BEBRAS 2023 Grade 5-6 Round 2 Ext

Beaver has three unusual dotted leaves as shown.



First, Beaver places leaf 2 on leaf 1 as shown and sprays both leaves with white paint.



Then Beaver places leaf 3 on top of those two leaves and sprays with gray paint.



Which leaf below could be leaf 1 after painting?

To visit Grandma, Beaver must cross the forest shown.



Which directions below lead Beaver to Grandma’s house?

Bob designs a robot beaver that follows these commands.



English translation: Tx = go to location x; G = pick up wood; D = put down wood. For example, T4T2GT4D sends the robot to the bridge, then location 2, to pick up wood, back to the bridge, and to put down the wood. The robot works in this forest:



Which program collects the wood at locations 1, 2 and 3 and puts it at location 5? To cross the river, the robot must use the bridge.

Nam, Việt, Mai and Mi each wear a hat, shirt and shoes in blue, red, purple or yellow. No two children have the same hat color, shirt color or shoe color.



The table columns are Name / Hat / Shirt / Shoes. Nam: blue / yellow / red. Việt: yellow / blue / purple. Mai: red / purple / yellow. Mi: unknown in all three columns. What colors are Mi’s hat, shirt and shoes?

Beaver brings a jar containing five candies to class. Each time, Beaver eats the candy at the top. The candies are eaten in this order:



Which jar did Beaver bring?

Beaver has logs of the lengths shown:



Beaver must join some logs to make a log 39 units long. What is the smallest number of logs needed? None of the given logs may be cut.

Beaver has a machine with trays A, B and C.



It operates as follows:
- Trays A and B contain some balls.
- If B is empty, the machine moves all balls from A to C.
- If B is not empty, the machine creates and adds to A as many new balls as are currently in B, then destroys one ball in B and repeats the process.
Initially A contains 2 balls and B contains 5. How many balls will C contain at the end?

The picture is a map of a mountain ski area.



Beaver can go uphill only by cable car and downhill only by skiing. Beaver wants to take every cable-car ride exactly once and ski down every distinct downhill path exactly once. At which point must Beaver start?

Beaver draws this diagram to remember the code for each key. For example, the first key has code 791 and the bottom key has code 243.



Alex has three keys with codes 73∗, 24∗∗, and 7∗1∗, where each ∗ is one hidden digit. Which group below could show Alex’s keys in that order?

Every year, the beavers choose a “lucky queen” as follows:
- Each participant receives a letter card. They stand clockwise in consecutive letter order around a circle.
- Starting with A, they count 1, 2, 3 clockwise. Anyone who says 3 is eliminated; the next remaining beaver restarts at 1. Continue until one remains.
- The last remaining beaver is the lucky queen.
This illustrates the case of four participants:



What if there are ten participants holding A, B, C, D, E, F, G, H, I, J?



Who becomes the lucky queen?

Beaver must distribute five items among Fuzzy, Wuzzy and Izzy. Each person values each item differently, as shown.



Columns mean Item 1, Item 2, Item 3, Item 4, Item 5. Fuzzy’s values are 10, 13, 8, 5, 3; Wuzzy’s are 2, 11, 4, 5, 6; Izzy’s are 6, 10, 6, 4, 8. For example, Fuzzy values item 1 at 10 units, while Izzy values it at only 6 units. If Wuzzy has items 4 and 5, Wuzzy has 11 units of value. Beaver A envies Beaver B if A considers B’s items to have greater total value than A’s own items. All five items must be distributed so that no one envies anyone else. What total value could Fuzzy receive? Select all valid options.

Beaver’s country has 36 cities arranged as the 6×6 square grid below. Two cities are adjacent if their squares share a vertex.



Each city is either densely populated (gray) or sparsely populated (white). Every year their populations change by these rules:
- A dense city remains dense if exactly 2 or 3 adjacent cities are dense; otherwise it becomes sparse.
- A sparse city becomes dense if exactly 3 adjacent cities are dense; otherwise it remains sparse.
How many dense cities are there after two years?

At a Bebras camp there are 20 boys and 22 girls. Some boys shake hands with some girls. Every girl shakes hands with exactly 10 boys, and every boy shakes hands with exactly A girls. What is A?

Two beavers play five rounds of darts with these rules:
- Each player throws three darts per round.
- The player with the dart nearest the center wins that round. The winner scores 1, 2 or 3 points according to how many of their darts are closer to the center than the opponent’s nearest dart.
The diagram shows all five rounds and the first-round scores.



“Lượt” means “Round”; “điểm” means “points”; A scores 1 and B scores 0 in round 1. What is the combined total score of the two players after five rounds?

La Pipopipette is a two-player game. Players take turns drawing a horizontal or vertical segment between consecutive dots, aiming to complete as many squares as possible. Whenever a player completes a 1×1 square, they put their own mark in it and immediately draw another segment. The result depends on the number of squares marked by each player. The picture illustrates a draw on a 3×3 dot grid.



“Lượt” means “Turn”; the dark line denotes A’s move and the gray line B’s move. Alice and Bob are now playing on a 2×3 dot grid. The next picture shows Alice’s first move. On each turn, either player may choose a segment randomly.



Which statement about the outcome is true?