PART 1
QUESTION 01
Let a point P move on a straight line according to the score shown on a fair dice that we throw by the following rules. P starts from the origin O.
- If the score is 6, then P returns to the origin O.
- If the score is 1, 2, or 3, then P moves 1 in a positive direction.
- If the score is 4 or 5, then P moves 1 in a negative direction.
When we throw the dice four times, the probability that the point P is at the origin O is \(\bbox[10px, border: 2px solid red]{[1-1]}\)
QUESTION 02
For a constant \(k\), we consider the number of distinct real solutions of equation \(x | x^2 - 3x + 21 | = k\). The range of \(k\) that the number of real solutions is maximum is \(\bbox[10px, border: 2px solid red]{[1-2]} < k < \bbox[10px, border: 2px solid red]{[1-3]}\), and the maximum number of real solutions is \(\bbox[10px, border: 2px solid red]{[1-4]}\).
QUESTION 03
Assume that \(0 < \theta < \pi\). For three points A\((1,0)\), B\((\cos \theta, \sin \theta)\), and C\((\cos 2 \theta, \sin 2\theta)\) on a unit circle, the area of Δ ABC is \(\bbox[10px, border: 2px solid red]{[1-5]}\) by using \(\theta\). When \(\theta = \bbox[10px, border: 2px solid red]{[1-6]}\), the maximum of the area of Δ ABC is \(\bbox[10px, border: 2px solid red]{[1-7]}\).
QUESTION 04
Let \(k\) be a positive integer and let \(p\) be a prime number that is greater than 2. The sum of all divisors of the number \(2^k \: p\) is
\(\left(\bbox[10px, border: 2px solid red]{[1-8]} - 1 \right) \left(1 + \bbox[10px, border: 2px solid red]{[1-9]} \right)\),
where all divisors include 1 and the number itself.
QUESTION 05
In a box, there are 10 cards and a number from 1 to 10 is written on each card. When three cards from the box are randomly taken at a time, we define X, Y, and Z according to three numbers in ascending order. The probability that X is less than or equal to 3 is \(\bbox[10px, border: 2px solid red]{[1-10]}\).
QUESTION 06
The n-th term of sequence 1, 4, 10, 19, 31, ... is \(\bbox[10px, border: 2px solid red]{[1-11]}\), and the sum of the first n terms of the sequence is \(\bbox[10px, border: 2px solid red]{[1-12]}\).
QUESTION 07
Let \(a\) and \(b\) be positive real numbers.
\(\dfrac {4a + b}{2a} + \dfrac {4a - 3b}{b}\)
is at minimum when \(b = \bbox[10px, border: 2px solid red]{[1-13]} \: a\). Its minimum value is \(\bbox[10px, border: 2px solid red]{[1-14]}\).
QUESTION 08
For a variable \(x\), we have
\(\displaystyle (x + 1)^n = \sum_{k = 1}^{n} \:\: _{n} C_{k} \bbox[10px, border: 2px solid red]{[1-15]}^{\bbox[10px, border: 2px solid red]{[1-16]}}\)
It follows that
\(\displaystyle \sum_{k = 1}^{n} \: _{n} C_{k} \: 2^k = \bbox[10px, border: 2px solid red]{[1-17]}^{\bbox[10px, border: 2px solid red]{[1-18]}}\)
By considering the derivatives of the first equality in this item with respect to \(x\), we have
\(\displaystyle \sum_{k = 0}^{n} \: _{n} C_{k} \: k \: 2^k = \dfrac {\bbox[10px, border: 2px solid red]{[1-19]}}{\bbox[10px, border: 2px solid red]{[1-20]}} \sum_{k = 0}^{n} \: _{n} C_{k} \: 2^k\)
QUESTION 09
For a positive integer \(n\), let \(x_k (k = 0, 1, \dotso , n)\) be an integer between 0 and 5.
We have
\(\displaystyle \sum_{k = 0}^{n} \: x_k \: 6^k = \bbox[10px, border: 2px solid red]{[1-21]} + \bbox[10px, border: 2px solid red]{[1-22]} \left(\sum_{k = 1}^{n} \: x_k \: \sum_{l = 0}^{k-1} \: 6^l \right)\)
so that a senary (base 6) number can be divided by \(\bbox[10px, border: 2px solid red]{[1-22]}\) with no remainder if and only if the sum of all of its digits can be divided by \(\bbox[10px, border: 2px solid red]{[1-23]}\) with no remainder.
QUESTION 10
It is clear that \(253x + 256y = 253(x + y) + 3y\). For a pair of integers \(x\) and \(y\) satisfying
\(253x + 256y = 1\),
the absolute value of \(x\) is minimum. Then, \(x = \bbox[10px, border: 2px solid red]{[1-24]}\) and \(y = \bbox[10px, border: 2px solid red]{[1-25]}\).
QUESTION 11
Translate the graph of the function \(y = 2x^2 + 3x + 1\) by \(2\) units in the x-direction and by \(-3\) units in the y-direction and express the resulting graph by
\(y = a_2 x^2 + a_1 x + a_0\).
Then we have \(a_2 = \bbox[10px, border: 2px solid red]{[1-26]}, a_1 = \bbox[10px, border: 2px solid red]{[1-27]}, a_0 = \bbox[10px, border: 2px solid red]{[1-28]}\)
PART 2
For a triangle ABC, take a point D on side AB such that side CD is orthogonal to side AB. We let \(\angle BAC = \dfrac {\pi}{12}\) and let the lengths of side AB and side AD be \(2 \sqrt{2}\) and \(\sqrt{6}\), respectively. Answer the following questions in the corresponding boxes on the answer sheet. They should be simplified as much as possible.
QUESTION 01
From \(\dfrac {\pi}{12} = \dfrac {\pi}{3} - \dfrac {\pi}{4}\), we have
\(\cos \dfrac {\pi}{12} = \dfrac{\bbox[10px, border: 2px solid red]{[2-1]} + \sqrt{2}}{4}\)
QUESTION 02
The length of side AC is \(\bbox[10px, border: 2px solid red]{[2-2]} - 2 \sqrt{3}\).
QUESTION 03
The square of the length of side BC, \((BC)^2\) , is \(\bbox[10px, border: 2px solid red]{[2-3]} - 32 \sqrt{3}\).
QUESTION 04
Thus, the length of side BC is \(\bbox[10px, border: 2px solid red]{[2-4]} - 2 \sqrt{6}\).
PART 3
For a quadratic function \(f(x)\), we define a function as follows:
\(\displaystyle F(x) = \int_0^x f(t) \: dt\)
Assume that \(a\) is a positive number and the function \(F(x)\) has extreme values at \(x = -2a, 2a\). Answer the following questions in the corresponding boxes on the answer sheet.
QUESTION 01
For any \(x\), it holds that
\(F(-x) = \bbox[10px, border: 2px solid red]{[3-1]} \: F(x)\)
QUESTION 02
All the values of \(x\) that satisfy \(F(x) + F(2a) = 0\) are \(\bbox[10px, border: 2px solid red]{[3-2]}\).
QUESTION 03
The local maximum value of function \(\dfrac {F(x)}{F'(0)}\) is \(\bbox[10px, border: 2px solid red]{[3-3]}\).