Found problems: 109
2005 Mid-Michigan MO, 7-9
[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square.
[b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$.
[b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor?
[b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.)
[img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img]
[b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game?
[b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits.
$\begin{tabular}{ccccc}
& & & a & b \\
* & & & c & d \\
\hline
& & c & e & f \\
+ & & a & b & \\
\hline
& c & f & d & f \\
\end{tabular}$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2006 Mid-Michigan MO, 5-6
[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] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply?
[b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert?
[b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 10-12, 2023
[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 .
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Mid-Michigan MO, 7-9
[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done?
[b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$.
[b]p3.[/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][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img]
[b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2003 Mid-Michigan MO, 10-12
[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.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 Mid-Michigan MO, 7-9
[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover:
a) greater than $9$ ?
b) greater than $15$?
[b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces?
[img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img]
[b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color.
[b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side?
[b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 Mid-Michigan MO, 5-6
[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?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2009 Mid-Michigan MO, 10-12
[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below.
( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] )
[b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done?
[b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$.
[b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern.
[b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2003 Mid-Michigan MO, 5-6
[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more?
[b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once?
[b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal?
[b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room.
[u]Bonus Problem [/u]
[b]p5.[/b] 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.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2012 Mid-Michigan MO, 5-6
[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family?
[b]p2.[/b] Solve each of the following problems.
(1) Find a pair of numbers with a sum of $11$ and a product of $24$.
(2) Find a pair of numbers with a sum of $40$ and a product of $400$.
(3) Find three consecutive numbers with a sum of $333$.
(4) Find two consecutive numbers with a product of $182$.
[b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive.
[b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$.
[b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map.
[img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img]
On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 Mid-Michigan MO, 10-12
[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?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2004 Mid-Michigan MO, 10-12
[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer 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?
[b]p2.[/b] Find the smallest positive whole number that ends with $17$, is divisible by $17$, and the sum of its digits is $17$.
[b]p3.[/b] Three consecutive $2$-digit numbers are written next to each other. It turns out that the resulting $6$-digit number is divisible by $17$. Find all such numbers.
[b]p4.[/b] Let $ABCD$ be a convex quadrilateral (a quadrilateral $ABCD$ is called convex if the diagonals $AC$ and $BD$ intersect). Suppose that $\angle CBD = \angle CAB$ and $\angle ACD = \angle BDA$ . Prove that $\angle ABC = \angle ADC$.
[b]p5.[/b] A circle of radius $1$ is cut into four equal arcs, which are then arranged to make the shape shown on the picture. What is its area?
[img]https://cdn.artofproblemsolving.com/attachments/f/3/49c3fe8b218ab0a5378ecc635b797a912723f9.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2004 Mid-Michigan MO, 7-9
[b]p1.[/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?
[b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes?
[b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$.
[b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks.
[b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 10-12, 2009
[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below.
( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] )
[b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done?
[b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$.
[b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern.
[b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 5-6, 2003
[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more?
[b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once?
[b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal?
[b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room.
[u]Bonus Problem [/u]
[b]p5.[/b] 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.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2017 Mid-Michigan MO, 10-12
[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends?
[b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots?
[b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$.
[b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$.
[b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions).
[b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 5-6, 2007
[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus?
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins?
[b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits?
$\begin{tabular}{ccccc}
& & & 9 & 5 \\
x & & & * & * \\
\hline
& & & * & * \\
+ & 1 & * & * & \\
\hline
& * & * & * & * \\
\end{tabular}$
Find all solutions.
[b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes.
[b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2007 Mid-Michigan MO, 10-12
[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks.
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins?
[b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct:
$\begin{tabular}{ccccc}
& & a & b & c \\
x & & & d & e \\
\hline
& f & a & c & c \\
+ & g & h & i & \\
\hline
f & f & f & c & c \\
\end{tabular}$
[b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning.
[b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 7-9, 2022
[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square.
[img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img]
[b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge).
[b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property.
(a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer.
(b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer.
[b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct.
[img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img]
[b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper
[b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 7-9, 2004
[b]p1.[/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?
[b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes?
[b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$.
[b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks.
[b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 10-12, 2017
[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends?
[b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots?
[b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$.
