Found problems: 109
2003 Mid-Michigan MO, 7-9
[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$?
[b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns.
[b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix}
2 & 3 & * & * \\
4 & * & * & *\\
* & * & * & *\\
* & * & * & *
\end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?)
[b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices?
[b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions.
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2015 Mid-Michigan MO, 10-12
[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square?
[b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$.
[b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$.
[b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle.
[b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy?
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Mid-Michigan MO, Grades 10-12, 2018
[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.)
[b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers.
[b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img]
[b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles?
[b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$.
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Mid-Michigan MO, Grades 7-9, 2019
[b]p1.[/b] Prove that the equation $x^6 - 143x^5 - 917x^4 + 51x^3 + 77x^2 + 291x + 1575 = 0$ has no integer solutions.
[b]p2.[/b] There are $81$ wheels in a storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that it can be detected with certainty after four measurements on a balance scale.
[b]p3.[/b] Rob and Ann multiplied the numbers from $1$ to $100$ and calculated the sum of digits of this product. For this sum, Rob calculated the sum of its digits as well. Then Ann kept repeating this operation until he got a one-digit number. What was this number?
[b]p4.[/b] Rui and Jui take turns placing bishops on the squares of the $ 8\times 8$ chessboard in such a way that bishops cannot attack one another. (In this game, the color of the rooks is irrelevant.) The player who cannot place a rook loses the game. Rui takes the first turn. Who has a winning strategy, and what is it?
[b]p5.[/b] The following figure can be cut along sides of small squares into several (more than one) identical shapes. What is the smallest number of such identical shapes you can get?
[img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img]
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Mid-Michigan MO, Grades 5-6, 2022
[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm.
[b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions.
[img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img]
[b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles.
[img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img]
[b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number?
[b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct.
$$ 4 \times 12 + 18 : 6 + 3 = 50$$
[b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square?
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2023 Mid-Michigan MO, 7-9
[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps?
[b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions?
[b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills.
(a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back.
(b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.)
[b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line.
(b) Do the same with $6$ points.
[b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$.
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Mid-Michigan MO, Grades 5-6, 2019
[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts?
[b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men?
[b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box.
[b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements.
[b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left:
(a) $8$ matches
(b) $4$ matches
[img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img]
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2010 Mid-Michigan MO, 10-12
[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccccc}
& & & a & b & c & d \\
x & & & & & a & b \\
\hline
& & c & d & b & d & b \\
+ & c & e & b & f & b & \\
\hline
& c & g & a & e & g & b \\
\end{tabular}$
[b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$?
[b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$
[b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish?
[b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture.
[img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img]
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2012 Mid-Michigan MO, 7-9
[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$.
[b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests.
"I wonder how many knights are among you?" he asked.
" Ask everyone a question and find out yourself" advised him one of the guests.
"Okay. Tell me one: Who are your neighbors?" asked the traveler.
This question was answered the same way by all the guests.
"This information is not enough!" said the traveler.
"But today is my birthday, do not forget it!" said one of the guests.
"Yes, today is his birthday!" said his neighbor.
Now the traveler was able to find out how many knights were at the table.
Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]?
[b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters?
[b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed?
[b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased?
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2006 Mid-Michigan MO, 7-9
[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccc}
& a & b & c & a \\
+ & & d & d & e \\
& & & d & e \\
\hline
d & f & f & d & d \\
\end{tabular}$
[b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$:
$$n^2 + m^2 = 111...111$$
[b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$.
[b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$
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2022 Mid-Michigan MO, 10-12
[b]p1.[/b] Consider a triangular grid: nodes of the grid are painted black and white. At a single step you are allowed to change colors of all nodes situated on any straight line (with the slope $0^o$ ,$60^o$, or $120^o$ ) going through the nodes of the grid. Can you transform the combination in the left picture into the one in the right picture in a finite number of steps?
