BEBRAS 2021 Grade 5-6 Round 2 Ext

Alex has five chests labeled with letter codes, as shown.



He has five numbered keys for these chests. Equal digits correspond to equal letters, and different digits to different letters. Unfortunately, one key has lost its number. The four remaining numbers are shown below.



Which number is missing?

Alice has five gift boxes distinguished by the symbols printed on them. The balances show the comparisons below.



Which of the five boxes is heaviest?

The village library has just one stack of books. Whenever someone borrows a book, they record their name and take the top book. Whenever they return it, they record their name and place the book on top. The picture shows the register on the left and the initial stack on the right.



The register, in order, says: Alba borrows; Felix borrows; Alba returns; Marta borrows; Felix returns; Cato borrows. Initially, from top to bottom, the books are Charlotte’s Web, Curious George, Go, Dog, Go!, The Hobbit, and Fox in Socks. Which book did Cato borrow?

Beavers record their choices at forks along wooden roads. They write R for a right turn and L for a left turn, relative to their direction of travel. In the example, the route from S to the butterfly is S R L R L.



One day a beaver leaves its home and stops at the position shown below.



How many R letters and how many L letters occur in its route record?

Editorial clarification: follow the route from the home to the pictured beaver without retracing any road.

Beavers use pictures in place of digits to label objects. The table gives the correspondence. Its row labels mean “beaver symbol” and “value”.



In the first symbol row, square, triangle, star, circle, ring, diamond, heart and cloud mean 1,2,3,4,5,6,7,8. In the second, cross, flower, crescent, fish, mountains, waves, drop and spiral mean 9,0,1,2,3,4,5,6. For example, the following sequence represents 56432.



Alex labels his favorite book with this sequence.



Which sequence below represents the same value?

The beavers’ path of light consists of 36 special tiles arranged in a 6×6 grid. Each tile has a lamp that changes state whenever a beaver passes over that tile. A walk starts at the entrance and ends at the exit; each step moves to a tile sharing an edge. After one walk, Alex obtains the pattern below. White lamps are on. “Vào” marks the entrance and “Ra” the exit.



Alex wants to start a new design and first needs to turn all the lamps on. Which of the following walks will do this?

A combination lock opens only when its number rings show the correct code. Each ring can turn in either direction. The lock currently shows 2719.



Turning the last ring up one position changes 9 to 0; turning it down one changes 9 to 8. What is the minimum total number of one-position turns needed to reach the correct code 4383?

A beaver engineer makes triangular windows by this procedure.
1. Make an equilateral triangular frame.
2. Join the midpoints of its sides to create four triangles.
3. Put gray glass in the central triangle.
4. Repeat steps 2 and 3 in any one pane that has not yet received glass.
Step 4 may be repeated any number of times. How many of these six windows can be made?

The beavers built a machine to clean the city’s water. To complete treatment, water must pass through filters N, P and C in that order.



Valve 1 is open and all the other valves are closed. Of the valves numbered 2–6, how many can the beavers open and be certain of receiving correctly treated water at that outlet?

Beaverland has 12 cities connected by highways A–O, as shown.



A group of cities is an economic community if any two cities in it are connected directly, or by a route through other cities in the same group. All 12 cities initially form one community. Because of a Covid-19 outbreak, the mayors decide to block exactly two highways to divide the country into three separate communities. To reduce economic disruption, they want the smallest resulting community to contain as many cities as possible. They find an optimal choice. How many cities are in the smallest community?

Beavers add letters by adding their positions in the alphabet, A=1 through Z=26.



The result is converted back to a letter. For example A+A=B and A+B=C. When the total exceeds 26, counting wraps around from A; for example Z+B=B and Y+C=B. Lenka encrypts a message as follows:
- Choose a random number, called the key.
- Write the message on one line, then write it again on the next line shifted right by the key number of positions.
- Add the letters in each column to obtain the encrypted message; a letter with no partner is unchanged.
For example, TEA with key 1 gives TYFA.



What does COMPUTER give with key 3?

Beavers encode the letters I,T,O,C,L using 3×3 number grids. First, the letters are drawn on 3×3 square grids as shown, in that order.



To encode one letter, examine each cell and write the number of the other four letters whose cell at the same position has the same color. For example, I is encoded as follows.



Which letter produces this number grid?

Kangaroo wants to get home by following the roads in the picture. It may move only horizontally (left or right) or vertically (up or down), and may cross a rock pile only if it contains at most two rocks.



One segment is the distance between two neighboring junctions. What is the minimum number of segments needed to reach the home at the bottom-right corner?

The beavers have three carving machines. Each accepts a block with the indicated symbol and adds other symbol blocks according to its illustrated rule, retaining existing symbols and the shown orientations. “Máy” means “Machine”.



The next picture gives an example of applying Machine 3 twice.



Starting from the single-circle block below, which result can be made using the three machines?

A beaver has nine trees of three different heights, three trees of each height. The three kinds are shown here.



Trees of the same kind have the same height, and different kinds have different heights. The beaver plants them in a 3×3 garden with each kind appearing exactly once in every row and column. Each sign around the garden records how many trees can be seen along that row or column from that side. A taller tree hides shorter trees behind it. The following example illustrates the signs and a side view.



In the actual garden, only some signs have been filled in, as shown below. What is the sum of all the numbers around the garden when all signs are filled in?