[b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$.
[b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions).
[b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2002 Mid-Michigan MO, 10-12
[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$.
[b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral.
[b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms?
[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?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2014 Mid-Michigan MO, 5-6
[b]p1.[/b] Find any integer solution of the puzzle: $WE+ST+RO+NG=128$
(different letters mean different digits between $1$ and $9$).
[b]p2.[/b] A $5\times 6$ rectangle is drawn on the piece of graph paper (see the figure below). The side of each square on the graph paper is $1$ cm long. Cut the rectangle along the sides of the graph squares in two parts whose areas are equal but perimeters are different by $2$ cm.
$\begin{tabular}{|l|l|l|l|l|l|}
\hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
\end{tabular}$
[b]p3.[/b] Three runners started simultaneously on a $1$ km long track. Each of them runs the whole distance at a constant speed. Runner $A$ is the fastest. When he runs $400$ meters then the total distance run by runners $B$ and $C$ together is $680$ meters. What is the total combined distance remaining for runners $B$ and $C$ when runner $A$ has $100$ meters left?
[b]p4.[/b] There are three people in a room. Each person is either a knight who always tells the truth or a liar who always tells lies. The first person said «We are all liars». The second replied «Only you are a liar». Is the third person a liar or a knight?
[b]p5.[/b] A $5\times 8$ rectangle is divided into forty $1\times 1$ square boxes (see the figure below). Choose 24 such boxes and one diagonal in each chosen box so that these diagonals don't have common points.
$\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& & & & & & & \\ \hline
& & & & & & & \\ \hline
& & & & & & & \\ \hline
& & & & & & & \\ \hline
\end{tabular}$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 7-9, 2018
[b]p1.[/b] Is it possible to put $9$ numbers $1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9$ in a circle in a way such that the sum of any three circularly consecutive numbers is divisible by $3$ and is, moreover:
a) greater than $9$ ?
b) greater than $15$?
[b]p2.[/b] You can cut the figure below along the sides of the small squares into several (at least two) identical pieces. What is the minimal number of such equal pieces?
[img]https://cdn.artofproblemsolving.com/attachments/8/e/9cd09a04209774dab34bc7f989b79573453f35.png[/img]
[b]p3.[/b] There are $100$ colored marbles in a box. It is known that among any set of ten marbles there are at least two marbles of the same color. Show that the box contains $12$ marbles of the same color.
[b]p4.[/b] Is it possible to color squares of a $ 8\times 8$ board in white and black color in such a way that every square has exactly one black neighbor square separated by a side?
[b]p5.[/b] In a basket, there are more than $80$ but no more than $200$ white, yellow, black, and red balls. Exactly $12\%$ are yellow, $20\%$ are black. Is it possible that exactly $2/3$ of the balls are white?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Mid-Michigan MO, Grades 10-12, 2012
[b]p1.[/b] A triangle $ABC$ is drawn in the plane. A point $D$ is chosen inside the triangle. Show that the sum of distances $AD+BD+CD$ is less than the perimeter of the triangle.
[b]p2.[/b] In a triangle $ABC$ the bisector of the angle $C$ intersects the side $AB$ at $M$, and the bisector of the angle $A$ intersects $CM$ at the point $T$. Suppose that the segments $CM$ and $AT$ divided the triangle $ABC$ into three isosceles triangles. Find the angles of the triangle $ABC$.
[b]p3.[/b] You are given $100$ weights of masses $1, 2, 3,..., 99, 100$. Can one distribute them into $10$ piles having the following property: the heavier the pile, the fewer weights it contains?
[b]p4.[/b] Each cell of a $10\times 10$ table contains a number. In each line the greatest number (or one of the largest, if more than one) is underscored, and in each column the smallest (or one of the smallest) is also underscored. It turned out that all of the underscored numbers are underscored exactly twice. Prove that all numbers stored in the table are equal to each other.
[b]p5.[/b] Two stores have warehouses in which wheat is stored. There are $16$ more tons of wheat in the first warehouse than in the second. Every night exactly at midnight the owner of each store steals from his rival, taking a quarter of the wheat in his rival's warehouse and dragging it to his own. After $10$ days, the thieves are caught. Which warehouse has more wheat at this point and by how much?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].