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1. 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) $1$
2. 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) $420$
3. Remove a portion of the cuboid to obtain the remaining geometric solid, as shown. Which option shows the left view of the solid?
   
4. $\{a_n\}$ 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) $\frac{10120}{3}$
5. 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) $\frac{3}{4}$
6. $O$ 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$\pi$
7. 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) $\frac{2}{3}$
8. 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
9. 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) $6075$
10. 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) $5$
11. 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) $\frac{1}{7}$
12. 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) $\frac{2024}{2025}$
13. 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) $150$
14. 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
15. 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) $\frac{7}{18}$
16. Find the number of consecutive $0$'s at the end of the result of the expression $2025 \times 20^{25} \times 25^{20}$.
17. If $a$, $b$, $c$, and $d$ are $4$ distinct prime numbers, and $(a - b)(a - c)(a - d) = -2025$, find the maximum value of $a + b + c + d$.
18. 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 $ABCD$.
    
19. Let $f(x) = x^{10} + 2x^9 - 2x^8 - 2x^7 + x^6 + 4x^2 + 8x + 1$. Find $f(\sqrt{2} - 1)$.
20. 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 $A \cap \{n \mid n \le 2025, n \in \mathbb{N}^*\}$.
21. 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 $p_1 = p_2$?
22. 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 $k$.
23. Find the number of positive integers $n$’s that satisfy the condition.
    
24. 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 $M$?
25. 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?
    

1. $a$ and $b$ are the solutions of the equation $x^2＋25x＋10＝0$. How many times does the digit 0 appear in the product of $(a^2＋2025a＋10)(b^2＋2025b＋10)$?
   A) 6
   B) 8
   C) 7
   D) 10
   E) 9
2. On the Cartesian coordinate plane, a straight line $L : y = -\frac{\sqrt{3}}{3}x + \frac{\sqrt{3}}{3}$ intersects the two axes at points $A$ and $B$. Make $\overline{OP} \perp \overline{AB}$, and find $\cos a$.
   
   A) $\frac{1}{2}$
   B) $\frac{\sqrt{2}}{2}$
   C) $\frac{\sqrt{3}}{2}$
   D) $\frac{\sqrt{3}}{3}$
   E) $\frac{\sqrt{3}}{4}$
3. The perimeters of two similar triangles are 210 and 70, respectively. The longest side of the triangle with a longer perimeter is 87. The shortest side of the triangle with a shorter perimeter is 20. Find the area of the triangle with a longer perimeter.
   A) 2140
   B) 1890
   C) 1344
   D) 2610
   E) 1560
4. If $m=1+\sqrt{2},n=1-\sqrt{2},(m^2-2m)(2n^2-4n-k)=5$, find $k$.
   A) -3
   B) 2
   C) 1
   D) -1
   E) -5
5. If a circle $O$ passes through three points (2, 2), (6, 2), and (4, 5), and the center of circle $O$ is $(m,n)$, find $m＋2n$.
   A) 10
   B) 11
   C) $\frac{31}{3}$
   D) $\frac{29}{3}$
   E) $\frac{21}{2}$
6. Given an incomplete glass cylindrical container with water. The radius of its bottom surface is $r$, and its height is $h$. When the container is placed upright, the water is $\frac{1}{2}h$ high. When the container is placed upside down, the water is $\frac{2}{3}h$ high. If the capacity of the container is $m\pi r^2h$, find $m$.
   
   A) $\frac{3}{4}$
   B) $\frac{7}{8}$
   C) $\frac{4}{5}$
   D) $\frac{5}{6}$
   E) $\frac{11}{12}$
7. As shown, $\triangle ABC$ is inscribed in circle $O$, $\overline{AH} \perp \overline{BC}$ at $H$, $\overline{AH} = 18$, $\overline{AC} = 24$, and circle $O$'s radius $\overline{OC} = 13$. Find $\overline{AB}$.
   
   A) 19
   B) $18\sqrt{2}$
   C) 20
   D) 21
   E) $\frac{39}{2}$
8. Given the net of a cube box without a lid. $A,B$, and $C$ are the three vertices on the net. Find $∠ABC$ in the cube box.
   
   A) $45^o$
   B) $60^o$
   C) $120^o$
   D) $90^o$
   E) $150^o$
9. At first, three boxes $A$, $B$, and $C$ each contain an equal number of balls. In box $A$, red balls are $\frac{1}{4}$ of the balls in the box. In box $B$, half of the balls are yellow, and the other half of the balls are white. In box $C$, red balls are $\frac{2}{3}$ of the balls in the box. Emilia pours the balls in boxes $B$ and $C$ into box $A$, and picks a ball from it. If the probability of each ball being picked is the same, find the probability that a red ball is picked.
   A) $\frac{3}{4}$
   B) $\frac{1}{6}$
   C) $\frac{5}{18}$
   D) $\frac{7}{12}$
   E) $\frac{11}{36}$
10. On the Cartesian coordinate plane, the side $\overline{AB}$ of the square $ABCD$ is on $x$-axis, the coordinates of points $C$ and $D$ are $(2, 1)$ and $(1, 1)$ respectively, the graph of the inverse proportional function $y = \frac{k}{x}$ intersects $\overline{BC}$ and $\overline{CD}$ at points $E$ and $F$ respectively. If $\overline{BE} : \overline{CE} = 4 : 1$, find $\overline{DF} : \overline{FC}$.
    
