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- Which of the following numbers has the largest value? $$2^{13},5^{14},6^{10},10^{11},18^7$$
A. $2^{13}$
B. $5^{14}$
C. $6^{10}$
D. $10^{11}$
E. $18^7$ - What is the smallest value of $\frac{a}{|a|}+\frac{|b|}{|b|}+\frac{|c|}{c}$?
A. 0
B. 1
C. 2
D. 3
E. None of the above - A triangle and a circle are drawn on a rectangular (with different dimensions) piece of paper. What is the greatest number of regions can be formed?
A. 8
B. 11
C. 13
D. 16
E. None of the above - How many prime numbers from 1 to 100 can be written as a sum of two perfect squares?
A. 10
B. 11
C. 12
D. 13
E. None of the above - A number is written on the board. It starts with one digit “1”, four digits “2”, nine digits “3”, …, 81 digits “9”, 100 digits “0”, 112 digits “1”, 122 digits “2” and so on. In other words, the number is as follow: $$12222\underset{9\text{ digits}}{\underbrace{333333333}}...\underset{81\text{ digits}}{\underbrace{999...999}}\text{ }\underset{100\text{ digits}}{\underbrace{000...000}}\text{ }\underset{11^2\text{ digits}}{\underbrace{111...111}}\text{ }\underset{12^2\text{ digits}}{\underbrace{222...222}}...$$ What is the 2019th digit of the number?
A. 5
B. 6
C. 7
D. 8
E. None of the above - The following calculations lead to 2 = −2:
(1) We rewrite 2, which is $\sqrt{4}$, as $\sqrt{(-1)(-1)\times 4}$.
(2) We rewrite the single root as product of multiple roots: $\sqrt{-1}\times\sqrt{-1}\times\sqrt{4}$
(3) Since $\sqrt{-1}\times\sqrt{-1}=-1$, the expression is simplified to $-\sqrt{4}$.
(4) Hence, $2=-2$.How many steps are wrong?
A. Only 1 step
B. Only 2 steps
C. Only 3 steps
D. All 4 steps
E. The proof is correct; 2 is equal to -2. - Extra air is pumped into a spherical balloon so that the volume of the balloon becomes 8 times larger. What is the percentage change of the surface area of the balloon?
A. 100%
B. 200%
C. 300%
D. 400%
E. None of the above - The letters “H”, “I”, “K”, “O” and “S” are rearranged to form all possible five-letter words. The list of words is then arranged in dictionary order. For example, the first word is HIKOS and the last word is SOKIH. What is the order of ‘SHIOK’ in the list?
A. 90
B. 98
C. 101
D. 120
E. None of the above - A number is said to be square-free if the largest power of a prime in its prime factorization is one. It is given that $$\sqrt{2+\sqrt{3}}=\frac{\sqrt{a}+\sqrt{b}}{2},$$ where 𝑎, 𝑏 are square-free positive integers, what is the value of $𝑎^2 + 𝑏^2$?
A. 5
B. 13
C. 40
D. 45
E. None of the above - The operator $\bullet$ acts on two numbers to give the following outcomes:
What is the value of 20 $\bullet$ 1?
A. 0
B. 19
C. 20
D. 32
E. None of the above - Chloe and Linda have been arrested for retail theft. If they both cooperate and confess to their crime, they will be sentenced to 3 months in jail. If only one of them confesses, the person who confesses will be sentenced to 2 months in jail and the other person will be sentenced to 10 months in jail. If neither of them confesses, they both will not be sentence to jail. Assuming they will not know the decision made by the other person, what should Chloe and Linda do to obtain the best possible outcome for themselves?
A. Chloe should confess and Linda should not confess.
B. Chloe should not confess, and Linda should confess.
C. Chloe should not confess, and Linda should not confess.
D. Chloe should confess and Linda should confess.
E. No matter what they do, the outcome will still be the same. - A standard dice is rolled five times. What is the probability that three rolls have a same face value and the other two rolls have a different same face value?
A. $\frac{1}{2}$
B. $\frac{25}{648}$
C. $\frac{5}{1296}$
D. $\frac{60}{7776}$
E. $\frac{18}{46656}$ - A group of knights, servants and traders joined a conference hosted by the King. Knights always tell the truth; servants always lie while traders alternate between telling the truth or lying. The King asked each person the 4 questions below in the following order:
1. Are you a knight?
2. Are you a trader?
3. Are you a servant?
4. Are you a knight?
Thirty-two people replied “Yes” to the first 2 questions, 18 people replied “Yes” to both the second and third questions, no one replied “Yes” to both questions 3 and 4 and 19 people replied “Yes” in both the first and last questions. How many knights are there?
A. 3
B. 4
C. 5
D. 6
E. None of the above - For any positive real number 𝑥 and positive integer 𝑛, which of the following options below is true?
A. $n^2<2^n$
B. $(1+x)^n\ge 1+nx$
C. $(1+2+...+n)^2>1^3+2^3+...+n^3$
D. $\frac{1+2+...+n}{n}<\sqrt[n]{1+2+...+n}$
E. None of the above - Five close friends each brought one gift to a Christmas party. They put all the five gifts on a table and distributed to each other such that no one receives his/her own gift. In how many ways can the gifts be distributed?
A. 12
B. 44
C. 120
D. 480
E. None of the above - Starting from the cell at row 0 and column 0, positive integers are written in a spiral way in an infinite table, as follow:
What is the number written on the cell of row 19 and column 19? - In the expansion of: $$(2x^2-3x+2)^3(x^2-2x+2)^3$$ Find the sum of digits in the sum of the coefficient of $x^{2n}$, where 𝑛𝑛 is a positive integer.
- What is the smallest positive integer 𝑛 such that the first 4 digits of the number 𝑛 × 2018 is 2017?
- Different letters represent different digits. Find the four-digit number “HOLE”.
- It is given that $$\text{sin}^2 0° + \text{sin}^2 5° + \text{sin}^2 10° + ⋯ + \text{sin}^2 85° + \text{sin}^2 90° = \frac{a}{b},$$ where $\frac{a}{b}$ is a fraction in simplest form. Find the value of 𝑎 + 𝑏.
- If 𝑃(𝑥) is a non-zero polynomial with all integer coefficients and has the root: $$𝑥 = 2019 + \sqrt{2} + \sqrt{3}$$ What is the minimum degree of 𝑃(𝑥)?
- Teacher Victor thinks of a four-digit “key” with different digits. He then asks his students to guess the “key”. In each turn, Victor’s students suggest a four-digit number with distinct digits. He then gives certain responses, as follow:
: One of the digits is in the “key” and in the correct position.
$\bullet$: One or more the digits are in the “key” but not in the correct position.
: One of the digits is not in the “key”.
For example, if the “key” is 1234 and the students guess the number 1345, the response will be
. If the students guess 7234, the response will be 
. Find the “key” teacher Victor is thinking of. - There are 20 diagonals in a regular octagon. When the diagonals intersect each other, how many distinct intersection points are there inside the octagon are there?
- Find the minimum value of $\sqrt{x^2-8x+25}+\sqrt{x^2-8x+97}$ where 𝑥 is a positive real number.
- A triangle ABC is inscribed in a circle with radius of 5 cm. Point D is the intersection point of the heights of the triangle ABC. If ∠BAC = 60°, find the length (in cm) of the segment AD.
Posted : 18/06/2026 6:17 am
