In a certain quiz show, there are the following rules. Every participant has 10 points at the beginning and has to answer 10 questions. For each correct answer, the participant earns 1 point. For each incorrect answer, the participant loses a point. Mrs. Smith had 10 points at the end of this quiz show. So how many correct answers did she give? We see that Mrs. Smith... she began with ten just like every participant and then at the end had four more than she started with so these four points came from answering some questions more questions were answered correctly than incorrect so four points has to be equal to the balance of correct answers subtract from that the number of incorrect answers okay and we also have that 10 points is equal to the total number of answers okay and from that we can we can calculate if 10 were correct and 0 were incorrect we would end up with 10 plus 10 or 20 points so that's way too much let's try 8 correct and if 8 were correct then 2 were incorrect and that gives us 10 plus 8 correct minus 2 incorrect and that comes out to 16 points which is still too much so let's keep going let's say there were severed 7 correct and if 7 were correct then 3 were incorrect so we have 10 points that we began with then plus 7 earned and minus 3 taken away for incorrect answers that gives us 10 plus 4 or 14 points and that is exactly what happened to mrs. Smith so now we know that she must have in fact out of the 10 questions that were asked answered 7 correctly and 3 incorrectly so that she earned the total of 4 points on top of her 10 so the answer here is a mrs. Smith answered 7 questions correctly question number 18 four friends Masha Sasha Dasha and Pasha were sitting on a bench first Masha changed places with Dasha then Dasha changed places with Pasha at the end the girls sat on the bench in the following order from left to right Masha Sasha Dasha and Pasha in what order from left to right were they sitting in the very beginning so here I have a larger version of our picture with the girls sitting on the bench let's label them with the first letters of their first names in the order that they are sitting at the end so Masha is first so I will just put a M over here then followed by Sasha Dasha is next and would be for Pasha and she is last and so our job is to follow the problem here backwards in time and say what was the initial order and we know that they change places in a certain order so we should do that backwards we know that first Masha changed places with Dasha so we'll do that last and then the second move was that Dasha changed places with Pasha so we should do that we should let Dasha sit where Pasha is sitting and interchange their places so let me move Pasha over to where Dasha was sitting and then let me move Dasha where Pasha was sitting like that so that would be the last move that happened but we do it first in reverse and then we know that the first thing that happened was that Dasha changed excuse me that Masha changed places with Dasha over here that's the first thing that happened so we should go and change D with M so M moves all the way to the end and D will move all the way forward like so and so that should be the very first thing that happened this is the initial position so Dasha is first then followed by Sasha then Pasha and Masha is at the end and that is exactly the order here that's presented in in C so that is going to be our answer C question number 19 the picture shows the distances between certain towns A B C D E and F in miles what is the distance between the towns here C and D that is not labeled so we are looking for the following distance from C to D where the distances here overlap so let me just label that as as an unknown let's say X unknown distance and then we already know that between the far towns A and F the distance is 34 miles so how could one travel from A to F well one way would be to go from A to D which is 21 miles and then go back to see this unknown distance X and then travel from C to F again which is known to be 23 miles so we can write that down in an equation we have 34 miles that's the distance from A to F and we can travel that in sort of a roundabout way we can say go from A to D which is 21 miles okay that's A to D and then go back X miles that is from D to C in reverse and then plus from C to F it is exactly 23 miles that's the distance from C to F okay and now we can solve for X so we have 34 miles on the left hand side of our equation that is equal to 21 plus 23 or 44 miles and we have to subtract X the distance we went back X miles and so we can solve for this we can rearrange the equation and have X miles is equal to 44 miles minus 34 miles and that gives us exactly the distance of 10 miles we have solved for the distance between towns C and D we obtain 10 miles and so the answer here should be C 10 question number 20 for identical dice have been arranged in a structure as shown in the figure here and I have enlarged that figure we know that the sum of the number of dots on any two opposite sides is always equal to 7 because these are regular dice what does this structure look like from behind so the first thing we should note is that we have a tower here of two dice that has the same face up front so in the back looking at this from behind we would see a tower where we would have the four on top and the four on the bottom and with that we immediately eliminate choice a where we have two fives and also we eliminate choice D where we have two fives okay and we also see that after looking at this picture from the other side we will not see the two dots on this face over here from the other side after rotation this becomes a five this value is five so right away we can also eliminate B where we see a tower of two fours but the same face as we see here that that cannot happen and we eliminate E because we cannot see the same face looking from the other side so C is the only remaining choice and that has to be our answer question number 21 Olga has three cards as shown in the picture she has a card with a nine a card with an eight and this looks like a six but it could be turned around and be a nine again using these cards all that can form different numbers for example 989 986 866 and so on how many different three digit numbers can she form using these three cards so we are not sure if the nine and six are going to stay a nine or a six they can be turned around and from a six we can have a nine from a nine we can have a six but the eight here will always be an eight whether we turn the card upside down or not so let's start working with an eight and list all the different numbers with eight