Found problems: 18
Kettering MO, 2016
[b]p1.[/b] Solve the equation $3^x + 9^x = 27^x$.
[b]p2.[/b] An equilateral triangle in inscribed in a circle of area $1$ m$^2$. Then the second circle is inscribed in the triangle. Find the radius of the second circle.
[b]p3.[/b] Solve the inequality: $2\sqrt{x^2 - 5x + 4} + 3\sqrt{x^2 + 2x - 3} \le 5\sqrt{6 - x - x^2}$
[b]p4.[/b] Peter and John played a game. Peter wrote on a blackboard all integers from $1$ to $18$ and offered John to choose $8$ different integers from this list. To win the game John had to choose 8 integers such that among them the difference between any two is either less than $7$ or greater than $11$. Can John win the game? Justify your answer.
[b]p5.[/b] Prove that given $100$ different positive integers such that none of them is a multiple of $100$, it is always possible to choose several of them such that the last two digits of their sum are zeros.
[b]p6.[/b] Given $100$ different squares such that the sum of their areas equals $1/2$ m$^2$ , is it possible to place them on a square board with area $1$ m$^2$ without overlays? Justify your answer.
PS. You should use hide for answers.
Kettering MO, 2018
[b]p1.[/b] Solve the equation: $\sqrt{x} +\sqrt{x + 1} - \sqrt{x + 2} = 0$.
[b]p2.[/b] Solve the inequality: $\ln (x^2 + 3x + 2) \le 0$.
[b]p3.[/b] In the trapezoid $ABCD$ ($AD \parallel BC$) $|AD|+|AB| = |BC|+|CD|$. Find the ratio of the length of the sides $AB$ and $CD$ ($|AB|/|CD|$).
[b]p4.[/b] Gollum gave Bilbo a new riddle. He put $64$ stones that are either white or black on an $8 \times 8$ chess board (one piece per each of $64$ squares). At every move Bilbo can replace all stones of any horizontal or vertical row by stones of the opposite color (white by black and black by white). Bilbo can make as many moves as he needs. Bilbo needs to get a position when in every horizontal and in every vertical row the number of white stones is greater than or equal to the number of black stones. Can Bilbo solve the riddle and what should be his solution?
[b]p5.[/b] Two trolls Tom and Bert caught Bilbo and offered him a game. Each player got a bag with white, yellow, and black stones. The game started with Tom putting some number of stones from his bag on the table, then Bert added some number of stones from his bag, and then Bilbo added some stones from his bag. After that three players started making moves. At each move a player chooses two stones of different colors, takes them away from the table, and puts on the table a stone of the color different from the colors of chosen stones. Game ends when stones of one color only remain on the table. If the remaining stones are white Tom wins and eats Bilbo, if they are yellow, Bert wins and eats Bilbo, if they are black, Bilbo wins and is set free. Can you help Bilbo to save his life by offering him a winning strategy?
[b]p6.[/b] There are four roads in Mirkwood that are straight lines. Bilbo, Gandalf, Legolas, and Thorin were travelling along these roads, each along a different road, at a different constant speed. During their trips Bilbo met Gandalf, and both Bilbo and Gandalf met Legolas and Thorin, but neither three of them met at the same time. When meeting they did not stop and did not change the road, the speed, and the direction. Did Legolas meet Thorin? Justify your answer.
PS. You should use hide for answers.
Kettering MO, 2007
[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane.
[b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.)
[img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img]
(a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs.
(b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$.
$\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer.
[b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected).
[img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img]
[b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$
(do not use a calculator and do not use approximations).
[b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle.
[b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order.
PS. You should use hide for answers.
Kettering MO, 2015
[b]p1.[/b] Solve the equation $\log_x (x + 2) = 2$.
[b]p2.[/b] Solve the inequality: $0.5^{|x|} > 0.5^{x^2}$.
[b]p3.[/b] The integers from 1 to 2015 are written on the blackboard. Two randomly chosen numbers are erased and replaced by their difference giving a sequence with one less number. This process is repeated until there is only one number remaining. Is the remaining number even or odd? Justify your answer.
[b]p4.[/b] Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer.
[b]p5.[/b] Prove that for any finite sequence of digits there exists an integer the square of which begins with that sequence.
[b]p6.[/b] The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$.