    A) 1:1
    B) 2:1
    C) 4:3
    D) 3:2
    E) 5:2
11. $△ABC$ is divided into 6 small triangles by three straight lines that pass through its three vertices. The areas of four of the small triangles are marked. Find the area of $△ABC$.
    
    A) 324
    B) 330
    C) 348
    D) 325
    E) 315
12. If $\sqrt{195 - n} - \sqrt{150 - n} = 3$, find $\sqrt{195 - n} + \sqrt{150 - n}$.
    A) 17
    B) 15
    C) 11
    D) 13
    E) 9
13. A semicircle intersects a rectangle $ABCD$ at three points, and $\overline{MC} = 12$. Find the area of the rectangle $ABCD$.
    
    A) 64
    B) 72
    C) 60
    D) 84
    E) 70
14. In a right triangle $\triangle ABC$, $\angle ACB = 90^\circ$, $\overline{AD}$ is an angle bisector of $\angle BAC$. Use $D$ as the center of the circle and $\overline{DC}$ as the radius to draw a circle that intersects $\overline{BC}$ at $E$. If $\overline{DC} = 6$, $\overline{BE} = 2$, find $\overline{AC}$.
    
    A) $6\sqrt{7}$
    B) 16
    C) 15
    D) $8\sqrt{3}$
    E) $12\sqrt{3}$
15. Fill each square of a $3×3$ grid with a number so that the sum of the three numbers in each row, column, and diagonal equals $S$. Find $S$.
    
    A) 4050
    B) 50
    C) 6000
    D) 6075
    E) 0
16. $a$, $b$, and $c$ are three different positive integers that greater than or equal to $20$ and less than or equal to $30$. If they satisfy $a^2 + b^2 + c^2 = 2025$, find $a + b + c$.
17. If $x$ is an integer that satisfies $$\frac{\sqrt{x + 20} + \sqrt{x - 25}}{\sqrt{x + 20} - \sqrt{x - 25}} = 5$$ find the value of $x$.
18. Tom and Judy play a win-or-lose game. Put five cards marked with King, Prime minister, General, and Civilian in an opaque bag. There is one card for the role of King, General, Civilian. There are two cards for the role of Prime minister. During the game, the two of them each pick a card to compete. Rules: King beats Prime minister and General. Prime minister beats General and Civilian. General beats Civilian. Civilian beats King. When the roles are the same, it results in a tie. Find the probability that Tom wins. (Without putting back the picked cards).
19. Let $f(x)=x^{10}+2x^9-2x^8-2x^7+x^6+4x^2+8x+1$. Find $f(\sqrt{2}-1)$.
20. Four natural numbers $a,b,c$, and $d$ form an arithmetic sequence, three of which are perfect squares. Find the minimum value of the number that is not a perfect square.
21. Among the three interior angles of a triangle, one is a prime number, and the other two happen to besquares of prime numbers. How many such triangles are there?
22. There is a rectangle $PQRS$ inside the square $ABCD$ that intersects $ABCD$ at points $P,Q,$ and $S$. If the area of $△APS$ is 27, the area of $△QRC$ is 30, find the area of $ABCD$.
    
23. What is the remainder when $(1 \times 1!) + (2 \times 2!) + (3 \times 3!) + \dots + (2025 \times 2025!)$ is divided by $2000$? (Here, $n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1$)
24. Refer to the figure. The sides of the square $ABCD$ are $42$, point $E$ is on $\overline{CD}$, $\overline{DE} = 2$, $\overline{BE}$ is connected. If the large circle is tangent to $\overline{AD}$, $\overline{AB}$, and $\overline{BE}$, $M$ is the point of tangency, the small circle and $\triangle BCE$ are tangent at point $N$, find $\overline{MN}$.
    
25. Fill integers 1～16 in a $4×4$ grid so that the sum of the ten sets of numbers in the four rows, fourcolumns, and two diagonals happens to be ten successive integers. If 9 numbers are already filled in, find $A＋2B＋3C＋4D$.
    

1. If $2025^2－2050×2025＋1025^2＝k＋999^2－1$, find $k$.
   A) 1000
   B) 2000
   C) 2025
   D) 999
   E) 1990
2. In a 3×3 grid, the product of the three real numbers in each row, column, and diagonal is the same. Find the product of $A,B,$ and $C$.
   
   A) $6\sqrt{6}$
   B) $12\sqrt{2}$
   C) $-12\sqrt{2}$
   D) $12$
   E) $-6\sqrt{3}$
3. Given that an integer $x$ satisfies $\frac{1}{\sqrt{3}=\sqrt{2}}<x<\frac{2}{\sqrt{6}-\sqrt{5}}$. How many such $x$'s are there?
   A) 8
   B) 7
   C) 6
   D) 5
   E) 4
4. If $x-\frac{2025}{x}=5$, find $\frac{(x-2)^3-(x-1)^2+1}{x-2}$.
   A) 2020
   B) 2029
   C) 2025
   D) 2021
   E) 2024
5. Several identical $△ABC$’s are placed together closely to obtain a closed shape, as shown. The figure on the left obtains a regular hexagon. If the figure on the right obtains a regular $n$-sided polygon, find $n$.
   
   A) 10
   B) 7
   C) 9
   D) 12
   E) 8
6. If a polynomial $(x－1)(x＋2)(x－4)(x－7)＋k$ is a perfect square expression, find $k$.
   A) 32
   B) 48
   C) 56
   D) 144
   E) 81
7. Fold a rectangular paper $ABCD$ along $\overline{EF}$ so that it meets $\overline{BC}$ at points $G$ and $F$, and points $C$ and $D$ correspond to points $C_1$ and $D_1$. Then, fold the paper along $\overline{GF}$ so that points $C_1$ and $D_1$ correspond to points $C_2$ and $D_2$. If $\angle FEG = 38^\circ$, find $\angle EFC_2$.
   
