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- Simplify $$\frac{2^{2^0} + 2^2 + 2^{-5}}{2^{0^5} + 2^{0^2} + 2^{0} - 5}$$
A) $-\frac{3}{2}$
B) $\frac{3}{2}$
C) $-\frac{7}{2}$
D) $-1$
E) $1$ - A positive integer $N$ has $20$ positive factors, and its prime factorization includes $3$ distinct prime factors. Find the minimum possible value of $N$.
A) $180$
B) $210$
C) $240$
D) $360$
E) $420$ - Remove a portion of the cuboid to obtain the remaining geometric solid, as shown. Which option shows the left view of the solid?
- $\{a_n\}$ is an arithmetic sequence, $a_1 + a_3 + a_5 = 20$, and $a_2 + a_4 + a_6 = 25$. Find $a_{2025}$.
A) $4051$
B) $4050$
C) $\frac{10130}{3}$
D) $\frac{10125}{3}$
E) $\frac{10120}{3}$ - Let $x = \frac{1 - \sqrt{2} + \sqrt{3}}{2}$ and $y = \frac{1 + \sqrt{2} - \sqrt{3}}{2}$. Find the value of $\left(\frac{x^2 - y^2}{2}\right)^2 + xy$.
A) $\frac{1}{2}$
B) $\frac{3}{8}$
C) $\frac{5}{8}$
D) $\frac{1}{4}$
E) $\frac{3}{4}$ - $O$ is the center of the circle. $\triangle OAB$ and $\triangle OCD$ are right triangles, and the lengths of their legs are $p$, $q$, $r$, and $s$, respectively. Points $A$ and $C$ are on the same straight line. Points $B$ and $D$ are on the circumference. If $p^2 + q^2 + r^2 + s^2 = 200$, find the circumference of the circle.
A) 200$\pi$
B) 20$\pi$
C) 100$\pi$
D) 50$\pi$
E) 25$\pi$ - There is an AI system. If a question is entered clearly, the probability that the AI gives the correct answer is $\frac{7}{8}$. If a question is entered unclearly, the probability that the AI gives the correct answer is $\frac{1}{2}$. Given that the probability of entering an unclear question is $\frac{1}{5}$, what is the probability that the AI gives the correct answer?
A) $\frac{4}{5}$
B) $\frac{9}{10}$
C) $\frac{3}{4}$
D) $\frac{7}{8}$
E) $\frac{2}{3}$ - Given that straight lines $L_1 : (k - 1)x + y - 1 = 0$ and $L_2 : (3k - 5)x + (k - 1)y - 3 = 0$. If $L_1 // L_2$, find the real number $k$.
A) 4
B) 2
C) 3
D) 2 or 4
E) 2 or 3 - If the real number $k > 1$ and satisfies $\log_9(\log_8 k) = 2025$. What is the value of $\log_3(\log_2 k)$?
A) $1379$
B) $4050$
C) $4051$
D) $2025$
E) $6075$ - Let the three sides of $\triangle ABC$ be three integers $a$, $b$, and $c$, where $c$ is the longest side ($c > a$, $c > b$). If $\frac{1}{a} + \frac{9}{b} = 1$, how many values of $c$ satisfy the conditions above?
A) $9$
B) $2$
C) $8$
D) $4$
E) $5$ - The school’s math competition includes 7 grade levels. Each grade level has 1 gold medal winner, 1 silver medal winner, and 1 bronze medal winner. Now, one award-winning student is selected from each grade level to stand in a row for a photo, so there will be 7 students in the photo. Given that at least 2 gold medal winners, 2 silver medal winners, and 2 bronze medal winners are selected, find the probability that winners of the same medal are placed together.
A) $\frac{1}{35}$
B) $\frac{1}{3}$
C) $\frac{2}{35}$
D) $\frac{1}{5}$
E) $\frac{1}{7}$ - If $|x_1 - 1| + (x_2 - 2)^2 + |x_3 - 3|^3 + (x_4 - 4)^4 + |x_5 - 5|^5 + \dots + (x_{2024} - 2024)^{2024} + |x_{2025} - 2025|^{2025} = 0$, find $$\frac{1}{x_1x_2} + \frac{1}{x_2x_3} + \frac{1}{x_3x_4} + \dots + \frac{1}{x_{2024}x_{2025}}$$
A) $1$
B) $\frac{1013}{1012}$
C) $\frac{1012}{1013}$
D) $\frac{2026}{2025}$
E) $\frac{2024}{2025}$ - The figure is formed with $5$ squares. Choose any $3$ vertices from the $11$ vertices of the small squares to form a triangle. How many triangles can be formed in total?
A) $120$
B) $153$
C) $159$
D) $121$
E) $150$ - Let the function $f : \mathbb{R} \rightarrow \mathbb{R}^+_{0}$ be defined as $f(x) = |x| + |x - 2|$. If $f(x) + f(y) \le 4$, the maximum and minimum values of $x + y$ are $M$ and $m$ respectively, find the value of $2M - 3m$.
A) 6
B) 2
C) 8
D) 4
E) 5 - Tom and Jerry play a game where they toss the die and move along the squares. Based on the number rolled on the die, move the piece forward $1 \sim 6$ squares. If the piece lands on a square with a number, it will be triggered to move forward or backward. For example, "$+2$" means to move $2$ squares forward, and "$-1$" means to move $1$ square backward. While the game is in process, their two pieces happen to land on the same square. Find the probability that, after both of them toss the die again, Tom's piece ends up closer to the Final than Jerry's piece.
A) $\frac{5}{18}$
B) $\frac{5}{12}$
C) $\frac{4}{9}$
D) $\frac{13}{36}$
E) $\frac{7}{18}$ - Find the number of consecutive $0$'s at the end of the result of the expression $2025 \times 20^{25} \times 25^{20}$.
- If $a$, $b$, $c$, and $d$ are $4$ distinct prime numbers, and $(a - b)(a - c)(a - d) = -2025$, find the maximum value of $a + b + c + d$.
- There is a rectangle $PQRS$ inside the square $ABCD$ that intersects $ABCD$ at points $P$, $Q$, and $S$. If the area of $\triangle APS$ is $27$, the area of $\triangle QRC$ is $30$, find the area of $ABCD$.
- Let $f(x) = x^{10} + 2x^9 - 2x^8 - 2x^7 + x^6 + 4x^2 + 8x + 1$. Find $f(\sqrt{2} - 1)$.
- A sequence $\{a_n\}$ is formed by natural numbers that leave a remainder of $2$ when divided by $3$, arranged from smallest to largest. A sequence $\{b_n\}$ is formed by natural numbers that leave a remainder of $1$ when divided by $4$, arranged from smallest to largest. If the common terms of the two sequences form a set $A$, find the number of elements in the set $A \cap \{n \mid n \le 2025, n \in \mathbb{N}^*\}$.
- 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 $p_1 = p_2$?
- 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,...). If $a_1＝1,c_k＝2024$, find $k$.
- Find the number of positive integers $n$’s that satisfy the condition.
- Let $a,b,c,d$, and $e$ be positive integers that satisfy $a＋b＋c＋d＋e＝2025$. If $M$ is the largest of the four sums $a＋b,b＋c,c＋d$, and $d＋e$, what is the smallest possible value of $M$?
- There are five $1×1$ squares numbered 1～5 on the $8×8$ grid paper below. Which numbered squares should be removed so that the remaining grid paper can be completely covered with twenty-one pieces of $1×3$ or $3×1$ rectangular paper?
Posted : 26/05/2026 4:31 am