PS. You should use hide for answers.
Kettering MO, 2001
[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root.
[b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it?
[b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins.
[b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$.
[b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself.
[img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img]
(b) Determine coordinates of the points $A$ and $B$.
[b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that
$$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$
(b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a).
(c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$
PS. You should use hide for answers.
Kettering MO, 2004
[b]p1.[/b] Find all real solutions of the system
$$x^5 + y^5 = 1$$
$$x^6 + y^6 = 1$$
[b]p2.[/b] The centers of three circles of the radius $R$ are located in the vertexes of equilateral triangle. The length of the sides of the triangle is $a$ and $\frac{a}{2}< R < a$. Find the distances between the intersection points of the
circles, which are outside of the triangle.
[b]p3.[/b] Prove that no positive integer power of $2$ ends with four equal digits.
[b]p4.[/b] A circle is divided in $10$ sectors. $90$ coins are located in these sectors, $9$ coins in each sector. At every move you can move a coin from a sector to one of two neighbor sectors. (Two sectors are called neighbor if they are adjoined along a segment.) Is it possible to move all coins into one sector in exactly$ 2004$ moves?
[b]p5.[/b] Inside a convex polygon several points are arbitrary chosen. Is it possible to divide the polygon into smaller convex polygons such that every one contains exactly one given point? Justify your answer.
[b]p6.[/b] A troll tried to spoil a white and red $8\times 8$ chessboard. The area of every square of the chessboard is one square foot. He randomly painted $1.5\%$ of the area of every square with black ink. A grasshopper jumped on the spoiled chessboard. The length of the jump of the grasshopper is exactly one foot and at every jump only one point of the chessboard is touched. Is it possible for the grasshopper to visit every square of the chessboard without touching any black point? Justify your answer.
PS. You should use hide for answers.
Kettering MO, 2014
[b]p1.[/b] Solve the equation $x^2 - x - cos y+1.25 =0$.
[b]p2.[/b] Solve the inequality: $\left| \frac{x - 2}{x - 3}\right| \le x$
[b]p3.[/b] Bilbo and Dwalin are seated at a round table of radius $R$. Bilbo places a coin of radius $r$ at the center of the table, then Dwalin places a second coin as near to the table’s center as possible without overlapping the first coin. The process continues with additional coins being placed as near as possible to the center of the table and in contact with as many coins as possible without overlap. The person who places the last coin entirely on the table (no overhang) wins the game.
Assume that $R/r$ is an integer.
(a) Who wins, Bilbo or Dawalin? Please justify your answer.
(b) How many coins are on the table when the game ends?
[b]p4.[/b] In the center of a square field is an orc. Four elf guards are on the vertices of that square. The orc can run in the field, the elves only along the sides of the square. Elves run $\$1.5$ times faster than the orc. The orc can kill one elf but cannot fight two of them at the same time. Prove that elves can keep the orc from escaping from the field.
[b]p5.[/b] Nine straight roads cross the Mirkwood which is shaped like a square, with an area of $120$ square miles. Each road intersects two opposite sides of the square and divides the Mirkwood into two quadrilaterals of areas $40$ and $80$ square miles. Prove that there exists a point in the Mirkwood which is an intersection of at least three roads.
PS. You should use hide for answers.
Kettering MO, 2006
[b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts:
(a) their ages are integers,
(b) the product of their ages is $36$,
(c) the sum of their ages is equal to the number of windows in the high school of the chemist and the mathematician.
The mathematician considered this problem and noted that there was not enough information to obtain a unique solution. The chemist then noted that his oldest son had red hair. The mathematician then announced that he had determined the ages of the three sons. Please (aspiring mathematicians) determine the ages of the chemists three sons and explain your solution.
[b]p2.[/b] A square is inscribed in a triangle. Two vertices of this square are on the base of the triangle and two others are on the lateral sides. Prove that the length of the side of the square is greater than and less than $2r$, where $r$ is a radius of the circle inscribed in the triangle.
[b]p3.[/b] You are given any set of $100$ integers in which none of the integers is divisible by $100$. Prove that it is possible to select a subset of this set of $100$ integers such that their sum is a multiple of $100$.
[b]p4.[/b] Find all prime numbers $a$ and $b$ such that $a^b + b^a$ is a prime number.