   A) $76^o$
   B) $62^o$
   C) $78^o$
   D) $60^o$
   E) $66^o$
8. $△ABC$ and $△DEF$ are equilateral triangles with sides of $2\sqrt{2}$, and $\overline{AD}＝\overline{DB}$. Find the area of $△DGC$.
   
   A) $\sqrt{3}$
   B) $\frac{3}{4}$
   C) $\frac{3\sqrt{3}}{4}$
   D) $1$
   E) $\frac{4\sqrt{6}}{5}$
9. $\Delta ABC$ is divided into three regions of equal area $S_1, S_2,$ and $S_3$ by $\overline{DE}$ and $\overline{FG}$. If $\overline{DE} \parallel \overline{FG} \parallel \overline{BC}$ and $\overline{BC} = \sqrt{6}$, find $\overline{FG} - \overline{DE}$.
   
   A) $\sqrt{2} - 1$
   B) $2 - \sqrt{2}$
   C) $\sqrt{6} - \sqrt{2}$
   D) $\sqrt{3} - \sqrt{2}$
   E) $\frac{\sqrt{6}}{6}$
10. $a$ is a positive integer, and the solution for $x$ in the equation $\frac{x+a}{x-2}+\frac{3a}{2-x}$ is a positive number. How many $a$’s satisfy the condition?
    A) 8
    B) 7
    C) 6
    D) 5
    E) 4
11. If the two solutions of $x^2－30x＋k＝0$ are prime numbers, how many such $k$’s are there?
    A) 1
    B) 2
    C) 3
    D) 4
    E) 5
12. If real numbers $a,b$ and $c$ satisfy $\sqrt{a+b+c}+\sqrt{(a^2+2025)(b-6)}+|10-2b|=2$, find $ab+bc$.
    A) 40
    B) -25
    C) 25
    D) -36
    E) 36
13. Given a sequence $a_1,a_2,a_3,...,a_n$. If $a_1=0,a_2=2a_1+1,a_3=2a_2+1,...,a_n=2a_{n-1}+1$, find the units digit of $a_{25}-a_{20}$.
    A) 2
    B) 4
    C) 0
    D) 6
    E) 8
14. Let $a,b$ and $c$ be prime numbers that satisfy $ab^bc+a=2025$. Find $a+b+c$.
    A) 46
    B) 48
    C) 56
    D) 60
    E) 50
15. The upper base and lower base of a trapezoid is 17cm and 51cm, respectively, and its two legs are 16cm and 30cm. Find the area of this trapezoid in $\text{cm}^2$.
    A) 442
    B) 408
    C) 512
    D) 480
    E) 476
16. If $xy = 2025$, find $$x\sqrt{\frac{y}{x}} + y\sqrt{\frac{x}{y}}$$
17. Let $f(x + \sqrt{x^2 + 1}) = x - \sqrt{x^2 + 1}$. What is the value of $$f\left(\frac{1}{20}\right) + f\left(\frac{1}{25}\right) - f\left(\frac{1}{2025}\right)?$$
18. If $a$ is a root of the quadratic equation $x^2 - x - 3 = 0$, find $$\frac{2025 + 2025a^3}{16a^5 - 16a^4 - 16a^3 + 16a^2}$$
19. $O$ is a circle with a radius of 2cm. The points on the circumference divide it into 12 equal parts. Find the area of the pentagon $ABCDE$ in $\text{cm}^2$.
    
20. Given that $a,b,c,d,$ and $e$ are five segments of integer length in cm, $a＜b＜c＜d＜e,a＝1$, and $e＝9$. If any three of these five segments cannot form a triangle, find the value of $d$ or the sum of the values of several possible $d$’s.
21. There are four triangles inside the square $ABCD$, in which $△AED,△EBF$, and $△DFC$ have equal areas. If $\overline{EB}＝12$, find $\overline{ED}$.
    
22. Below is an arithmetic sequence. If the number of trailing 0’s in the product of the first $n－1$ numbers is 3 less than the number of trailing 0’s in the product of the first $n$ numbers, find the minimum value of $n$.
    
23. Three different positive integers $a,b$, and $c$, where $a＞b＞c$, satisfy $a＋b＋c＝133$, and $a＋b,b＋c$,and $c＋a$ are perfect squares. Find the multi-digit number $\overline{abc}$.
24. Stack three squares with sides of 16cm together, with two of them are on the straight line $L$. If the distance between the vertex $P$ of the middle square and $L$ is (24＋5$\sqrt{3}$) cm, find the length of $x$ in cm.
    
25. Fill integers 1～16 in a $4×4$ grid so that the sum of the ten sets of numbers in the four rows, four columns, and two diagonals happens to be ten successive integers. If 9 numbers are already filled in, find $A＋2B＋3C＋4D$.
    

1. Let $p$ be an even prime number. How many prime numbers are there in the values of the integral expressions $p＋1,p－1,p^3＋20,p^2＋25$, and $2025－p$?
   A) 0
   B) 1
   C) 2
   D) 3
   E) 4
2. If $\left\{ \begin{array}{cl}
   20x+25y=415 \\
   25x-20y=-45
   \end{array} \right.$, find $x-y$.
   A) 9
   B) 13
   C) 3
   D) -3
   E) -4
3. There is a number at each of the four vertices of the square $ABCD$. The number in the circle is the sum of the numbers at the two vertices on each side of $ABCD$. Find “?.”
   
