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16 problems match “case analysis”. Clear

Tournament with two dropouts ●●●○○ 50 ptsIn a tournament, each participant was supposed to play exactly one game with each of the other participants. Two participants dropped out during the tournament, each having played only 4 games. In the end, the total number of games played was 62. How many participants were there in total?
Children with siblings in the class ●●●○○ 50 ptsIn a class, exactly 18 fathers and exactly 24 mothers attended a parent-teacher meeting, and at least one parent of each student attended. Exactly 10 boys and exactly 8 girls had both parents present. For exactly 4 boys and exactly 3 girls only the mother was present, and for exactly 1 boy and exactly 1 girl only the father was present. There are no children in the class with step-parents or step-siblings. Determine the number of children in this class who have siblings in the same class.
Centroid distance to a plane ●●●○○ 50 ptsThe vertices $A$ , $B$ , $C$ of an equilateral triangle $ABC$ with side length $6$ are at distances $1$ , $2$ and $3$ from a plane $\alpha$ . Find all possible distances from the centroid $G$ of triangle $ABC$ to the plane $\alpha$ . Round each value to two decimal places and give them in increasing order.
Placing plants under colored lamps ●●●○○ 50 ptsRachel has two identical basil plants and one aloe plant. She has two identical white lamps and two identical red lamps. Each plant is placed under exactly one lamp, and more than one plant may share a lamp. How many ways are there for Rachel to place her plants under her lamps?
Piecewise function with equal values ●●●○○ 50 ptsLet $a \neq 0$ be a real number, and define the function $f(x)$ by $f(x)=\begin{cases}2x+a, & x<1,\\ -x-2a, & x\geq 1.\end{cases}$ If $f(1-a)=f(1+a)$ , find the value of $a$ . Give your answer as a decimal.
Possible values of the largest removed integer ●●●●○ 100 ptsYann writes down the first $n$ consecutive positive integers, $1,2,3,4,\ldots,n-1,n$ . He removes four different integers $p,q,r,s$ from the list. At least three of $p,q,r,s$ are consecutive and $100<p<q<r<s$ . The average of the integers remaining in the list is $89.5625$ . What is the number of possible values of $s$ ?