MathDB

7

Part of 2015 BMT Spring

Problems(5)

BMT 2015 Spring - Geometry 7

Source:

12/30/2021
Define A=(1,0,0)A = (1, 0, 0), B=(0,1,0)B = (0, 1, 0), and PP as the set of all points (x,y,z)(x, y, z) such that x+y+z=0x+y+z = 0. Let PP be the point on PP such that d=AP+PBd = AP + P B is minimized. Find d2d^2.
geometry
2015 points, equal segments, equal angles 2015 BMT Team 7

Source:

1/4/2022
X1,X2,...,X2015X_1, X_2, . . . , X_{2015} are 20152015 points in the plane such that for all 1≤i,j≤20151 \le i, j \le 2015, the line segment XiXi+1=XjXj+1X_iX_{i+1} = X_jX_{j+1} and angle ∠XiXi+1Xi+2=∠XjXj+1Xj+2\angle X_iX_{i+1}X_{i+2} = \angle X_jX_{j+1}X_{j+2} (with cyclic indices such that X2016=X1X_{2016} = X_1 and X2017=X2X_{2017} = X_2). Given fixed X1X_1 and X2X_2, determine the number of possible locations for X3X_3.
geometryequal anglesequal segments
2015 BMT Spring Analysis 7

Source:

1/20/2022
Evaluate ∑k=037(−1)k(752k)\sum_{k=0}^{37}(-1)^k\binom{75}{2k}.
Summationbinomial coefficientsalgebra
2015 BMT Spring Discrete 7

Source:

1/20/2022
At Durant University, an A grade corresponds to raw scores between 9090 and 100100, and a B grade corresponds to raw scores between 8080 and 9090. Travis has 33 equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a 90%90\% chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.
probabilityalgebracombinatorics
max angle of triangle by touchpoints of incircle 2015 BMT Individual 7

Source:

1/3/2022
In △ABC \vartriangle ABC, ∠B=46o\angle B = 46^o and ∠C=48o\angle C = 48^o . A circle is inscribed in △ABC \vartriangle ABC and the points of tangency are connected to form PQRPQR. What is the measure of the largest angle in △PQR\vartriangle P QR?
geometryincircleangles