[img]https://cdn.artofproblemsolving.com/attachments/3/a/957b199149269ce1d0f66b91a1ac0737cf3f89.png[/img]
[b]p2.[/b] Find $x$ satisfying $\sqrt{x\sqrt{x \sqrt{x ...}}} = \sqrt{2022}$ where it is an infinite expression on the left side.
[b]p3.[/b] $179$ glasses are placed upside down on a table. You are allowed to do the following moves. An integer number $k$ is fixed. In one move you are allowed to turn any $k$ glasses .
(a) Is it possible in a finite number of moves to turn all $179$ glasses into “bottom-down” positions if $k=3$?
(b) Is it possible to do it if $k=4$?
[b]p4.[/b] An interval of length $1$ is drawn on a paper. Using a compass and a simple ruler construct an interval of length $\sqrt{93}$.
[b]p5.[/b] Show that $5^{2n+1} + 3^{n+2} 2^{n-1} $ is divisible by $19$ for any positive integer $n$.
[b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=1-z \\ \dfrac{yz}{y+z}=2-x \\ \dfrac{xz}{x+z}=2-y \end{cases}$$
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Mid-Michigan MO, Grades 5-6, 2015
[b]p1.[/b] To every face of a given cube a new cube of the same size is glued. The resulting solid has how many faces?
[b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's?
[b]p3.[/b] A radio transmitter has $4$ buttons. Each button controls its own switch: if the switch is OFF the button turns it ON and vice versa. The initial state of switches in unknown. The transmitter sends a signal if at least $3$ switches are ON. What is the minimal number of times you have to push the button to guarantee the signal is sent?
[b]p4.[/b] $19$ matches are placed on a table to show the incorrect equation: $XXX + XIV = XV$. Move exactly one match to change this into a correct equation.
[b]p5.[/b] Cut the grid shown into two parts of equal area by cutting along the lines of the grid.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/7f2f284acf3709c2f6b1bea08835d2fb409c44.png[/img]
[b]p6.[/b] A family of funny dwarfs consists of a dad, a mom, and a child. Their names are: $A$, $R$, and $C$ (not in order). During lunch, $C$ made the statements: “$R$ and $A$ have different genders” and “$R$ and $A$ are my parents”, and $A$ made the statements “I am $C$'s dad” and “I am $R$'s daughter.” In fact, each dwarf told truth once and told a lie once. What is the name of the dad, what is the name of the child, and is the child a son or a daughter?
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Mid-Michigan MO, Grades 10-12, 2010
[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccccc}
& & & a & b & c & d \\
x & & & & & a & b \\
\hline
& & c & d & b & d & b \\
+ & c & e & b & f & b & \\
\hline
& c & g & a & e & g & b \\
\end{tabular}$
[b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$?
[b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$
[b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish?
[b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture.
[img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 5-6, 2009
[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax.
[b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.)
[b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order.
[b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs?
[b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img]
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Mid-Michigan MO, Grades 5-6, 2002
[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one.
[b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$.
[b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that
(a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color;
(b) there are no two neighboring blue chips.
[b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers?
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2023 Mid-Michigan MO, 10-12
[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months?
[b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$.
[b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$.
[b]p4.[/b] Prove that $\cos 1^o$ is irrational.
[b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal .
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2019 Mid-Michigan MO, 5-6
[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts?
[b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men?
[b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box.
[b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements.
[b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left:
(a) $8$ matches
(b) $4$ matches
[img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img]
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2014 Mid-Michigan MO, 7-9
[b]p1.[/b] (a) Put the numbers $1$ to $6$ on the circle in such way that for any five consecutive numbers the sum of first three (clockwise) is larger than the sum of remaining two.
(b) Can you arrange these numbers so it works both clockwise and counterclockwise.
[b]p2.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace:
$\bullet$ two candies in the box with one chocolate bar,
$\bullet$ two muffins in the box with one chocolate bar,
$\bullet$ two chocolate bars in the box with one candy and one muffin,
$\bullet$ one candy and one chocolate bar in the box with one muffin,
$\bullet$ one muffin and one chocolate bar in the box with one candy.