   A) $\frac{1}{2}$
   B) $-9$
   C) $1$
   D) $\frac{1}{4}$
   E) $-1\frac{3}{4}$
4. Let $a,b,c,d$, and $e$ be rational numbers, and $ab^2c^3d^4e^5＜0$. Which product in the options below must be a negative number?
   A) $abcde$
   B) $ab^2cde$
   C) $ab^2cd^4e$
   D) $abcd^4e$
   E) $abc^3de^5$
5. Enam titik $A, B, C, D, E,$ dan $F$ terletak pada sebuah garis bilangan dari kiri ke kanan. Diketahui bahwa $\overline{CD} = \frac{1}{3}\overline{AD} = \frac{1}{4}\overline{CF}$, $B$ adalah titik tengah dari $\overline{AD}$, dan $E$ adalah titik tengah dari $\overline{CF}$. Jika $\overline{BE} = 2025$ dan $\overline{AD} = K$, carilah jumlah dari semua digit (angka) dari $K$.
   A) 9
   B) 18
   C) 10
   D) 12
   E) 7
6. $\overline{AB}$ adalah sebuah bilangan yang terdiri dari 2 digit (angka). Hasil kali dari kedua digitnya adalah dua kali lipat dari jumlah kedua digitnya. Jika jumlah dari $\overline{AB}$ dan 9 adalah dua kali lipat dari $\overline{BA}$, carilah selisih antara kedua digit dari $\overline{AB}$.
   A) 3
   B) 4
   C) 1
   D) 6
   E) 7
7. Arrange positive integers $5, 6, 6,x,$ and $y$ from smallest to largest. The median of this set of numbers is 5, and the only mode is 6. How many possible values are there for $x＋y$?
   A) 7
   B) 6
   C) 5
   D) 10
   E) 8
8. Place six identical squares on the Cartesian coordinate plane, as shown. The coordinates of points $M$ and $N$ are (3, 9) and (12, 9), respectively. If a straight line $L$ passes through point $A$ and happens to divide these six squares into two figures with the same area, find the coordinates where the straight line $L$ and the $x$-axis intersect.
   
   A) (18, 0)
   B) ($\frac{39}{2}$, 0)
   C) (20, 0)
   D) ($\frac{79}{4}$, 0)
   E) (19, 0)
9. There are black and white Go stones in a box. First, add some white stones to the box to make the ratio of the number of black stones to the number of white stones 2:5. Then, add some black stones to the box to make the ratio of the number of black stones to the number of white stones 3:5. If the ratio of the number of black stones to the number of white stones added to the box is 3:7, find the original ratio of the number of black stones to the number of white stones.
   A) 5:7
   B) 1:4
   C) 3:4
   D) 4:5
   E) 2:3
10. Holiday Hotel has three types of rooms: double rooms, triple rooms, and quadruple rooms. A soccer team of 25 people plans to stay in a total of 8 rooms of these three types at the same time. If each room must be fully occupied, how many accommodation options do they have?
    A) 1
    B) 2
    C) 3
    D) 4
    E) 5
11. If $|a|＝5,|b|＝2,|c|＝6,|a＋b|＝－(a＋b),|a＋c|＝a＋c$, find $a－b＋c$.
    A) 9
    B) 13
    C) 3
    D) 1 or－1
    E) - 1 or 3
12. The sum of the numbers on the opposite faces of the cube is 7. Stack several such cubes to form the shape shown on the right. If the sum of the numbers on the two touching faces is 8, find the number that “＃” represents.
    
    A) 1
    B) 2
    C) 5
    D) 6
    E) 3
13. Let $a,b,$ and $c$ be positive integers. If $2025^{2025}－2025^{2023}＝2024^a×2025^b×2026^c$, find $a＋b＋c$.
    A) 2026
    B) 2025
    C) 2024
    D) 2023
    E) None of these
14. If $abc\neq 0, \frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=k$, which quadrants does the straight line $y＝kx－k＋1$ certainly pass through?
    A) I, III, IV
    B) I, II, III
    C) I, II, IV
    D) I, IV
    E) II, III
15. Hexadecimal (base-16) is a counting system where 1 is carried over at every 16. It uses digits $0～9$ and letters $A～F$ as numerical symbols. The table below shows how these symbols correspond to the numbers of Decimal (Base-10). If $4＋A＝E$ and $4＋E＝12$, find $A×C$.
    
    A) $6F$
    B) $78$
    C) $CA$
    D) $B0$
    E) $85$
16. If $a=\frac{15^4}{3^{44}},b=\frac{5^4}{3^{40}}$, find $2025^{a-b}$.
17. Let $a,b,c$, and $d$ be four distinct integers, and min$(a,b)$ and max$(a,b)$ represent the minimum value and the maximum value of two numbers $a$ and $b$, respectively. Suppose
    min$(a,b)＝2$
    min$(b,c)＝0$
    max$(a,c)＝2$
    max$(c,d)＝5$
    Find the fourth minimum possible value of $a＋b＋c＋d$.
18. Three non-negative real numbers $a, b,$ and $c$ satisfy $\begin{cases} 3a + 2b + c = 4 \\ 2a + b - 3c = 1 \end{cases}$. If $S = 6a - 3b + 2$, find the sum of the maximum value $S_{\text{max}}$ and minimum value $S_{\text{min}}$ of $S$.
19. Given that $n！＝n×(n－1)×(n－2)×...×2×1$. For example, $4!＝4×3×2×1＝24$. How many consecutive 0’s are there in the last digits of $2026!－2025!$?
20. A moving point $P$(1, 0) moves in a regular pattern within Quadrant I. In the 1st second, it moves from (1, 0) to (2, 0). Directed by the arrows, the point moves through (1, 0) → (2, 0) → (2, 1) → (1, 1) → ..., and it moves 1 unit every second. In how many seconds will $P$ reach (20, 25)?
    
