MathDB

2023 LMT Spring

Part of LMT

Subcontests

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2023 LMT Spring Guts Round p16-p27- Lexington Mathematical Tournament

Round 6
p16. Triangle ABCABC with AB<ACAB < AC is inscribed in a circle. Point DD lies on the circle and point EE lies on side ACAC such that ABDEABDE is a rhombus. Given that CD=4CD = 4 and CE=3CE = 3, compute AD2AD^2.
p17. Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of 11 m/s. At any positive time tt (in seconds),Wam walks with a speed of 11 m/s with a direction of tt radians clockwise of the positive xx-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly 5π5\pi seconds.
p18. Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to 88 times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins.
Round 7
p19. Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are 44 rounds, of which everyone must attend exactly 22. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly 33 must attend every round. In how many different ways can Ephram organize the teams like this?
p20. For some 44th degree polynomial f(x)f (x), the following is true: ∙\bullet f(−1)=1f (-1) = 1. ∙\bullet f(0)=2f (0) = 2. ∙\bullet f(1)=4f (1) = 4. ∙\bullet f(−2)=f(2)=f(3)f (-2) = f (2) = f (3). Find f(4)f (4).
p21. Find the minimum value of the expression 5x2−16x+16+5x2−18x+29\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29} over all real xx.
Round 8
p22. Let OO and II be the circumcenter and incenter, respectively, of △ABC\vartriangle ABC with AB=15AB = 15, BC=17BC = 17, and CA=16C A = 16. Let X≠AX \ne A be the intersection of line AIAI and the circumcircle of △ABC\vartriangle ABC. Find the area of △IOX\vartriangle IOX.
p23. Find the sum of all integers xx such that there exist integers yy and zz such that x2+y2=3(2016z)+77.x^2 + y^2 = 3(2016^z )+77.
p24. Evaluate ⌊∑i=120221i⌋=⌊11+12+13+...+12022⌋ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor
Round 9
p25.Either: 1. Submit −2-2 as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within 11 of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points.
p26. Jeff is playing a turn-based game that starts with a positive integer nn. Each turn, if the current number is nn, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to nn 2. The number becomes n−an-a, where aa is the largest digit in nn Let S(k)S(k) be the least number of turns Jeff needs to get from the starting number kk to 00. Estimate ∑k=12023S(k).\sum^{2023}_{k=1}S(k). If your estimation is EE and the actual answer is AA, you will receive max⁡(⌊10−∣E−A6000∣⌋,0)\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right) points.
p27. Estimate the smallest positive integer n such that if NN is the area of the nn-sided regular polygon with circumradius 100100, 10000π−N<110000\pi -N < 1 is true. If your estimation is EE and the actual answer is AA, you will receive max⁡⌊(10−∣10⋅log⁡3(AE)∣∣,0⌋. \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor. points.
PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2023 LMT Spring Guts Round p1-p15- Lexington Mathematical Tournament

Round 1
p1. Solve the maze https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png
p2. Billiam can write a problem in 3030 minutes, Jerry can write a problem in 1010 minutes, and Evin can write a problem in 2020 minutes. Billiam begins writing problems alone at 3:003:00 PM until Jerry joins himat 4:004:00 PM, and Evin joins both of them at 4:304:30 PM. Given that they write problems until the end of math team at 5:005:00 PM, how many full problems have they written in total?
p3. How many pairs of positive integers (n,k)(n,k) are there such that (nk)=6{n \choose k}= 6?
Round 2
p4. Find the sumof all integers b>1b > 1 such that the expression of 143143 in base bb has an even number of digits and all digits are the same.
p5. Ιni thinks that a#b=a2−ba \# b = a^2 - b and a&b=b2−aa \& b = b^2 - a, while Mimi thinks that a#b=b2−aa \# b = b^2 - a and a&b=a2−ba \& b = a^2 - b. Both Ini and Mimi try to evaluate 6&(3#4)6 \& (3 \# 4), each using what they think the operations &\& and #\# mean. What is the positive difference between their answers?
p6. A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count.
Round 3
p7. Maya wants to buy lots of burgers. A burger without toppings costs $4\$4, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy 11 burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter.
p8. Consider square ABCDABCD and right triangle PQRPQR in the plane. Given that both shapes have area 11, PQ=QRPQ =QR, PA=RBPA = RB, and PP, AA, BB and RR are collinear, find the area of the region inside both square ABCDABCD and △PQR\vartriangle PQR, given that it is not 00.
p9. Find the sum of all nn such that nn is a 33-digit perfect square that has the same tens digit as n\sqrt{n}, but that has a different ones digit than n\sqrt{n}.
Round 4
p10. Jeremy writes the string: LMTLMTLMTLMTLMTLMTLMTLMTLMTLMTLMTLMT on a whiteboard (“LMTLMT” written 66 times). Find the number of ways to underline 33 letters such that from left to right the underlined letters spell LMT.
p11. Compute the remainder when 12202212^{2022} is divided by 13311331.
p12. What is the greatest integer that cannot be expressed as the sum of 55s, 2323s, and 2929s?
Round 5
p13. Square ABCDABCD has point EE on side BCBC, and point FF on side CDCD, such that ∠EAF=45o\angle EAF = 45^o. Let BE=3BE = 3, and DF=4DF = 4. Find the length of FEFE.
p14. Find the sum of all positive integers kk such that ∙\bullet kk is the power of some prime. ∙\bullet kk can be written as 5654b5654_b for some b>6b > 6.
p15. If x3+y3=2\sqrt[3]{x} + \sqrt[3]{y} = 2 and x+y=20x + y = 20, compute max⁡ (x,y)\max \,(x, y).
PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2023 LMT Spring Speed Round - Lexington Mathematical Tournament

