Notifications
Clear all
Topic starter
- Set $A = \{ x \mid 20 < x < 24 \}$, Set $B = \{ x \mid a < x < 25 \}$, and $A \cup B = \{ x \mid 20 < x < 25 \}$. Which option can be the domain of the real number $a$?
A) $[20, 24)$
B) $[20, 24]$
C) $[20, 25)$
D) $[20, 25]$
E) $(24, 25)$ - Given that $m$ is an integer, and $f(x) = \begin{cases} x + 88 & x > 0 \\ x^2 & x \le 0 \end{cases}$. If $f(f(f(m))) = 2025$, find the sum of the maximum value and the minimum value of $m$.
A) $1804$
B) $1694$
C) $1718$
D) $1318$
E) $1716$ - Suppose the complex number $z = (\sin\alpha - \frac{5}{13}) + (\cos\alpha - \frac{12}{13})i$ is a pure imaginary number. Find $\tan\alpha$.
A) $-\frac{5}{12}$
B) $\frac{12}{13}$
C) $-\frac{13}{5}$
D) $-\frac{5}{24}$
E) $\frac{5}{12}$ - Which of the following $(x, y, z)$ satisfies $x^a = y^b = z^c$ and $\frac{1}{a} - \frac{2}{b} + \frac{3}{c} = 0$?
A) $(4, 3, 2)$
B) $(2, 6, 3)$
C) $(9, 6, 1)$
D) $(18, 12, 2)$
E) $(3, 9, 2)$ - If $\frac{1 - \cos 2\alpha + \sin 2\alpha}{1 + \cos 2\alpha + \sin 2\alpha} = 3$, find $\tan 2\alpha$.
A) $0$
B) $-\frac{4}{3}$
C) $\frac{3}{4}$
D) $\frac{4}{3}$
E) $-\frac{3}{4}$ - The three sides of a triangle are $a$, $b$, and $\sqrt{a^2 + b^2 + \sqrt{3}ab}$, respectively. Find the largest interior angle of this triangle.
A) $105^\circ$
B) $120^\circ$
C) $150^\circ$
D) $135^\circ$
E) $90^\circ$ - $ABCD$ is a square with sides of $2$. $E$ and $F$ are the midpoints of $\overline{BC}$ and $\overline{CD}$, respectively. Fold the square along $\overline{AE}$, $\overline{AF}$, and $\overline{EF}$ to form a tetrahedron that makes three points $B$, $C$, and $D$ coincide at point $G$. Find the volume of the circumscribed sphere of the tetrahedron $G\text{-}AEF$.
A) $2\sqrt{6}\pi$
B) $4\sqrt{3}\pi$
C) $2\sqrt{3}\pi$
D) $6\sqrt{6}\pi$
E) $\sqrt{6}\pi$ - The common difference of an arithmetic sequence $\{a_n\}$ is $d$. If $d \in (0, \frac{\pi}{2})$, and $$\frac{\sin^2 a_3 - \sin^2 a_7}{\sin(a_3 + a_7)} = -1,$$ find $d$.
A) $\frac{\pi}{6}$
B) $\frac{\pi}{10}$
C) $\frac{\pi}{8}$
D) $\frac{\pi}{5}$
E) $\frac{\pi}{4}$ - Any point $P(x, y)$ on the conic section $C$ satisfies $\sqrt{x^2 + y^2 + 2y + 1} + \sqrt{x^2 + y^2 - 2y + 1} = 2\sqrt{2}$. What are the coordinates of the point on $C$ that is closest to the straight line $L : 2x - y - 4 = 0$?
A) $(\frac{\sqrt{2}}{3}, -\frac{\sqrt{2}}{3})$
B) $(\frac{2\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$
C) $(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2})$
D) $(-\frac{\sqrt{6}}{3}, \frac{\sqrt{6}}{3})$
E) $(\frac{\sqrt{6}}{3}, -\frac{\sqrt{6}}{3})$ - $a$ and $b$ are positive real numbers, $\vec{M} = (a, a - 4)$, and $\vec{N} = (b, 1 - b)$. If $\vec{M} \parallel \vec{N}$, find the minimum value of $a + b$.
A) $\frac{5}{2}$
B) $\frac{7}{2}$
C) $1$
D) $\frac{9}{2}$
E) $2$ - The common ratio of a geometric sequence $\{a_n\}$ is $2$, and $a_1 + a_2 = 3$. If $a_k + a_{k+1} + a_{k+2} + \dots + a_{k+9} = 2^{15} - 2^5$, find the positive integer $k$.