21. Let $f(x)$ be a polynomial with real coefficients such that forall the real numbers $x$’s,
    $2＋2f(x)＝f(x＋1)＋f(x－1)$. If $f(2)＝32,f(0)＝8$, find the value of
    $f(25)＋f(24)＋f(23)＋...＋f(1)$.
22. Refer to the figure. $ABJI$ is a rectangle. $D,C,G,$ and $H$ are on the sides of $ABJI$. Points $E$ and $F$ are on $\overline{CD}$. If the area of the square $ABCD$ is 2025, the total area of the shaded regions is 825, find the area of the square $EFGH$.
    
23. Let $n$ be a positive integer, and $n$’s five smallest positive factors are ordered as: $a＜b＜c＜d＜e$. If $n＝a^2＋b^2＋c^2＋d^2$, find $e$.
24. Fill in the six □’s below with “＋”, “－,” or “×” to make the equation established. If a total of $a$“＋’$s$, ”$b$“－’$s$”, and $c$“ × ’$s$” are used, find $a^2＋bc$.
    
25. In a $4×4$ grid, 4 squares are painted. Fill the empty white squares with numbers. Given that 1 and 2 are already filled in. Try to fill in 3, 5, 6, 7, 8, 9, 10, 11, 12, and 14 so that the sum of the numbers in each row and column is the same. Find the maximum value of $A＋2B＋3C＋4D$.
    

1. If $$\frac{2024}{2025}\times 2026=\frac{2024}{2025}+A=2025-B,$$ find $A\times B$.
   A) 1
   B) $2025\times 2026$
   C) $\frac{2024}{2025}$
   D) 2
   E) $\frac{2026}{2025}$
2. If $p$ is a prime number, $q$ is a composite number, which option below must be a composite number?
   A) $p+q$
   B) $p+q+2$
   C) $\frac{p\times q}{2}$
   D) $(p-2)\times q$
   E) $(p+2)\times q$
3. $Cecil$ tosses a regular 6-sided die with numbers 1, 2, 3, 4, 5, and 6. What is the probability that the number on the vertical lateral face of the die is 5?
   A) $\frac{1}{6}$
   B) $\frac{1}{3}$
   C) $\frac{1}{4}$
   D) $\frac{2}{3}$
   E) $\frac{1}{2}$
4. The ratio of the average number of the two numbers $A$ and $B$ to $C$ is $\frac{11}{5}$. Find the ratio of the average number of the three numbers $A,B,$ and $C$ to $C$.
   A) $\frac{27}{5}$
   B) $\frac{33}{5}$
   C) $\frac{6}{5}$
   D) $\frac{9}{5}$
   E) $\frac{12}{5}$
5. If $\frac{a}{b}$ can be simplified to $\frac{1}{25},\frac{a+20}{b+20}$ can be simplified to $\frac{1}{20}$, find $a+b$.
   A) 2028
   B) 1976
   C) 2132
   D) 1274
   E) 1404
6. Compute $0^3＋1^3＋2^3＋3^3＋4^3＋5^3＋6^3＋7^3＋8^3＋9^3＋10^3$. (Hint: $2^3＝2×2×2$)
   A) 2035
   B) 2025
   C) 2125
   D) 3025
   E) 4025
7. Daniel and Nancy worked on a report together. If the two of them had worked on the report in a 1:3 ratio as planned, they would have finished it simultaneously. When they started doing the report, Nancy’s efficiency decreased by 50％ compared to the plan because she didn’t feel well, and Daniel’s efficiency was as scheduled. After Daniel finished his part, he helped Nancy immediately, and they completed the report 120 minutes later. If Daniel had worked on the report alone from the beginning, how many minutes would he have needed to finish the report?
   A) 480
   B) 800
   C) 640
   D) 400
   E) 720
8. A cuboid with dimensions $6×6×3$ is cut from another cuboid with dimensions $12×9×6$, as shown on the left. Combine two such cuboids together, as shown on the right. What is its surface area?
   
   A) 640
   B) 540
   C) 792
   D) 720
   E) 848
9. Company W imports goods from Asia to sell in the US. The original profit is $x$％. Due to tariff increase, import costs have risen by 25％. If the selling price remains unchanged, the profit will dropto ($x-60$)％. Find $x$.
   A) 85
   B) 150
   C) 105
   D) 200
   E) 185
10. Three circular boards each with a radius of 9cm $A,B,$ and $C$ are placed in a straight line. Both $A$ and $C$ rotate around $B$ without sliding. It takes $A$ 36 seconds to rotate counterclockwise once. $C$ rotates clockwise at 2 times the speed of $A$. If they rotate simultaneously, how many seconds later will $A$ collide with $C$?
    
    A) 6
    B) 3
    C) 4
    D) 2
    E) 8
11. The area of a regular hexagon $ABCDEF$ is $18\text{ cm}^2$. Each black point is the midpoint of each side. Find the area of the shaded region in $\text{cm}^2$.
    
