MathDB

General

Part of 2020 MOAA

Problems(1)

2020 MOAA General Round - Math Open At Andover

Source:

9/28/2023
p1. What is 20×20−19×1920\times 20 - 19\times 19?
p2. Andover has a total of 14401440 students and teachers as well as a 1:51 : 5 teacher-to-student ratio (for every teacher, there are exactly 55 students). In addition, every student is either a boarding student or a day student, and 70%70\% of the students are boarding students. How many day students does Andover have?
p3. The time is 2:202:20. If the acute angle between the hour hand and the minute hand of the clock measures xx degrees, find xx. https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png
p4. Point PP is located on segment ACAC of square ABCDABCD with side length 1010 such that AP>CPAP >CP. If the area of quadrilateral ABPDABPD is 7070, what is the area of △PBD\vartriangle PBD?
p5. Andrew always sweetens his tea with sugar, and he likes a 1:71 : 7 sugar-to-unsweetened tea ratio. One day, he makes a 100100 ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a 1:21 : 2 sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness?
p6. Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly 20202020 meters. He wants to raise the entire track 66 meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of 22 meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground.
p7. Mr. DoBa writes the numbers 1,2,3,...,201, 2, 3,..., 20 on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining 1818 numbers is exactly 1111. What is the maximum possible value of the larger of the two numbers that Will erased?
p8. Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number?
p9. Let SS be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, 11, 2424, and 369369 are all elements of SS, while 2020 and 667667 are not. If the elements of SS are written in increasing order, what is the 100100th number written?
p10. Find the largest prime factor of the expression 220+216+212+28+24+12^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1.
p11. Christina writes down all the numbers from 11 to 20202020, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down?
p12. Triangle ABCABC has side lengths AB=AC=10AB = AC = 10 and BC=16BC = 16. Let MM and NN be the midpoints of segments BCBC and CACA, respectively. There exists a point P≠AP \ne A on segment AMAM such that 2PN=PC2PN = PC. What is the area of △PBC\vartriangle PBC?
p13. Consider the polynomial P(x)=x4+3x3+5x2+7x+9.P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9. Let its four roots be a,b,c,da, b, c, d. Evaluate the expression (a+b+c)(a+b+d)(a+c+d)(b+c+d).(a + b + c)(a + b + d)(a + c + d)(b + c + d).
p14. Consider the system of equations ∣y−1∣=4−∣x−1∣|y - 1| = 4 -|x - 1| ∣y∣=∣k−x∣.|y| =\sqrt{|k - x|}. Find the largest kk for which this system has a solution for real values xx and yy.
p16. Let Tn=1+2+...+nT_n = 1 + 2 + ... + n denote the nnth triangular number. Find the number of positive integers nn less than 100100 such that nn and TnT_n have the same number of positive integer factors.
p17. Let ABCDABCD be a square, and let PP be a point inside it such that PA=4PA = 4, PB=2PB = 2, and PC=22PC = 2\sqrt2. What is the area of ABCDABCD?
p18. The Fibonacci sequence {Fn}\{F_n\} is defined as F0=0F_0 = 0, F1=1F_1 = 1, and Fn+2=Fn+1+FnF_{n+2}= F_{n+1} + F_n for all integers n≥0n \ge 0. Let S=1F6+1F6+1F8+1F8+1F10+1F10+1F12+1F12+... S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... Compute 420S420S.
p19. Let ABCDABCD be a square with side length 55. Point PP is located inside the square such that the distances from PP to ABAB and ADAD are 11 and 22 respectively. A point TT is selected uniformly at random inside ABCDABCD. Let pp be the probability that quadrilaterals APCTAPCT and BPDTBPDT are both not self-intersecting and have areas that add to no more than 1010. If pp can be expressed in the form mn\frac{m}{n} for relatively prime positive integers mm and nn, find m+nm + n.
Note: A quadrilateral is self-intersecting if any two of its edges cross.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
MOAAalgebrageometrycombinatoricsnumber theory