[b]p5.[/b] $N$ airports are connected by airlines. Some airports are directly connected and some are not. It is always possible to travel from one airport to another by changing planes as needed. The board of directors decided to close one of the airports. Prove that it is possible to select an airport to close so that the remaining airports remain connected.
[b]p6.[/b] (A simplified version of the Fermat’s Last Theorem). Prove that there are no positive integers $x, y, z$ and $z \le n$ satisfying the following equation: $x^n + y^n = z^n$.
PS. You should use hide for answers.
Kettering MO, 2008
[b]p1.[/b] The case of Mr. Brown, Mr. Potter, and Mr. Smith is investigated. One of them has committed a crime. Everyone of them made two statements.
Mr. Brown: I have not done it. Mr. Potter has not done it.
Mr. Potter: Mr. Brown has not done it. Mr. Smith has done it.
Mr. Smith: I have not done it. Mr. Brown has done it.
It is known that one of them told the truth both times, one lied both times, and one told the truth one time and lied one time. Who has committed the crime?
[b]p2.[/b] Is it possible to draw in a plane $1000001$ circles of the radius $1$ such that every circle touches exactly three other circles?
[b]p3.[/b] Consider a circle of radius $R$ centered at the origin. A particle is “launched” from the $x$-axis at a distance, $d$, from the origin with $0 < d < R$, and at an angle, $\alpha$, with the $x$-axis. The particle is reflected from the boundary of the circle so that the [b]angle of incidence[/b] equals the [b]angle of reflection[/b]. Determine the angle $\alpha$ so that the path of the particle contacts the circle’s interior at exactly eight points. Please note that $\alpha$ should be determined in terms of the qunatities $R$ and $d$.
[img]https://cdn.artofproblemsolving.com/attachments/e/3/b8ef9bb8d1b54c263bf2b68d3de60be5b41ad0.png[/img]
[b]p4.[/b] Is it possible to find four different real numbers such that the cube of every number equals the square of the sum of the three others?
[b]p5. [/b]The Fibonacci sequence $(1, 2, 3, 5, 8, 13, 21, . . .)$ is defined by the following formula:
$f_n = f_{n-2} + f_{n-1}$, where $f_1 = 1$, $f_2 = 2$. Prove that any positive integer can be represented as a sum of different members of the Fibonacci sequence.
[b]p6.[/b] $10,000$ points are arbitrary chosen inside a square of area $1$ m$^2$ . Does there exist a broken line connecting all these points, the length of which is less than $201$ m$^2?
PS. You should use hide for answers.
Kettering MO, 2002
[b]p1.[/b] The expression $3 + 2\sqrt2$ can be represented as a perfect square: $3 +\sqrt2 = (1 + \sqrt2)^2$.
(a) Represent $29 - 12\sqrt5$ as a prefect square.
(b) Represent $10 - 6\sqrt3$ as a prefect cube.
[b]p2.[/b] Find all values of the parameter $c$ for which the following system of equations has no solutions.
$$x+cy = 1$$
$$cx+9y = 3$$
[b]p3.[/b] The equation $y = x^2 + 2ax + a$ represents a parabola for all real values of $a$.
(a) Prove hat each of these parabolas pass through a common point and determine the coordinates of this point.
(b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its equation.
[b]p4.[/b] Miranda is a $10$th grade student who is very good in mathematics. In fact she just completed an advanced algebra class and received a grade of A+. Miranda has five sisters, Cathy, Stella, Eva, Lucinda, and Dorothea. Miranda made up a problem involving the ages of the six girls and dared Cathy to solve it.
Miranda said: “The sum of our ages is five times my age. (By ’age’ throughout this problem is meant ’age in years’.) When Stella is three times my present age, the sum of my age and Dorothea’s will be equal to the sum of the present ages of the five of us; Eva’s age will be three times her present age; and Lucinda’s age will be twice Stella’s present age, plus one year. How old are Stella and Miranda?”
“Well, Miranda, could you tell me something else?”
“Sure”, said Miranda, “my age is an odd number”.
[b]p5.[/b] Cities $A,B,C$ and $D$ are located in vertices of a square with the area $10, 000$ square miles. There is a straight-line highway passing through the center of a square. Find the sum of squares of the distances from the cities of to the highway.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/1f53d81d3bc2a465387ff64de15f7da0949f69.png[/img]
[b]p6.[/b] (a) Among three similar coins there is one counterfeit. It is not known whether the counterfeit coin is lighter or heavier than a genuine one (all genuine coins weight the same). Using two weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin?