    A) 5
    B) 4
    C) $4\frac{1}{2}$
    D) 6
    E) $5\frac{1}{2}$
12. Mary accidentally erases the symbol of a repeating decimal, and it becomes 0.987654321. If the 2025th digit after the decimal point of the original repeating decimal is 5, and the repeating period consists of at least 2 digits, which digit is the initial digit of the repeating period of the original repeating decimal?
    A) 9
    B) 8
    C) 7
    D) 6
    E) 5
13. On the table are three cylindrical cups $A,B$, and $C$ with base areas of $100\text{ cm}^2$, $120\text{ cm}^2$, and $140\text{ cm}^2$ respectively. Each cup is 25 cm deep and contains 10 cm of water. Dave pours some of the water from cups $A$ and $B$ into cup $C$ so that the water levels in the three cups achieve a ratio of 4:3:6. Disregarding the thickness of the cups, find the water height in cup $A$ in cm.
    A) 9
    B) 9.1
    C) 7.5
    D) 8
    E) 8.6
14. The result of the expression $\frac{7}{6}+\frac{13}{12}+\frac{21}{20}+\cdots +\frac{2025\times 2026+1}{2025\times 2026}$ is between an integer $n$ and $n+1$. Find $n$.
    A) 2026
    B) 2025
    C) 2024
    D) 4050
    E) 4048
15. Four people Mary (SAP Entertainment), Lilly (RR Entertainment), Cindy (RR Entertainment), and Daisy (JYK Entertainment) form a girl group, and they belong to different agencies. A program today requires two of them to participate in filming. SAP says, “If Mary goes, Lilly must go.” RR says, “If Cindy can’t go, Lilly can’t go, either.” JYK says, “If Cindy goes, Daisy will not go.” In the end, which two people participate in filming?
    A) Mary, Lilly
    B) Mary, Daisy
    C) Cindy, Daisy
    D) Lilly, Cindy
    E) Mary, Cindy
16. If $\frac{a}{3}=\frac{b}{4}=\frac{c}{5}=202320242025$, find $\frac{3a+2b-2c}{2c-b+3a}$.
17. Divide a cube into 3 cuboids $A,B,$ and $C$. The surface areas of $A,B,$ and $C$ are in a ratio of 7:6:5. Find the ratio of the volumes of $A,B,$ and $C$.
    
18. How many values of $n$’s less than 2025 can make the units digit of the result of this expression $$\underbrace{2023 \times 2023 \times \dots \times 2023}_{n\text{'s}} + \underbrace{2025 \times 2025 \times \dots \times 2025}_{n\text{'s}} + \underbrace{2027 \times 2027 \times \dots \times 2027}_{n\text{'s}} + \underbrace{2029 \times 2029 \times \dots \times 2029}_{n\text{'s}}$$ equal to 4?
19. Use ten digits 0～9 to form a 1-digit number, a 2-digit number, a 3-digit number, and a 4-digit number which are all non-zero perfect squares. For example, 7056, 324, 81, and 9. Find the maximum value of the sum of these four perfect squares.
20. Two circles with radii of 1cm and 2cm are in a rectangle $ABCD$. If the two circles remain attached to the sides of the rectangle and move around its interior once, find the area in the rectangle that only a circle can pass through in $\text{cm}^2$. ($\pi＝3.14$)
    
21. In a quadrilateral $ABCD,\overline{AF}=\frac{1}{4}\overline{AD},\overline{EB}=\frac{1}{4}\overline{AB}$, the area of the quadrilateral $AEOF$ is 64. Find the area of the parallelogram $OBCD$.
    
22. Compute $2025\times (1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{2024})-$.
23. The sum of 8 different positive integers is 2025. Find the maximum value of the greatest common factor of these 8 positive integers.
24. A puzzle is shown. Enter from Start and move toward Goal. How many ways are there?
    1. When entering into a square with an arrow, you have to follow its direction.
    2. From a colored square (including Start), you can move to any adjacent square above, below, left, or right.
    3. You can’t enter the square that you visited before.
    
25. In order to celebrate the 13th WMI, the teacher designs a number puzzle. Fill in the □’s below with 1～23 to make the sum of the numbers in each row and column 60. A few numbers have been filled in. What number is ★?
    

1. Compute $\frac{2025-202.5-20.25}{2.025}$
   A). 990
   B) 199
   C) 1000
   D) 890
   E) 89
2. Lilly used a calculator to compute 50.625－0.4, but she got a wrong answer 20.25. Which incorrect input did she enter into the calculator instead of the correct one? (The others remained the same)
   A) 30 instead of 50
   B) 40 instead of 0.4
   C) 20 instead of 50
   D) $\div$ instead of $-$
   E) $\times$ instead of $-$
3. The square $ABCD$ is divided into two different types of small squares. What fraction of the entire figure is covered by the shaded region?
   
   A) $\frac{5}{16}$
   B) $\frac{1}{3}$
   C) $\frac{11}{36}$
   D) $\frac{7}{18}$
   E) $\frac{13}{36}$
4. Several cards are on the table. Each card contains a number, and it is a multiple of either 3 or 4. $\frac{2}{3}$ of the numbers are multiples of 3, and $\frac{3}{4}$ of the numbers are multiples of 4. If 30 cards contain multiples of 12, how many cards contain multiples of 3?
   A) 40
   B) 18
   C) 48
   D) 24
   E) 54
5. A regular pentagon and a regular hexagon lie on a straight line, as shown. Find $x$.
   
