Olympiad problems

2025 AMC 8, 22

A classroom has a row of $35$ coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least $1$ coat and at least $1$ empty hook. How many different numbers of coats can satisfy Paulina's pattern? $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$\\ (need visuals)

2025 AMC 8, 21

Tags: AMC 8 , 2025 AMC 8

The Konigsberg School has assigned grades $1$ through $7$ to pods $A$ through $G$, one grade per pod. The school noticed that each pair of connected pods has been assigned grades differing by $2$ or more grade levels. (For example, grades $1$ and $2$ will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E,$ and $F$? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 16$\\

2025 AMC 8, 9

Tags: AMC 8 , 2025 AMC 8

Nigli looks at the $6$ pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting $6$ numbers? [asy] import graph; size(8cm); // Draw the outer circle draw(circle((0,0), 1)); // Add the hour notches for (int i = 1; i <= 12; ++i) { real angle = (90 - i * 30) * pi / 180; pair outer = (cos(angle), sin(angle)); // Outer point of the notch pair inner = 0.9 * outer; // Inner point of the notch draw(inner -- outer); // Draw the notch // Add the hour numbers pair textPos = 1.15 * outer; // Position slightly outside the circle label(format("%d", i), textPos, align=(0,0)); } // Calculate the positions for 2 and 8 real angle2 = (90 - 2 * 30) * pi / 180; // 2 o'clock position real angle8 = (90 - 8 * 30) * pi / 180; // 8 o'clock position pair pos2 = (cos(angle2), sin(angle2)); // Position for 2 o'clock pair pos8 = (cos(angle8), sin(angle8)); // Position for 8 o'clock // Draw a dashed line from 2 to 8 draw(pos2 -- pos8, dashed); [/asy] $\textbf{(A) }5 \qquad\textbf{(B) } 6.5\qquad\textbf{(C) }8\qquad\textbf{(D) }9.5 \qquad\textbf{(E) }12$\\

2025 AMC 8, 19

Tags: AMC 8 , 2025 AMC 8

Two towns, $A$ and $B$, are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B$, the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B$, start moving toward each other at the same time. They drive exactly the speed limit in each portion of the road. How far from town $A$, in miles, will the two cars meet? $\textbf{(A) }7.75 \qquad\textbf{(B) }8 \qquad\textbf{(C) }8.25\qquad\textbf{(D) }8.5 \qquad\textbf{(E) }8.75$

2025 AMC 8, 1

Tags: AMC 8 , geometry , 2025 AMC 8 , MAA , America

The eight pointed star is a popular quilting pattern. What percent of the entire 4-by-4 grid is covered by the star? $(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$

2025 AMC 8, 3

Tags: AMC 8 , 2025 AMC 8

Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and $3$ of her friends play Buffalo Shuffle-o, each player is dealt $15$ cards. Suppose $2$ more friends join the next game. How many cards will be dealt to each player? $\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$ ngl easily silliable

2025 AMC 8, 13

Tags: AMC 8 , 2025 AMC 8

Each of the even numbers $2, 4, 6, \ldots, 50$ is divided by $7$. The remainders are recorded. Which histogram displays the number of times each remainder appears? [img]https://i.imgur.com/f1oQExa.png[/img]

2025 AMC 8, 14

Tags: AMC 8 , 2025 AMC 8

A number N is inserted into the list 2, 6, 7, 7, 28. The mean is now twice as great as the median. What is N? $\textbf{(A) } 7\qquad\textbf{(B) } 14\qquad\textbf{(C) } 20\qquad\textbf{(D) } 28\qquad\textbf{(E) } 34$

2025 AMC 8, 15

Tags: AMC 8 , 2025 AMC 8

Kei draws a $6\times 6$ grid. He colors $13$ of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let $m$ and $M$ the least and greatest possible number of gold-on-gold pairs, respectively. What is $m + M?$ $\textbf{(A) } 12 \qquad\textbf{(B) }14 \qquad\textbf{(C) }16\qquad\textbf{(D) }18 \qquad\textbf{(E) }20$\\

2025 AMC 8, 10

Tags: AMC 8 , 2025 AMC 8

In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^{\circ}$ clockwise about the midpoint of side $\overline{DC}$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles? [img]https://i.imgur.com/NyhZpL6.png[/img] $\textbf{(A) }21 \qquad\textbf{(B) }22.25 \qquad\textbf{(C) }23\qquad\textbf{(D) }23.75 \qquad\textbf{(E) }25$

2025 AMC 8, 11

Tags: AMC 8 , 2025 AMC 8

A [i]tetromino[/i] consists of four squares connected along their edges. There are five possible tetromino shapes, I, O, L, T, S, shown below, which can be rotated or flipped over. Three tetrominos are used to completely cover a $3\times 4$ rectangle. At least one of the titles is an S tile. What are the other two tiles? [img]https://i.imgur.com/9Nxq4y6.png[/img] $\textbf{(A) } \text{I and L} \qquad\textbf{(B) }\text{I and T} \qquad\textbf{(C) }\text{L and L}\qquad\textbf{(D) }\text{L and S} \qquad\textbf{(E) }\text{O and T}$\\

2025 AMC 8, 20

Tags: AMC 8 , Edmits , orz , Cheese , 2025 AMC 8

Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total? $\hspace*{5mm}\text{(A) } \frac{4}{7} \quad \text{(B) } \frac{3}{5} \quad \text{(C) } \frac{2}{3} \quad \text{(D) } \frac{3}{4} \quad \text{(E) } \frac{7}{8}$