(b) There is one counterfeit coin among $12$ similar coins. It is not known whether the counterfeit coin is lighter or heavier than a genuine one. Using three weightings on a pan balance, how can the counterfeit be identified and in process determined to be lighter or heavier than a genuine coin?
PS. You should use hide for answers.
Kettering MO, 2005
Today was the 5th Kettering Olympiad - and here are the problems, which are very good intermediate problems.
1. Find all real $x$ so that $(1+x^2)(1+x^4)=4x^3$
2. Mark and John play a game. They have $100$ pebbles on a table. They take turns taking at least one at at most eight pebbles away. The person to claim the last pebble wins. Mark goes first. Can you find a way for Mark to always win? What about John?
3. Prove that
$\sin x + \sin 3x + \sin 5x + ... + \sin 11 x = (1-\cos 12 x)/(2 \sin x)$
4. Mark has $7$ pieces of paper. He takes some of them and splits each into $7$ pieces of paper. He repeats this process some number of times. He then tells John he has $2000$ pieces of paper. John tells him he is wrong. Why is John right?
5. In a triangle $ABC$, the altitude, angle bisector, and median split angle $A$ into four equal angles. Find the angles of $ABC.$
6. There are $100$ cities. There exist airlines connecting pairs of cities.
a) Find the minimal number of airlines such that with at most $k$ plane changes, one can go from any city to any other city.
b) Given that there are $4852$ airlines, show that, given any schematic, one can go from any city to any other city.
Kettering MO, 2020
[b]p1.[/b] Darth Vader urgently needed a new Death Star battle station. He sent requests to four planets asking how much time they would need to build it. The Mandalorians answered that they can build it in one year, the Sorganians in one and a half year, the Nevarroins in two years, and the Klatoonians in three years. To expedite the work Darth Vader decided to hire all of them to work together. The Rebels need to know when the Death Star is operational. Can you help the Rebels and find the number of days needed if all four planets work together? We assume that one year $= 365$ days.
[b]p2.[/b] Solve the inequality: $\left( \sin \frac{\pi}{12} \right)^{\sqrt{1-x}} > \left( \sin \frac{\pi}{12} \right)^x$
[b]p3.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 3x - 6$
[b]p4.[/b] Solve the system of inequalities on $[0, 2\pi]$:
$$\sin (2x) \ge \sin (x)$$
$$\cos (2x) \le \cos (x)$$
[b]p5.[/b] The planet Naboo is under attack by the imperial forces. Three rebellian camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle by a defensive field. What is the maximal area that they may need to cover?
[b]p6.[/b] The Lake Country on the planet Naboo has the shape of a square. There are nine roads in the country. Each of the roads is a straight line that divides the country into two trapezoidal parts such that the ratio of the areas of these parts is $2:5$. Prove that at least three of these roads intersect at one point.
PS. You should use hide for answers.
Kettering MO, 2010
[b]p1.[/b] Find the value of the parameter $a$ for which the following system of equations does not have solutions:
$$ax + 2y = 1$$
$$2x + ay = 1$$
[b]p2.[/b] Find all solutions of the equation $\cos(2x) - 3 \sin(x) + 1 = 0$.
[b]p3.[/b] A circle of a radius $r$ is inscribed into a triangle. Tangent lines to this circle parallel to the sides of the triangle cut out three smaller triangles. The radiuses of the circles inscribed in these smaller triangles are equal to $1,2$ and $3$. Find $r$.
[b]p4.[/b] Does there exist an integer $k$ such that $\log_{10}(1 + 49367 \cdot k)$ is also an integer?
[b]p5.[/b] A plane is divided by $3015$ straight lines such that neither two of them are parallel and neither three of them intersect at one point. Prove that among the pieces of the plane obtained as a result of such division there are at least $2010$ triangular pieces.
PS. You should use hide for answers.
Kettering MO, 2017
[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy?
[b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$
[b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations:
$$x^y = y^{x-y}$$
$$x^x = y^{12y}$$
[b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$.
[b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation:
$$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$
[b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer.