   A) 87
   B) 93
   C) 97
   D) 85
   E) 95
6. Divide the trapezoid into three triangles of equal area $A,B$, and $C$. What is the length of the upper base of the trapezoid $x$ in cm?
   
   A) 16
   B) 19
   C) 17
   D) 20
   E) 18
7. A ball with a radius of 1 cm is placed in a sealed cuboid box which is 4 cm long, 5 cm wide, and 6 cm high. The ball can roll inside the box at will, and the box can be flipped. Find the maximum sum of the areas of the six faces inside the box that the ball is able to touch in $\text{cm}^2$.
   A) 54
   B) 60
   C) 52
   D) 94
   E) 148
8. Three people Amy, Bill, and Celia take an exam. The average score of two people Amy and Bill is 2.5 points higher than the average score of three people. The average score of two people Bill and Celia is 1 point lower than the average score of three people. Given that Amy gets 93 points. How many points does Celia get?
   A) 94
   B) 86
   C) 90
   D) 88
   E) 82
9. Leon wants to paint a 3×3 grid to make the squares in each row, column, and diagonal have three different colors. How many colors does Leon need to use the least?
   
   A) 3
   B) 6
   C) 5
   D) 9
   E) 4
10. Three consecutive positive integers $n$, $n＋1$, and $n＋2$ have the same number of factors. Which number might $n$ be?
    A) 25
    B) 57
    C) 98
    D) 85
    E) 104
11. $A,B$, and $C$ are the three faces of a cuboid. The areas of $A,B,$ and $C$ are in the ratio of 8:6:3. If the perimeter of $C$ is 42 cm, find the perimeter of $A$ in cm.
    
    A) 66
    B) 72
    C) 84
    D) 77
    E) 80
12. 5 students each write a 6-digit number (its units digit is not a 0). Move the units digit of each 6-digit number to the left most position to form a new 6-digit number, and add the original 6-digit number to it. If the results are written in the options below, which answer is correct?
    A) 766666
    B) 172535
    C) 826354
    D) 1019456
    E) 620708
13. Compute $\frac{5}{6}\times\frac{1}{13}+\frac{5}{9}\times\frac{2}{13}+\frac{5}{18}\times\frac{6}{13}$.
    A) $\frac{5}{18}$
    B) $\frac{5}{13}$
    C) $1$
    D) $5$
    E) $\frac{13}{18}$
14. Daniel and Nancy set off at the two ends of a bridge and walked toward each other. Nancy had walked 2025m before Daniel set off. If the speeds of the two of them were in the ratio of 5:4; by the time they met, the distances that they had walked were in the ratio of 4:5, find the full length of the bridge in meters.
    
    A) 8100
    B) 10125
    C) 9650
    D) 10375
    E) 12150
15. A cuboid has three colored parts. Each part is formed with 6 closely connected ’s of the same color. Which option is Part 3?
    
    
16. $(0.123456＋4)×2＋(3－0.123456)×0＋(2－0.123456)×2＋(1－0.123456)×5＝?$ (Round to twodecimal places)
17. Today is July 26, 2025. The 7-digit number $\overline{ABCDEFG}$ happens to satisfy the following conditions: The first 5 digits $\overline{ABCDE}$ is a multiple of 2025. The last 5 digits $\overline{CDEFG}$ is a multiple of 726. Find the minimum value of the 4-digit number $\overline{ABFG}$.
18. Given a mixed fraction. If its whole number part is reduced to $\frac{1}{2}$ of the original value, the fraction becomes $3\frac{5}{6}$. If its whole number part is reduced to $\frac{2}{7}$ of the original value, the fraction becomes $2\frac{1}{3}$. Find this mixed fraction.
19. Tony uses identical small cubes to build a castle and notices that the images he sees from the four faces are identical. How many small cubes does he use the most?
    
20. Cars in a parking lot are parked either perpendicular or parallel. Perpendicular cars can only move forward or backward. Parallel cars can only move left or right. Each time, only one car can move 1 square, and that takes 1 second. A car can only enter a square when it is empty. If the square is occupied by another car, the car cannot enter it. How many seconds does it take the least for the blue car to leave the parking lot and reach the EXIT square completely?
    
21. An integer has 16 positive factors, and its units digit is 7. What is the possible minimum value of this integer?
22. In the rectangle $PQRS$ are 3 identical squares. They form two overlapping parts $A$ and $B$, and the uncovered parts are $C,D$, and $E$. All of them are rectangles. If the ratio of the areas $A：B：C：D：E＝3：4：5：6：7$, find the ratio of the length to the width of $PQRS$.
    
23. 12 black or white magnetic beads can form a bracelet. When 3, 4, 5, or 6 consecutive beads are removed, rotated, and placed back to the bracelet, it is called 1 “turn.” If Vivian wants to make the 6 black beads of the bracelet below link together, and make the 6 white beads link together, how many “turns” are needed the least?
    
24. Fill in the nine □’s below with “＋” or “－” to make the equation established. How many ways arethere?
    
25. A puzzle is shown. Enter from Start and move toward Goal. How many ways are there?
    1. When entering into a square with an arrow, you have tofollow its direction.
    2. From a colored square (including Start), you can move to any adjacent square above, below, left, or right.
    3. You can’t enter the square that you visited before.
    