and the first position so if eight is first we can have two sixes we can have a six and a nine and if we do that the nine and the six can switch positions or we can have an eight and two nines now if we suppose that eight is in the middle we can have again two sixes like that we can have a six and a nine we can have a nine and a six but their positions switched and we can have two nines like that and eight is always in the middle and finally we can say what happens if the eight is on the other side well then we would have two sixes first and then a six and a nine but we can do that in two different ways and then we can finally have two nines and so that's it so there are four numbers in each column and all together if we add a four with a four with the four that gives us twelve so that would be the answer here she can make twelve different three-digit numbers and the answer is E question number 22 Andrea formed the ornament shown in the picture here to the right and here in my enlarged version using several identical pieces these pieces cannot overlap or cover each other which of the following pieces could not have been used by Andrea to create the ornament so what we have to do to check is just find copies of the pieces shown and the ornament to show that it can be constructed using each piece without overlaps so here I have found a copy of piece a and I can find two more copies that would not overlap like that so here are the pieces without any overlap so we see that a can definitely be used to make the ornament and now a quick glance here tells us that E can also be used because I can simply cut in half these pieces that look like the one in A and obtain exactly the pieces in E so we see that E also works so let's move on let me erase here and look at the piece in B and it turns out that I can also find that in the ornament it is exactly this piece here like that and now I have to find more of him without any overlaps so here is going to be the second one like that on the side and you see that the remaining top portion can be exactly built with a third copy of the piece in B so B is fine as well so let me erase and move on to C. C is pretty complicated we can find a copy of C here in the lower portion of the of the diagram like so so we need to then move over here like that okay that's a that's the outline of C and then moving on we need to find another copy of it okay so here we have that rounded part and and there is another outline of that piece and the remaining portion is exactly another copy so C is also possible so then the process of elimination tells us that well in fact D has to be the answer and we can check that no matter how we arrange the piece in D we will not be able to obtain the ornament here so by the process of elimination this is the piece that cannot be used. Question number 23. Picture 1 over here shows a castle built out of identical cubes when we look at the same castle from above we see picture 2 so every cube lies either on the ground which from above is displayed in red or on top of another cube which from above has this outline here in blue. How many cubes were used to build the castle? Now looking at the perspective picture 1 we see that there are essentially two types of towers there are towers with three cubes stacked on top of one another and then there are towers with just two cubes stacked on top of one another so let's mark on the diagram here how many of these are the three block towers and those are the corner ones so we have the corners being the tallest and also the one in the middle on each wall like that so we can see that there are going to be exactly 3 plus 3, 8 tall towers and then the remaining towers are short towers so how many do we have? We would have 2, 4, 6, 8, 10, 12, 14, 16 of them we would have 16 short towers if we count all the all the towers here in blue 2, 4, 6, 8, 10, 12, 14, and 16 so we have 16 short towers and so that gives us 16 times 2 blocks for each of those towers which comes out to 32 and for the red towers we have 8 times 3 blocks or 24 blocks and then all together we add so and get 24 plus 32 that gives us exactly 56 blocks so that's how many blocks are used to build this castle and the answer therefore has to be a 56. Question number 24. Chris wrote the numbers 6 and 7 in the circles as shown in the picture here. Here we have a 6, here we have a 7. He will then write each of the numbers 1, 2, 3, 4, 5, and 8 into the remaining 6 circles without the number so that the sum of the numbers on each of the sides here 3 on the side will always be equal to 13. The question is what will the sum of the numbers in the shaded yellow circles be when this is complete? So we have two of the numbers given and so we see that together the yellow circles on the right face they have to add up to 7 because 7 plus 6 is 13 and likewise the two values that are missing over here will have to add up to 6. Now we have to use the number 8 somewhere and if it's anywhere on the right side or on the bottom side then it will exceed the given sum of 13 so we have to place 8 over here. This is something we can say right away and then we just try. We see what happens if we try to write 7 as a 5 plus 2 or we can write 7 as a 3 plus 4. We can also write 6 as a 4 plus 2 or a 5 plus 1. So let's begin by saying okay 7 is going to be 5 plus 2 and let's put a 2 over here. So that is a 2. If that is a 2 then we have to have a 4 over here and we have to have a 5 over here. So we see that 8 plus 5 is already 13 and we cannot put any number in here. So that was an incorrect choice. Let's see if we can still use a 5 and a 2 on the right face but just switch their positions. So I will erase and try things in the other order. Now instead of putting a 2 on the bottom let's put a 5 on the bottom and a 2 on top. In that case I will have to choose a 1 where the 4 is now. So let me erase the 4 and write in a 1 over here like that. And now we have the correct sum on the right and on the bottom. We then go to the top and we see that we have an 8 and a 2 so a 3 is missing and on the left we have an 8 and a 1 so that gives us a 9 and a 4 is missing. Now in this case the sum of the shaded circles is going to be 8 plus 2 plus 1 plus 5 which gives us 16. Now the question was what is the sum of the numbers in the shaded circles and that is exactly what we have found here. So we can answer the question right away. We can say that it is 16 because we have found a possibility. There is no reason to suspect that there a different arrangement will give us a different value for the shaded circles. So the answer is E.

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