A) $10$ Â Â
B) $6$
C) $4$
D) $5$
E) $8$ - The net of the lateral surface of a cone is a semicircle. Its axial cross section (a cross section formed by a plane passing through the apex of the cone and the center of its base) is an isosceles triangle whose apex is $A$. If $\angle A$, $\angle B$, and $\angle C$ are the three interior angles of this triangle, find $$\tan \frac{B}{3} + \tan \frac{2B}{3} + \tan \frac{3B}{3} + \tan \frac{B}{3} \cdot \tan \frac{2B}{3} \cdot \tan \frac{3B}{3}.$$
A) $\sqrt{3}$
B) $2\sqrt{3}$
C) $3\sqrt{3}$
D) $0$
E) $1$ - If $\sqrt[3]{\frac{68 - 27\sqrt{6}}{4}}$ and $\sqrt[3]{\frac{68 + 27\sqrt{6}}{4}}$ are the roots of the equation $x^2 + ax + b = 0$, find the value of $2025ab$.
A) $-10125$
B) $-20250$
C) $-8100$
D) $-13500$
E) $0$ - Bag $A$ has two $10$-dollar bills and four $1$-dollar bills. Bag $B$ has six $5$-dollar bills. Toss a fair die. If the number shown is $n$, then $n$ bills are drawn from each bag. Find the probability that the total value of the bills drawn from bag $A$ is greater than that from bag $B$.
A) $\frac{7}{30}$
B) $\frac{23}{90}$
C) $\frac{3}{10}$
D) $\frac{1}{3}$
E) $\frac{5}{18}$ - Given that a complex number $z = x + yi$ ($x, y \in \mathbb{R}$). If $|z| = 1$, find the minimum value of $\frac{x^4}{x^2 + 1} + \frac{y^4}{y^2 + 1}$.
A) $\frac{1}{3}$
B) $\frac{1}{5}$
C) $\frac{1}{4}$
D) $\frac{1}{2}$
E) $\frac{1}{8}$ - $\vec{a} = (20, -25)$, $\vec{b} = (\cos A, \sin A)$, and $\vec{a} \perp \vec{b}$. Find $\frac{\sin A - \cos A}{\sin A + \cos A}$.
- Today is Saturday. What day will it be after $2025^{2025}$ days?
- Given that the domain of a function $f(x)$ is a real number $\mathbb{R}$, $f(0) = 0$, $f(x) + f(1 - x) = 1$, and $f(\frac{x}{5}) = \frac{1}{2}f(x)$. If $0 \le x_1 < x_2 \le 1$, $f(x_1) \le f(x_2)$. Find $f(\frac{1}{2025})$.
- Let $x = 1 + \sqrt[5]{2} + \sqrt[5]{4} + \sqrt[5]{8} + \sqrt[5]{16}$. Find $(\frac{x + 1}{x})^{55}$
- The two adjacent terms $a_n$ and $a_{n+1}$ in the sequence $\{a_n\}$ are the two roots of an equation $x^2 - 3nx + c_n = 0$ (where $n = 1, 2, 3, \dots$). If $a_1 = 1$, $c_k = 2024$, find $k$.
- Two balls are externally tangent to each other inside a cube with edges of $6$. Each of the two balls is tangent to three faces of the cube. If the centers of the two balls lie on the same diagonal of the cube, find the sum of their radii.
- Find all the real number pairs $(x, y)$ that satisfy the equation $x^2 + \frac{xy}{2} + \frac{y^2}{4} - \frac{y}{4} + 1 = \sqrt{3x^2 - y}$. What is the minimum value of $x - 2y$?
- An unfair die is given. The probabilities of rolling faces $1, 2, 3, 4, 5,$ and $6$ form an arithmetic sequence, with each face corresponding to a term in the sequence one by one. Toss the die independently twice, and record the number of dots on the die as $p_1$ and $p_2$, respectively. If the probability that $p_1 + p_2 = 7$ is $\frac{4}{27}$, what is the probability that $p_1 = p_2$?
- Let $\{a_n\}$ be an arithmetic sequence with a first term of $2$ and a common difference of $1$, and $\{b_n\}$ be a geometric sequence with a first term of $1$ and a common ratio of $2$. If $M_n = a_{b_1} + a_{b_2} + \dots + a_{b_n}$, find the number of terms in $\{M_n\}$ that do not exceed $2025$.
- There are five $1 \times 1$ squares numbered $1 \sim 5$ on the $8 \times 8$ grid paper below. Which numbered squares should be removed so that the remaining grid paper can be completely covered with twenty-one pieces of $1 \times 3$ or $3 \times 1$ rectangular paper?
Posted : 28/05/2026 2:30 am