p1. Evaluate (2−0)2⋅3+202+3(2-0)^2 \cdot 3+ \frac{20}{2+3} .
p2. Let x=11⋅99x = 11 \cdot 99 and y=9⋅101y = 9 \cdot 101. Find the sumof the digits of x⋅yx \cdot y.
p3. A rectangle is cut into two pieces. The ratio between the areas of the two pieces is3:1 3 : 1 and the positive difference between those areas is 2020. What’s the area of the rectangle?
p4. Edgeworth is scared of elevators. He is currently on floor 5050 of a building, and he wants to go down to floor 11. Edgeworth can go down at most 44 floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor 11?
p5. There are 2020 people at a party. Fifteen of those people are normal and 55 are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party?
p6. Wam and Sang are chewing gum. Gum comes in packages, each package consisting of 1414 sticks of gum. Wam eats 66 packs and 99 individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy?
p7. At Lakeside Health School (LHS), 40%40\% of students are male and 60%60\% of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology?
p8. Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with 1010 mL of iced tea and a 20002000 mL cup of water with 10%10\% raspberry iced tea. If he fills up the cup with 2020 more mL of 10%10\% raspberry iced tea water, what percent of the solution will be iced tea?
p9. Tree 11 starts at height 220220 m and grows continuously at 33 m per year. Tree 22 starts at height 2020 m and grows at 55 m during the first year, 77 m per during the second year, 99 m during the third year, and in general (3+2n)(3+2n) m in the nth year. After which year is Tree 22 taller than Tree 11?
p10. Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability 499999\frac{499}{999} , tails with probability 499999\frac{499}{999} , and it lands on its side with probability 1999\frac{1}{999} . For each flip of the coin, Leo agrees to give Chris 44 dollars if it lands on heads, nothing if it lands on tails, and 22 dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin 1717 times?
p11. Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of 77, there are 55 balls that don’t get divided into any pile. If he divides the balls into piles of 1111, there are 99 balls that aren’t in any pile. If he divides the balls into piles of 1313, there are 1111 balls that aren’t in any pile. What is the minimumnumber of balls Ephram has?
p12. Let △ABC\vartriangle ABC be a triangle such that AB=3AB = 3, BC=4BC = 4, and CA=5C A = 5. Let FF be the midpoint of ABAB. Let EE be the point on ACAC such that EF∥BCEF \parallel BC. Let CF and BEBE intersect at DD. Find ADAD.
p13. Compute the sum of all even positive integers n≤1000n \le 1000 such that: lcm(1,2,3,...,(n−1))≠lcm(1,2,3,,...,n)lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n).
p14. Find the sum of all palindromes with 66 digits in binary, including those written with leading zeroes.
p15. What is the side length of the smallest square that can entirely contain 33 non-overlapping unit circles?
p16. Find the sum of the digits in the base 77 representation of 62500006250000. Express your answer in base 1010.
p17. A number nn is called sus if n4n^4 is one more than a multiple of 5959. Compute the largest sus number less than 20232023.
p18. Michael chooses real numbers aa and bb independently and randomly from (0,1)(0, 1). Given that aa and bb differ by at most 14\frac14, what is the probability aa and bb are both greater than 12\frac12 ?
p19. In quadrilateral ABCDABCD, AB=7AB = 7 and DA=5DA = 5, BC=CDBC =CD, ∠BAD=135o\angle BAD = 135^o and ∠BCD=45o\angle BCD = 45^o. Find the area of ABCDABCD.
p20. Find the value of ∑i∣210∑j∣i⌊i+1j⌋\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor
p21. Let ana_n be the number of words of length nn with letters {A,B,C,D}\{A,B,C,D\} that contain an odd number of AAs. Evaluate a6a_6.
p22. Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are 6969 people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is?
p23. Find ∑n=2∞2n+10n3+4n2+n−6.\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.
p24. Let △ABC\vartriangle ABC be a triangle with circumcircle ω\omega such that AB=1AB = 1, ∠B=75o\angle B = 75^o, and BC=2BC =\sqrt2. Let lines ℓ1\ell_1 and ℓ2\ell_2 be tangent to ω\omega at AA and CC respectively. Let DD be the intersection of ℓ1\ell_1 and ℓ2\ell_2. Find ∠ABD\angle ABD (in degrees).
p25. Find the sum of the prime factors of 146+2714^6 +27.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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2023 Spring LMT Team #3

Beter Pai wants to tell you his fastest 4040-line clear time in Tetris, but since he does not want Qep to realize she is better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements, given that the correct time is t seconds: ∙\bullet t<100t < 100. ∙\bullet tt is prime. ∙\bullet t−1t -1 has 5 proper factors. ∙\bullet all prime factors of t+1t +1 are single digits. ∙\bullet t−2t -2 is a multiple of 33. ∙\bullet t+2t +2 has 22 factors. Find t.

2023 LMT Spring Accuracy Problem 3

Phoenix is counting positive integers starting from 11. When he counts a perfect square greater than 11, he restarts at 11, skipping that square the next time. For example, the first 1010 numbers Phoenix counts are 11, 22, 33, 44, 11, 22, 33, 55, 66, 77, ...... How many numbers will Phoenix have counted after counting 10000 for the first time?
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