1. Compute 20×45＋25×45.
   A) 2225
   B) 2025
   C) 3025
   D) 2205
   E) 2525
2. Which value in the options is the closest to half of the sum of 45674567 and 67896789?
   A) 55556666
   B) 56565656
   C) 56775677
   D) 56756756
   E) 55000000
3. When Omar calculated an expression, he mistook a number multiplied by 4, then plus 20, for a number divided by 4, then minus 20. If the result he got was 2025, find the result of the correct expression.
   A) 32660
   B) 32740
   C) 32720
   D) 32640
   E) 32700
4. July 26, 2025, is a Saturday. In which month of 2025 does the 26th day also fall on a Saturday?
   A) February
   B) May
   C) October
   D) April
   E) November
5. If the sum of 10 consecutive integers is 255, find the sum of the next 10 consecutive integers.
   A) 510
   B) 1255
   C) 555
   D) 265
   E) 355
6. Given two sets of numbers. The first set has 9 numbers, and their sum is 63. The second set has 3 numbers. If the average number of the numbers in the two sets is 8, find the average number of the numbers in the second set.
   A) 13
   B) 8
   C) 11
   D) 6
   E) 9
7. Leo puts ribbons in red, yellow, and green together. Ribbons of the same color have the same length. What fraction of the length of the green ribbon is the length of each yellow ribbon?
   
   A) $\frac{2}{3}$
   B) $\frac{3}{4}$
   C) $\frac{2}{5}$
   D) $\frac{3}{8}$
   E) $\frac{2}{7}$
8. 3-digit number addition: 4□3＋126＝5△9. If 5△9 is a multiple of 9, find □＋△.
   A) 4
   B) 8
   C) 11
   D) 6
   E) 9
9. The composite figure consists of 2 equilateral triangles, 1 square, 1 regular pentagon, and 1 regular hexagon. If the perimeter of the square is 48cm, find the perimeter of this composite figure in cm?
   
   A) 72
   B) 63
   C) 84
   D) 64
   E) 88
10. A 9×9 small square is placed in a 10×10 square and overlaps with it at the top right. If the oblique lines are their diagonals, find the area of the shaded region.
    
    A) 8.5
    B) 9
    C) 7
    D) 6.5
    E) 8
11. David says to Henry, “My age is 7 times your age now. In a few years, my age will be 6 times your age. In a few more years, my age will be 5 times your age. In a few more years, my age will be 4 times, then 3 times your age.” How old is David this year?
    A) 28
    B) 42
    C) 35
    D) 49
    E) 70
12. Andy wants to paint the three letters $WMI$, but the two adjacent letters must be in different colors. The paints are in 5 colors: yellow, green, red, purple, and blue. How many different color combinations are there for $WMI$?
    
    A) 96
    B) 60
    C) 80
    D) 64
    E) 100
13. 4 of the 12 identical squares are painted while the other 8 squares are numbered 1～8. Find the sum of the numbers on the unpainted squares that can be folded into a lidless cube with the four painted squares.
    
    A) 31
    B) 27
    C) 24
    D) 17
    E) 30
14. Fill in the □’s below with appropriate numbers so that each number equals the product of the two numbers that are linked together beneath it. Find the sum of the five □’s.
    
    A) 61
    B) 56
    C) 47
    D) 51
    E) 55
15. $a,b,c,d$, and $e$ are five positive integers that are larger than 1. Their product $a×b×c×d×e＝2025$. Find the difference between the maximum value and the minimum value of their sums.
    A) 4
    B) 8
    C) 12
    D) 15
    E) 14
16. $\frac{2024}{2+0+2+4}-\frac{2025}{2+0+2+5}=?$
17. Four people Nath, Lal, Kumar, and Datt are of different ages, and the sum of their ages does not exceed 70. Nath notices that the ages of three of them are perfect squares. Fifteen years later, the ages of three of them will still be perfect squares. Find the age difference between the oldest and the youngest of the four people.
18. The 8-digit number is a multiple of 4, 5, 6, 7, 8, 9, 10, and 11. Find this 8-digit number.
19. Form a 4-digit number with two consecutive 2-digit numbers. For example, 2425 or 3534. Find the average value of all these 4-digit numbers.
20. Each side of a regular hexagon $ABCDEF$ is 1 cm. Extend the two sides of this regular hexagon $\overline{AF}$ and $\overline{CD}$ equally to form a new hexagon with an area 4 times the original. What is the length of the side $\overline{AF}$ in this new hexagon in cm?
    
21. Use matchsticks to form nine digits 1～9. For certain numbers, the sum of the digits equals the number of matchsticks that are used. For example, the sum of the digits of 18 is 1＋8＝9, and the number of matchsticks that are used is 2＋7＝9. What is the largest 5-digit number among these certain numbers?
    
22. Cut a large cube into 27 identical small cubes, and paint the squares on the faces of the large cube as shown in the figures. How many small cubes only have one face painted?
    
23. Mary accidentally erases the symbol of a repeating decimal, and it becomes 0.987654321. If the 2025th digit after the decimal point of the original repeating decimal is 1, and the repeating period consists of at least 2 digits, how many possible values can this repeating decimal have? (An example of a repeating decimal $0.1\overline{23}＝0.1232323...$)
24. A bag contains one red card for each number:1, 2, 3, ..., 135. Take several cards from the bag at will, divide the sum of the numbers on the cards by 17, write the remainder on a yellow card, and place it in the bag. After repeating the process for several times, only two red cards marked 15 and 99 and a yellow card are left in the bag. What number is written on the yellow card?
25. In the equation below, different letters represent different digits, the same letter represents the same digit, and each letter is not a 0. If the equation is established, find the maximum value of the 4-digit number $\overline{MATH}$.