MathDB

2020 Stanford Mathematics Tournament

Part of Stanford Mathematics Tournament

Subcontests

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SMT 2020 Geometry #3 square

Square ABCDABCD has side length 44. Points PP and QQ are located on sides BCBC and CDCD, respectively, such that BP=DQ=1BP = DQ = 1. Let AQAQ intersect DPDP at point XX. Compute the area of triangle PQXP QX.

SMT 2020 Geometry Tiebreaker #3

Three cities that are located on the vertices of an equilateral triangle with side length 100100 units. A missile flies in a straight line in the same plane as the equilateral triangle formed by the three citiies. The radar from City AA reported that the closest approach of the missile was 2020 units. The radar from City BB reported that the closest approach of the missile was 6060 units. However, the radar for city CC malfunctioned and did not report a distance. Find the minimum possible distance for the closest approach of the missile to city CC.
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2020 SMT Team Round - Stanford Math Tournament

p1. Find the sum of the largest and smallest value of the following function: f(x)=∣x−23∣+∣32−x∣−∣12+x∣f(x) = |x -23| + |32 -x| - |12 + x| where the function has domain [−37,170][-37,170].
p2. Given a convex equiangular hexagon with consecutive side lengths of 9,a,10,5,5,b9, a, 10, 5, 5, b, where aa and bb are whole numbers, find the area of the hexagon.
p3. Let §=1,2,...,100\S = {1, 2, ... , 100}. Compute the minimum possible integer nn such that, for any subset T⊂ST \subset S with size nn, every integer aa in SS satisfies the relation a≡bcmod  101a \equiv bc \mod 101, for some choice of integers b,cb, c in TT.
p4. Let CC be the circle of radius 22 centered at (4,4)(4, 4) and let LL be the line x=−2x = -2. The set of points equidistant from CC and from LL can be written as ax2+by2+cxy+dx+ey+f=0ax^2 + by^2 + cxy + dx + ey + f = 0 where a,b,c,d,e,fa, b, c, d, e, f are integers and have no factors in common. What is ∣a+b+c+d+e+f∣|a + b + c + d + e + f|?
p5. If aa is picked randomly in the range (14,34)\left(\frac{1}{4},\frac{3}{4}\right) and bb is chosen such that intab1x2dx=1int_a^b \frac{1}{x^2}dx = 1, compute the expected value of b−ab - a.
p6. Let A1,A2,..,A2020A_1, A_2, .., A_{2020} be a regular 20202020-gon with a circumcircle C of diameter 11. Now let PP be the midpoint of the small-arc A1−A2A_1 - A_2 on the circumcircle CC. Then find ∑2020i=1∣PAi∣2\sum_{2020}^{i=1} |PA_i|^2.
p7. A certain party of 20202020 people has the property that, for any 44 people in the party, there is at least one person of those 44 that is friends with the other three (assume friendship is mutual). Call a person in the party a politician if they are friends with the other 20192019 people in the party. If nn is the number of politicians in the party, compute the sum of the possible values of nn.
p8. Let SS be an nn-dimensional hypercube of sidelength 11. At each vertex draw a hypersphere of radius 1/21/2 , let Ω\Omega be the set of these hyperspheres. Consider a hypersphere Γ\Gamma centered at the center of the cube that is externally tangent to all the hyperspheres in Ω\Omega. For what value of nn does the volume of Γ\Gamma equal to the sum of the volumes of the hyperspheres in Ω\Omega.
p9. Solve for CC: 2π3=∫011Cx−x2dx\frac{2 \pi}{3} = \int_0^1 \frac{1}{\sqrt{Cx -x^2}}dx
p10. Nathan and Konwoo are both standing in a plane. They each start at (0,0)(0,0). They play many games of rock-paper-scissors. After each game, the winner will move one unit up, down, left, or right, chosen randomly. The outcomes of each game are independent, however, Konwoo is twice as likely to win a game as Nathan. After 66 games, what is the probability that Konwoo is located at the same point as Nathan? (For example, they could have each won 33 games and both be at (1,2)(1,2).)
p11. Suppose that x,y,zx, y, z are real positive numbers such that (1+x4y4)ez+(1+81e4z)x4e−3z=12x3y(1 + x^4y^4)e^z +(1 + 81e^{4z})x^4e^{-3z} = 12x^3y. Find all possible values of x+y+zx + y + z.
p12. Given a large circle with center (x0,y0)(x_0, y_0), one can place three smaller congruent circles with centers (x1,y1)(x_1, y_1),(x2,y2)(x_2, y_2),(x3,y3)(x_3, y_3) that are pairwise externally tangent to each other and all internally tangent to the outer circle. If this placement makes x0=x1x_0 = x_1 and y1>y0 y_1 > y_0, we call this an “up-split”. Otherwise, if the placement makes x0=x1x_0 = x_1 and y1<y0y_1 < y_0, we call it a “down-split.” Alice starts at the center of a circle CC with radius 11. Alice first walks to the center of the upper small circle CuC_u of an up-slitting of CC. Then, Alice turns right to walk to the center of the upper-right small circle of a down-splitting of CdC_d. Alice continues this process of turning right and walking to the center of a new circle created by alternatingly up- and down-splitting. Alice’s path will form a spiral converging to (xA,yA)(x_A, y_A). On the otherhand, Bob always up-splits the circle he is in the center of, turns right and finds the center of the next small circle. His path will converges to (xB,yB)(x_B, y_B). Compute ∣1xA−1xB∣\left| \frac{1}{x_A} - \frac{1}{x_B}\right|.
p13. Compute the sum of all natural numbers bb less than 100100 such that bb is divisible by the number of factors of the base-1010 representation of 2020b2020_b.
p14. Iris is playing with her random number generator. The number generator outputs real numbers from 00 to 11. After each output, Iris computes the sum of her outputs, if that sum is larger than 22, she stops. What is the expected number of outputs Iris will receive before she stops?
p15. Evaluate ∫0π2ln⁡(9sin⁡2θ+121cos⁡2θ)dθ\int_0^{\frac{\pi}{2}} \ln (9 \sin^2 \theta + 121 \cos^2 \theta) d\theta
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.