Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.

Let $k,x,y$ be postive integers. The quotients of $k$ divided by $x^2, y^2$ are $n,n+148$ respectively.($k$ is divisible by $x^2$ and $y^2$) (a) If $\gcd(x,y)=1$, then find $k$. (b) If $\gcd(x,y)=4$, then find $k$.

Let $a,b$ be positive integers satisfying $$\sqrt{\dfrac{ab}{2b^2-a}}=\dfrac{a+2b}{4b}$$. Find $|10(a-5)(b-15)|+8$.

If we fill $1\sim 16$ into $4\times4$ chessboard randomly. What is the possibility of the sum of each rows and columns are all even?

Let (i) $a_1,a_2,a_3$ is an arithmetic progression and $a_1+a_2+a_3=18$ (ii) $b_1,b_2,b_3$ is a geometric progression and $b_1b_2b_3=64$ If $a_1+b_1,a_2+b_2,a_3+b_3$ are all positive integers and it is a ageometric progression, then find the maximum value of $a_3$.

Let $ABCD$ be an unit aquare.$E,F$ be the midpoints of $CD,BC$ respectively. $AE$ intersects the diagonal $BD$ at $P$. $AF$ intersects $BD,BE$ at $Q,R$ respectively. Find the area of quadrilateral $PQRE$.

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?