- If the hands on a circular clock start at midnight, what number will the hour hand point to 1000 hours later?
$(A)\ 2 \ \ \ \ \ \ \ \ \ (B)\ 4 \ \ \ \ \ \ \ \ \ (C)\ 8 \ \ \ \ \ \ \ \ \ \ (D)\ 10 \ \ \ \ \ \ \ \ \ (E)\ 12$
2. If $x$ is an integer, what is the least possible value of $|20-7x|$?
$(A)\ 1 \ \ \ \ \ \ \ \ \ (B)\ 2 \ \ \ \ \ \ \ \ \ (C)\ 3 \ \ \ \ \ \ \ \ \ (D)\ 4 \ \ \ \ \ \ \ \ \ (E)\ 5 $
3. If Sy can shovel snow from half of a driveway in 2 hours, and Ty can shovel snow from one quarter of the driveway in 2 hours, how many $minutes$ would it take to shovel the whole driveway working together at their respective constant rates?
$(A)\ 120 \ \ \ \ \ \ \ \ \ (B)\ 160 \ \ \ \ \ \ \ \ \ (C)\ 180 \ \ \ \ \ \ \ \ \ (D)\ 240 \ \ \ \ \ \ \ \ \ (E)\ 360 $
4. Of the bottles that Viola collects, $80%$ are green. Of the green bottles, $30%$ held perfume and $45%$ held spices. If the remaining 25 green bottles held pills, how many bottles are in Viola’s collection?
$(A)\ 75 \ \ \ \ \ \ \ \ \ (B)\ 100 \ \ \ \ \ \ \ \ \ (C)\ 120 \ \ \ \ \ \ \ \ \ (D)\ 125 \ \ \ \ \ \ \ \ \ (E)\ 150 $
5. If $x\ne 0$ and $2x-\dfrac{y-3x^2}{x}=\dfrac{4}{x}$, then $y=$
$(A)\ 4-x^2 \ \ \ \ \ \ \ \ \ (B)\ 4+x^2 \ \ \ \ \ \ \ \ \ (C)\ 5x^2-4 \ \ \ \ \ \ \ \ \ (D)\ 4-5x^2 \ \ \ \ \ \ \ \ \ (E)\ 4x-5x^2 $
6. Don and Juan had a total of $x$ cherries, but then Don ate 27 fewer than $x$ cherries and Juan ate 11 fewer than $x$ cherries. If they each ate at least 10 cheeries, and there was at least one cheery that was’nt eaten, then $x=$
$(A)\ 37 \ \ \ \ \ \ \ \ \ (B)\ 38 \ \ \ \ \ \ \ \ \ (C)\ 39 \ \ \ \ \ \ \ \ \ (D)\ 49 \ \ \ \ \ \ \ \ \ (E)\ 50 $
7. Of the 200 pets for sale at Pip’s Pets, $a$ have scales, $b$ have gills, and $c$ have both. How many of them have neither scales nor gills?
$(A)\ 200-a-b \ \ \ \ \ \ \ \ \ (B)\ 200-c \ \ \ \ \ \ \ \ \ (C)\ 200-a-b-c \ \ \ \ \ \ \ \ \ (D)\ 200-a-b+c \ \ \ \ \ \ \ \ \ (E)\ 200-a-b+2c $
8. The product of two numbers is 144, and the lesser of the two is 6 less than three times of the greater. What is the greater of the two numbers?
$(A)\ 18 \ \ \ \ \ \ \ \ \ (B)\ 8 \ \ \ \ \ \ \ \ \ (C)\ -6 \ \ \ \ \ \ \ \ \ (D)\ -24 \ \ \ \ \ \ \ \ \ (E)\ -30 $
9. If $x$ and $y$ are positive numbers and $x+y=2$, which of the following could be teh value of $20x+50y$?
$(A)\ 35 \ \ \ \ \ \ \ \ \ (B)\ 65 \ \ \ \ \ \ \ \ \ (C)\ 105 \ \ \ \ \ \ \ \ \ (D)\ 140 \ \ \ \ \ \ \ \ \ (E)\ 150 $
10. Iko’s rectangular vegetable garden is $2x$ $m$ wide and $3x$ $m$ long. She wants to plant flowers to form a border of uniform width around the vegetable garden, and measures that the border will cover $14x^2$ $m^2$. How wider is the border of flowers going to be?
