1. A deck of cards has only red cards. The probability of a randomly chosen card being red is $$$\frac{1}{3}$$$ . When 4 black cards are added to the deck, the probability of choosing red becomes $$$\frac{1}{4}$$$ . How many cards were in the deck originally?

2. The graph of $$$y=x^2+{2x}-{15}$$$ intersects the x-axis at A and C and the y-axis at point B. What is tan(∠ABC) ?

3. For how many values of the constant k will the polynomial $$$x^2+{kx}+{36}$$$ have two distinct integer roots?

4. Among 64 students, 28 of them like Science, 41 like Mathematics and 31 like English. 18 of them like both Mathematics and English. 15 students like both Science and English. 11 students like both Science and Mathematics. How many students like all the three subjects?

5. When a student multiplied the number 66 by the repeating decimal, where $$${a}$$$ and $$${b}$$$ are digits, he did not notice the notation and just multiplied 66 times Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit number

6. The point (-1,-2) is rotated 270° counterclockwise about the point (3,1). What are the coordinates of its new position?

7. Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are 3cm and 6cm. Into each cone is dropped a spherical marble of radius 1cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?

8. Suppose x and y are positive real numbers such that $$$x^y=2^{64} and (log2x)^{log2y}=2^7$$$ What is the greatest possible value of $$$log2y$$$ ？

9. If Mike ran with the speed equaling the square of time for the first 3 seconds (speed is measure in meter/second), how far would Mike run within the first 3 seconds?

10. Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $$$x^2+y^2=4$$$，$$$x^2+y^2=64$$$，and $$$(x-5)^2+y^2=3$$$. What is the sum of the areas of all circles in S?