first group, paid a total of 460 dollars. How many children were in the first group?
$8$
$6$
$10$
In a right-angled triangle $ABC$, the right angle is at $B$. Point $D$ is on the hypotenuse $AC$ such that $BD$ is perpendicular to $AC$. If $AD = x^2 - 7$ and $CD = 3x - 3$, and the length of the altitude $BD$ is 6, what is the length of the hypotenuse $AC$?
$6\sqrt{5}$
$5\sqrt{6}$
$20$
On a plane, there are 10 points. Exactly 4 of these points are collinear (lie on the same straight line), and no other set of three or more points is collinear. How many distinct triangles can be formed by choosing 3 of these 10 points as vertices?
$116$
$119$
$120$
$110$
$236$
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