$(A)\ 0.5x\ m \ \ \ \ \ \ \ \ \ (B)\ x\ m \ \ \ \ \ \ \ \ \ (C)\ 1.5x\ m \ \ \ \ \ \ \ \ \ (D)\ 2x\ m \ \ \ \ \ \ \ \ \ (E)\ 2.5x\ m $
11. If a $3\times 4$ rectangle is split into eight congruent triangles as shown, what is the perimeter of one of these eight triangles?
$(A)\ 4 \ \ \ \ \ \ \ \ \ (B)\ 6 \ \ \ \ \ \ \ \ \ (C)\ 8 \ \ \ \ \ \ \ \ \ (D)\ 10 \ \ \ \ \ \ \ \ \ (E)\ 12 $
12. If $a$, $b$, and $c$ are 1-digit non-negative integers, not necessarily distinct, how many differnt values are possible from the sum $a+b+c$?
$(A)\ 27 \ \ \ \ \ \ \ \ \ (B)\ 28 \ \ \ \ \ \ \ \ \ (C)\ 29 \ \ \ \ \ \ \ \ \ (D)\ 30 \ \ \ \ \ \ \ \ \ (E)\ 31 $
13. What is the area of an isosceles triangle with side lengths of $22$ and $61$?
$(A)\ 660 \ \ \ \ \ \ \ \ \ (B)\ 682 \ \ \ \ \ \ \ \ \ (C)\ 1320 \ \ \ \ \ \ \ \ \ (D)\ 1342 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
14. How many different ordered pairs of positive integers are there each of whose squares sum to $9797$? For example, for this ordered pair of positive integers, $(10,11)$, its squares sum to $10^2+11^2=121$. [Hint: The identity $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$ can help].
$(A)\ 2 \ \ \ \ \ \ \ \ \ (B)\ 4 \ \ \ \ \ \ \ \ \ (C)\ 6 \ \ \ \ \ \ \ \ \ (D)\ 8 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
15. What is the least prime number which can be written as a sum of two composite numbers?
$(A)\ 7 \ \ \ \ \ \ \ \ \ (B)\ 11 \ \ \ \ \ \ \ \ \ (C)\ 13 \ \ \ \ \ \ \ \ \ (D)\ 5 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
16. If we define the $separation$ between two points in the $x-y$ plane as the length of the shortest path from one point to another along the axes and/or along lines parallel to axes, then there are exactly four points with integer coordinates whose separation from origin is 1. How many points with integral coordinates have a separation from the origin of 5?
$(A)\ 4 \ \ \ \ \ \ \ \ \ (B)\ 5 \ \ \ \ \ \ \ \ \ (C)\ 20 \ \ \ \ \ \ \ \ \ (D)\ 25 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
17. From a point inside an equilateral triangle, if the distances to the three sides are $2\sqrt{3}$, $4\sqrt{3}$ and $5\sqrt{3}$, what is the area of the equilateral triangle?
$(A)\ 22\sqrt{3} \ \ \ \ \ \ \ \ \ (B)\ 120 \ \ \ \ \ \ \ \ \ (C)\ 120\sqrt{3} \ \ \ \ \ \ \ \ \ (D)\ 124\sqrt{3} \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
18. If $a$ and $c$ are rational, and if $x^3+cx^2-5x+a=(x-c)(x+c)(x+\dfrac{a}{c^2})$, what are all possible values of $a$?
$(A)\ 1,2 \ \ \ \ \ \ \ \ \ (B)\ \pm 3 \ \ \ \ \ \ \ \ \ (C)\ \pm 5 \ \ \ \ \ \ \ \ \ (D)\ \pm 6 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
19. The length of the sides of hexagon $H$ are 1, 2, 3, 4, 5, and 6. If no two consecutive sides of $H$ have consecutive-integer lengths, what is the maximum sum of the lenths of thrww consecutive sides of $H$?