PS. You should use hide for answers.
Kettering MO, 2003
[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer.
$$x + y = 3$$
$$3xy -z^2 = 9$$
[b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much?
[b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles.
[b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$
[b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer.
[b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile
PS. You should use hide for answers.
Kettering MO, 2009
[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$
[b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$.
[b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$.
[b]p4.[/b] Find the irrational number: $$A =\sqrt{ \frac12+\frac12 \sqrt{\frac12+\frac12 \sqrt{ \frac12 +...+ \frac12 \sqrt{ \frac12}}}}$$ ($n$ square roots).
[b]p5.[/b] The Math country has the shape of a regular polygon with $N$ vertexes. $N$ airports are located on the vertexes of that polygon, one airport on each vertex. The Math Airlines company decided to build $K$ additional new airports inside the polygon. However the company has the following policies:
(i) it does not allow three airports to lie on a straight line,
(ii) any new airport with any two old airports should form an isosceles triangle.
How many airports can be added to the original $N$?
[b]p6.[/b] The area of the union of the $n$ circles is greater than $9$ m$^2$(some circles may have non-empty intersections). Is it possible to choose from these $n$ circles some number of non-intersecting circles with total area greater than $1$ m$^2$?
PS. You should use hide for answers.
Kettering MO, 2019
[b]p1.[/b] At $8$ AM Black Widow and Hawkeye began to move towards each other from two cities. They were planning to meet at the midpoint between two cities, but because Black Widow was driving $100$ mi/h faster than Hawkeye, they met at the point that is located $120$ miles from the midpoint. When they met Black Widow said ”If I knew that you drive so slow I would have started one hour later, and then we would have met exactly at the midpoint”. Find the distance between cities.
[b]p2.[/b] Solve the inequality: $\frac{x-1}{x-2} \le \frac{x-2}{x-1}$.
[b]p3.[/b] Solve the equation: $(x - y - z)^2 + (2x - 3y + 2z + 4)^2 + (x + y + z - 8)^2 = 0$.
[b]p4.[/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]p5.[/b] $N$ regions are located in the plane, every pair of them have a nonempty 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]p6.[/b] Numbers $1, 2, . . . , 100$ 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 numbers are written in decreasing order and denoted $b_1$, $b_2$, $...$, $b_{50}$. Thus, $a_1 < a_2 < ... < a_{50}$ and $b_1 > b2_ > ... > b_{50}$. Evaluate $|a_1 - b_1| + |a_2 - b_2| + ... + |a_{50} - b_{50}|$.
PS. You should use hide for answers.
Kettering MO, 2012
[b]p1.[/b] Solve the equation $$\frac{\sqrt{x^2 - 2x + 1}}{x^2 - 1}+\frac{x^2 - 1}{\sqrt{x^2 - 2x + 1}}=\frac52.$$
[b]p2.[/b] Solve the inequality: $\frac{1 - 2\sqrt{1-x^2}}{x} \le 1$.
[b]p3.[/b] Let $ABCD$ be a convex quadrilateral such that the length of the segment connecting midpoints of the two opposite sides $AB$ and $CD$ equals $\frac{|AD| + |BC|}{2}$. Prove that $AD$ is parallel to $BC$.
[b]p4.[/b] Solve the equation: $\frac{1}{\cos x}+\frac{1}{\sin x}= 2\sqrt2$.
[b]p5.[/b] Long, long ago, far, far away there existed the Old Republic Galaxy with a large number of stars. It was known that for any four stars in the galaxy there existed a point in space such that the distance from that point to any of these four stars was less than or equal to $R$. Master Yoda asked Luke Skywalker the following question: Must there exist a point $P$ in the galaxy such that all stars in the galaxy are within a distance $R$ of the point $P$? Give a justified argument that will help Like answer Master Yoda’s question.
[b]p6.[/b] The Old Republic contained an odd number of inhabited planets. Some pairs of planets were connected to each other by space flights of the Trade Federation, and some pairs of planets were not connected. Every inhabited planet had at least one connections to some other inhabited planet. Luke knew that if two planets had a common connection (they are connected to the same planet), then they have a different number of total connections. Master Yoda asked Luke if there must exist a planet that has exactly two connections. Give a justified argument that will help Luke answer Master Yoda’s question.
PS. You should use hide for answers.