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- $a$ 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 - 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) $\frac{\sqrt{3}}{4}$ - 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 - 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 - 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) $\frac{21}{2}$ - 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) $\frac{11}{12}$ - 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) $\frac{39}{2}$ - 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) $150^o$ - 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) $\frac{11}{36}$ - 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 - $△ABC$ 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 - 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 - 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 - 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) $12\sqrt{3}$ - 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 - $a$, $b$, 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 $a + b + c$.
- 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 $x$.
- 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).
- Let $f(x)=x^{10}+2x^9-2x^8-2x^7+x^6+4x^2+8x+1$. Find $f(\sqrt{2}-1)$.
- 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.
- 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?
- 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 $ABCD$.
- What is the remainder when $(1 \times 1!) + (2 \times 2!) + (3 \times 3!) + \dots + (2025 \times 2025!)$ is divided by $2000$? (Here, $n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1$)
- 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 $\overline{MN}$.
- 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 $A＋2B＋3C＋4D$.
Posted : 26/05/2026 3:34 am