$(A)\ 12 \ \ \ \ \ \ \ \ \ (B)\ 13 \ \ \ \ \ \ \ \ \ (C)\ 14 \ \ \ \ \ \ \ \ \ (D)\ 15 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
20. What is the least $k\ge 2$ for which there exist $k$ consecutive integers whose sum is $1000$?
$(A)\ 3 \ \ \ \ \ \ \ \ \ (B)\ 4 \ \ \ \ \ \ \ \ \ (C)\ 5 \ \ \ \ \ \ \ \ \ (D)\ 6 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
21. If $f(2t+7)=12t+37$ for all real numbers $t$, what are all values of $x$ which satisfy $f(x)=x^2$?
$(A)\ 2,3 \ \ \ \ \ \ \ \ \ (B)\ 2,7\ \ \ \ \ \ \ \ \ (C)\ \pm 5 \ \ \ \ \ \ \ \ \ (D)\ \pm 6 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
22. Three congruent circles have their centers on the same diagonal of a square, with two of the circles each tangent to two sides of the square, and the third circle externally tangent to the other two circles, all as shown. If the length of a side of the square is $8$, what is the length of a radius of the one of the circles?
$(A)\ \sqrt{2} \ \ \ \ \ \ \ \ \ (B)\ \dfrac{8\sqrt{2}}{7}\ \ \ \ \ \ \ \ \ (C)\ 4\sqrt{2}-4 \ \ \ \ \ \ \ \ \ (D)\ 8-4\sqrt{2} \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
23. Two of the numbers I wrote on my paper are the greatest prime numbers less than 100 that differ by 4. What is their sum?
$(A)\ 162 \ \ \ \ \ \ \ \ \ (B)\ 172\ \ \ \ \ \ \ \ \ (C)\ 182 \ \ \ \ \ \ \ \ \ (D)\ 192 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
24. For what integer $k$ will $2^k-1$ be the greatest divisor of $2^{22}-2$ that is less than $2^{22}-2$?
$(A)\ 1 \ \ \ \ \ \ \ \ \ (B)\ 20\ \ \ \ \ \ \ \ \ (C)\ 21 \ \ \ \ \ \ \ \ \ (D)\ 22 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
25. The area of the parallelogram shown is $44$. If the total area of the shaded region is $14$, what is the area of the region common to the two large unshaded triangles that share a common base?
$(A)\ 14 \ \ \ \ \ \ \ \ \ (B)\ 15\ \ \ \ \ \ \ \ \ (C)\ 20 \ \ \ \ \ \ \ \ \ (D)\ 21 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
26. What is the only value of $x$ for which there are 24 positive integers (not necessarily distinct) whose sum is $x$ and whose product is $x$?
$(A)\ 12 \ \ \ \ \ \ \ \ \ (B)\ 24\ \ \ \ \ \ \ \ \ (C)\ 36 \ \ \ \ \ \ \ \ \ (D)\ 48 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
27. What is the largest integer $n$ for which $2020-n$, 2020, $2020+n$ could be the length of sides of a triangle?
$(A)\ 1009 \ \ \ \ \ \ \ \ \ (B)\ 1010\ \ \ \ \ \ \ \ \ (C)\ 2019 \ \ \ \ \ \ \ \ \ (D)\ 2020 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
28. The only possible scores on an exam are the 16 integers from 0 to 15. The most frequent score earned by the 100 students who took the exam was 0 (a score achieved by $k$ students). If no other score was earned as frequently, what the least poissble value of $k$?
$(A)\ k\le 10 \ \ \ \ \ \ \ \ \ (B)\ 10\lt k\le 20\ \ \ \ \ \ \ \ \ (C)\ 10\lt k\le 30 \ \ \ \ \ \ \ \ \ (D)\ k\gt 30 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
29. Two isosceles triangles with supplementary vertex angles share a common base. The lengths of the legs of one triangle are 12 and the other triangle are 5. What is the sum of the lengths of the altitudes that can be drawn to the common base of the triangles?
$(A)\ 12 \ \ \ \ \ \ \ \ \ (B)\ 13\ \ \ \ \ \ \ \ \ (C)\ 14 \ \ \ \ \ \ \ \ \ (D)\ 15 \ \ \ \ \ \ \ \ \ (E)\ none\ of\ above $
30. Two numbers are called reversal numbers if one is obtained from the other by reversing the order of digits. For example, 123 and 321. Are there two reversal numbers whose product is $92,565$?
$(A)\ Yes \ \ \ \ \ \ \ \ \ (B)\ No\ \ \ \ \ \ \ \ \ $