Is it possible that after some time it remains only one object in the box?
[b]p3.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$ (different letters mean different digits between $1$ and $9$).
[b]p4.[/b] Two consecutive three‐digit positive integer numbers are written one after the other one. Show that the six‐digit number that is obtained is not divisible by $1001$.
[b]p5.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points?
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Mid-Michigan MO, Grades 10-12, 2003
[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer.
[b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$.
[b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer.
[b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$.
[b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions.
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2004 Mid-Michigan MO, 5-6
[b]p1.[/b] On the island of Nevermind some people are liars; they always lie. The remaining habitants of the island are truthlovers; they tell only the truth. Three habitants of the island, $A, B$, and $C$ met this morning.
$A$ said: “All of us are liars”.
$B$ said: “Only one of us is a truthlover”.
Who of them is a liar and who of them is a truthlover?
[b]p2.[/b] Pinocchio has $9$ pieces of paper. He is allowed to take a piece of paper and cut it in $5$ pieces or $7$ pieces which increases the number of his pieces. Then he can take again one of his pieces of paper and cut it in $5$ pieces or $7$ pieces. He can do this again and again as many times as he wishes. Can he get $2004$ pieces of paper?
[b]p3.[/b] In Dragonland there are coins of $1$ cent, $2$ cents, $10$ cents, $20$ cents, and $50$ cents. What is the largest amount of money one can have in coins, yet still not be able to make exactly $1$ dollar?
[b]p4.[/b] Find all solutions $a, b, c, d, e$ if it is known that they represent distinct
digits and satisfy the following:
$\begin{tabular}{ccccc}
& a & b & c & d \\
+ & a & c & a & c \\
\hline
c & d & e & b & c \\
\end{tabular}$
[b]p5.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy?
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Mid-Michigan MO, Grades 7-9, 2012
[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$.
[b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests.
"I wonder how many knights are among you?" he asked.
" Ask everyone a question and find out yourself" advised him one of the guests.
"Okay. Tell me one: Who are your neighbors?" asked the traveler.
This question was answered the same way by all the guests.
"This information is not enough!" said the traveler.
"But today is my birthday, do not forget it!" said one of the guests.
"Yes, today is his birthday!" said his neighbor.
Now the traveler was able to find out how many knights were at the table.
Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]?
[b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters?
[b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed?
[b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased?
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2006 Mid-Michigan MO, 10-12
[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible?
[img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img]
[b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$.
[b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon.
[img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img]
[b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller?
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Mid-Michigan MO, Grades 10-12, 2019
[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$.
[b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$?
[b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal.
[b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$.
[b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box?
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2010 Mid-Michigan MO, 5-6
[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run?
[b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle?
[b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region?
[img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img]
[b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$?
[b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true?
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Mid-Michigan MO, Grades 7-9, 2008
[b]p1.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. His drink contains $45\%$ of orange juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $60\%$ of orange juice?
[b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img]
[b]p3.[/b] For one particular number $a > 0$ the function f satisfies the equality $f(x + a) =\frac{1 + f(x)}{1 - f(x)}$ for all $x$. Show that $f$ is a periodic function. (A function $f$ is periodic with the period $T$ if $f(x + T) = f(x)$ for any $x$.)
[b]p4.[/b] If $a, b, c, x, y, z$ are numbers so that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ and $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}= 0$. Show that $\frac{x^2}{a^2} +\frac{y^2}{b^2} +\frac{z^2}{c^2} = 1$
[b]p5.[/b] Is it possible that a four-digit number $AABB$ is a perfect square?
(Same letters denote the same digits).
[b]p6.[/b] A finite number of arcs of a circle are painted black (see figure). The total length of these arcs is less than $\frac15$ of the circumference. Show that it is possible to inscribe a square in the circle so that all vertices of the square are in the unpainted portion of the circle.
[img]https://cdn.artofproblemsolving.com/attachments/2/c/bdfa61917a47f3de5dd3684627792a9ebf05d5.png[/img]
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