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- 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 - 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 - 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) $-1\frac{3}{4}$ - 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) $abc^3de^5$ - 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 - $\overline{AB}$ 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 - 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 - 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) - 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 - 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 - 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 - 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 - 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 - 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 - 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) $85$ - If $a=\frac{15^4}{3^{44}},b=\frac{5^4}{3^{40}}$, find $2025^{a-b}$.
- 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 $a＋b＋c＋d$. - 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 $S$.
- 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 $2026!－2025!$?
- 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)?
- 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
$f(25)＋f(24)＋f(23)＋...＋f(1)$. - 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 $EFGH$.
- 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 $e$.
- 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 $a^2＋bc$.
- 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 $A＋2B＋3C＋4D$.
This topic was modified 23 jam ago by Admin dot
Posted : 26/05/2026 2:10 am
