Subcontests
(3)2019 Fall LMT Individual Round - Lexington Mathematical Tournament
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. 2019 LMT Fall Team Round - Lexington Math Tournament
p1. What is the smallest possible value for the product of two real numbers that differ by ten?
p2. Determine the number of positive integers n with 1≤n≤400 that satisfy the following:
∙ n is a square number.
∙ n is one more than a multiple of 5.
∙ n is even.
p3. How many positive integers less than 2019 are either a perfect cube or a perfect square but not both?
p4. Felicia draws the heart-shaped figure GOAT that is made of two semicircles of equal area and an equilateral triangle, as shown below. If GO=2, what is the area of the figure?
https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png
p5. For distinct digits A,B, and C:
\begin{tabular}{cccc}
& A & A \\
& B & B \\
+ & C & C \\
\hline
A & B & C \\
\end{tabular} Compute A⋅B⋅C.
p6 What is the difference between the largest and smallest value for lcm(a,b,c), where a,b, and c are distinct positive integers between 1 and 10, inclusive?
p7. Let A and B be points on the circumference of a circle with center O such that ∠AOB=100o. If X is the midpoint of minor arc AB and Y is on the circumference of the circle such that XY⊥AO, find the measure of ∠OBY .
p8. When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in 6 hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in 4 hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates?
p9. How many positive integer divisors n of 20000 are there such that when 20000 is divided by n, the quotient is divisible by a square number greater than 1?
p10. What’s the maximum number of Friday the 13th’s that can occur in a year?
p11. Let circle ω pass through points B and C of triangle ABC. Suppose ω intersects segment AB at a point D=B and intersects segment AC at a point E=C. If AD=DC=12, DB=3, and EC=8, determine the length of EB.
p12. Let a,b be integers that satisfy the equation 2a2−b2+ab=18. Find the ordered pair (a,b).
p13. Let a,b,c be nonzero complex numbers such that a−b1=8,b−c1=10,c−a1=12.
Find abc−abc1 .
p14. Let △ABC be an equilateral triangle of side length 1. Let ω0 be the incircle of △ABC, and for n>0, define the infinite progression of circles ωn as follows:
∙ ωn is tangent to AB and AC and externally tangent to ωn−1.
∙ The area of ωn is strictly less than the area of ωn−1.
Determine the total area enclosed by all ωi for i≥0.
p15. Determine the remainder when 132020+112020 is divided by 144.
p16. Let x be a solution to x+x1=1. Compute x2019+x20191 .
p17. The positive integers are colored black and white such that if n is one color, then 2n is the other color. If all of the odd numbers are colored black, then how many numbers between 100 and 200 inclusive are colored white?
p18. What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once?
p19. Let △ABC have side lengths AB=19, BC=2019, and AC=2020. Let D,E be the feet of the angle bisectors drawn from A and B, and let X,Y to be the feet of the altitudes from C to AD and C to BE, respectively. Determine the length of XY .
p20. Suppose I have 5 unit cubes of cheese that I want to divide evenly amongst 3 hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese?
PS. You had better use hide for answers.