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- If $2025^2－2050×2025＋1025^2＝k＋999^2－1$, find $k$.
A) 1000
B) 2000
C) 2025
D) 999
E) 1990 - 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) $-6\sqrt{3}$ - 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 - 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 - 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 - 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 - 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) $66^o$ - $△ABC$ 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) $\frac{4\sqrt{6}}{5}$ - $\Delta ABC$ 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) $\frac{\sqrt{6}}{6}$ - $a$ 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 - 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 - 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 - 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 - 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 - 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 - If $xy = 2025$, find $$x\sqrt{\frac{y}{x}} + y\sqrt{\frac{x}{y}}$$
- Let $f(x + \sqrt{x^2 + 1}) = x - \sqrt{x^2 + 1}$. What is the value of $$f\left(\frac{1}{20}\right) + f\left(\frac{1}{25}\right) - f\left(\frac{1}{2025}\right)?$$
- If $a$ is a root of the quadratic equation $x^2 - x - 3 = 0$, find $$\frac{2025 + 2025a^3}{16a^5 - 16a^4 - 16a^3 + 16a^2}$$
- $O$ 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 $\text{cm}^2$.
- 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.
- There are four triangles inside the square $ABCD$, in which $△AED,△EBF$, and $△DFC$ have equal areas. If $\overline{EB}＝12$, find $\overline{ED}$.
- 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 $n$.
- 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 $\overline{abc}$.
- 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.
- 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 $A＋2B＋3C＋4D$.
Posted : 26/05/2026 3:00 am
