answer:1. **Define the given elements and their properties:** - Let ( k ) be a circle centered at ( M ). - Let ( t ) be a tangent line to ( k ) at point ( T ). - Let ( P ) be a point on ( t ). - Let ( g neq t ) be a line through ( P ) intersecting ( k ) at points ( U ) and ( V ). - Let ( S ) be the point on ( k ) bisecting the arc ( UV ) that does not contain ( T ). - Let ( Q ) be the image of ( P ) under reflection over ( ST ). 2. **Establish the cyclic quadrilateral:** - Let ( L ) be the intersection of lines ( QP ) and ( TS ). - Let ( K ) be the intersection of lines ( VP ) and ( MS ). - Notice that ( angle SKP = angle SLP = 90^circ ) because ( SK ) and ( SL ) are perpendicular to the tangents at ( K ) and ( L ) respectively. - Therefore, quadrilateral ( SKLP ) is cyclic. 3. **Analyze the angles:** - Since ( SKLP ) is cyclic, we have: [ angle QPV = angle LPK ] - By the properties of cyclic quadrilaterals, we know: [ angle LPK = angle LSK ] - Since ( S ) is the midpoint of the arc ( UV ) not containing ( T ), we have: [ angle LSK = angle TSM ] - Since ( TS ) is a diameter of the circle, we have: [ angle TSM = 90^circ - angle SVT ] - Since ( angle SVT = angle PTS ), we get: [ angle TSM = 90^circ - angle PTS ] - Since ( angle PTS = angle PTL ), we have: [ angle TSM = 90^circ - angle PTL ] - Therefore: [ angle QPV = angle TPL ] - Since ( Q ) is the reflection of ( P ) over ( ST ), we have: [ angle TPQ = angle TQP ] 4. **Conclude the trapezoid property:** - Since ( angle TPQ = angle TQP ), ( QP parallel UV ). - Therefore, ( Q, T, U, ) and ( V ) form a trapezoid. (blacksquare)

answer:To find the average number of calls Jean answers per day, we need to add up the total number of calls she answered from Monday to Friday and then divide that total by the number of days she worked. Total calls answered = 35 (Monday) + 46 (Tuesday) + 27 (Wednesday) + 61 (Thursday) + 31 (Friday) Total calls answered = 200 Number of days worked = 5 (Monday to Friday) Average number of calls per day = Total calls answered / Number of days worked Average number of calls per day = 200 / 5 Average number of calls per day = 40 So, Jean answers an average of boxed{40} calls per day.

answer:Let's denote the total number of people in the group as ( T ), and the number of people with an age of less than 20 years as ( L ). According to the problem, the probability of selecting a person with an age of less than 20 years is 0.4. This can be expressed as: [ frac{L}{T} = 0.4 ] We also know that the number of people with an age of more than 30 years is 90, so the rest of the people must have an age of less than 20 years. This can be expressed as: [ L = T - 90 ] Now we can substitute ( L ) from the second equation into the first equation: [ frac{T - 90}{T} = 0.4 ] Now we can solve for ( T ): [ T - 90 = 0.4T ] [ T - 0.4T = 90 ] [ 0.6T = 90 ] [ T = frac{90}{0.6} ] [ T = 150 ] So, there are boxed{150} people in the group.

answer:According to the property of non-negative numbers, (x + y)^2 geq 0. Therefore, the maximum value of 2005 - (x + y)^2 is 2005, when x + y = 0. Since (x + y)^2 geq 0, it follows that 2005 - (x + y)^2 leq 2005. Therefore, the maximum value of 2005 - (x + y)^2 is 2005, when x + y = 0 or x and y are opposite numbers. That is: x = -y Thus, the relationship between x and y when the expression reaches its maximum value is boxed{x = -y}.