p1. For positive real numbers x,y, the operation ⊗ is given by x⊗y=x2−y and the operation ⊕ is given by x⊕y=x2+y. Compute (((5⊗4)⊕3)⊗2)⊕1.
p2. Janabel is cutting up a pizza for a party. She knows there will either be 4, 5, or 6 people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices?
p3. If the numerator of a certain fraction is added to the numerator and the denominator, the result is 1920 . What is the fraction?
p4. Let trapezoid ABCD be such that AB∥CD. Additionally, AC=AD=5, CD=6, and AB=3. Find BC.
p5. AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream?
p6. Find the minimum possible value of the expression ∣x+1∣+∣x−4∣+∣x−6∣.
p7. How many 3 digit numbers have an even number of even digits?
p8. Given that the number 1a99b67 is divisible by 7, 9, and 11, what are a and b? Express your answer as an ordered pair.
p9. Let O be the center of a quarter circle with radius 1 and arc AB be the quarter of the circle’s circumference. Let M,N be the midpoints of AO and BO, respectively. Let X be the intersection of AN and BM. Find the area of the region enclosed by arc AB, AX,BX.
p10. Each square of a 5-by-1 grid of squares is labeled with a digit between 0 and 9, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by 3. How many such labelings are possible if each digit can be used more than once?
p11. A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is 5, how many different possible values of the units digit are there?
p12. There are 2019 red balls and 2019 white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red?
p13. Let ABCD be a square with side length 2. Let ℓ denote the line perpendicular to diagonal AC through point C, and let E and F be themidpoints of segments BC and CD, respectively. Let lines AE and AF meet ℓ at points X and Y , respectively. Compute the area of △AXY .
p14. Express 21−66+21+66 in simplest radical form.
p15. Let △ABC be an equilateral triangle with side length two. Let D and E be on AB and AC respectively such that ∠ABE=∠ACD=15o. Find the length of DE.
p16. 2018 ants walk on a line that is 1 inch long. At integer time t seconds, the ant with label 1≤t≤2018 enters on the left side of the line and walks among the line at a speed of t1 inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when t=2019 seconds.
p17. Determine the number of ordered tuples (a1,a2,...,a5) of positive integers that satisfy a1≤a2≤...≤a5≤5.
p18. Find the sum of all positive integer values of k for which the equation gcd(n2−n−2019,n+1)=k has a positive integer solution for n.
p19. Let a0=2, b0=1, and for n≥0, let
an+1=2an+bn+1,
bn+1=an+2bn+1.
Find the remainder when a2019 is divided by 100.
p20. In △ABC, let AD be the angle bisector of ∠BAC such that D is on segment BC. Let T be the intersection of ray CB and the line tangent to the circumcircle of △ABC at A. Given that BD=2 and TC=10, find the length of AT.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. LMTalgebrageometrycombinatoricsnumber theory