Algebra, Equations
An equation is a statement that two mathematical expressions are equal. Solving an equation involves finding the values of variables that make the statement true. Questions cover various types: linear, quadratic, polynomial, rational, radical, and systems of equations.
Diophantine Equations-
Average of Averages 3
Consider the following diagram:
Each number in the diagram, connected to 4 other numbers, must be equal to their average:
Sources:
What is the number in the circle marked with a question mark? -
Cyclic Quadrilaterals
Given two triangles ACE, BDF
intersecting at 6 points: G,H,I,J,K,L
as shown in the figure. It is given that in each of the quadrilaterals
EFGI, DELH, CDKG, BCJL, ABIK a circle can be inscribed.
Is it possible that a circle can also be inscribed in quadrilateral FAHJ?
Sources:Topics:Geometry -> Solid Geometry / Geometry in Space Geometry -> Plane Geometry -> Circles Algebra -> Equations Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation- Gillis Mathematical Olympiad, 2019-2020 Question 5
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Quadratic Equations
Solve the equations:
a: `({x^2+6}/{x^2-4})^2=({5x}/{4-x^2})^2`
b: `7(x+1/x)-2(x^2+1/x^2)=9`
c: `sqrt{x+2sqrt{x-1}}-sqrt{x-2sqrt{x-1}}=2`
Sources: -
Relationships Between the Roots of Quadratic Equations
Given a quadratic equation `ax^2+bx+c=0` whose solutions are `x_{1,2}={-b+-sqrt{b^2-4ac}}/{2a}`.
A: Show that Vieta's formulas hold: `x_1x_2=c/a` `x_1+x_2=-b/a,`.
B: Express the following in terms of a, b, c: `1/{x_1^2}+1/{x_2^2}` `1/x_1+1/x_2, ` `x_1^3+x_2^3,` `x_1^2+x_2^2`.
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Enlarging a Rectangle
A rectangle is given in the plane. Is it possible that after each side of the rectangle is increased by 1 cm, the area increases by 1 square meter? Provide an example or prove that it is impossible.
(If the rectangle is 1x5, it becomes 2x6 and no side can be 0)
Sources:- Grossman Math Olympiad, 2017, Juniors Question 1
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Integer Expression
Find all integers n for which the expression `{(n+2)^4}/{n-1}` is defined and an integer.
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A POST-OFFICE PERPLEXITY
In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post-office when a gentleman entered and deposited a crown on the counter with this request: "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence-halfpenny stamps." For a moment she seemed bewildered, then her brain cleared, and with a smile she handed over stamps in exact fulfilment of the order. How long would it have taken you to think it out? Sources: -
AT A CATTLE MARKET
Three countrymen met at a cattle market. "Look here," said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that," said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here."
No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge, and Durrant must have taken to the cattle market.
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THE BEANFEAST PUZZLE
A number of men went out together on a bean-feast. There were four parties invited鈥攏amely, `25` cobblers, `20` tailors, `18` hatters, and `12` glovers. They spent altogether 拢`6, 13`s. It was found that five cobblers spent as much as four tailors; that twelve tailors spent as much as nine hatters; and that six hatters spent as much as eight glovers. The puzzle is to find out how much each of the four parties spent. Sources: -
A QUEER COINCIDENCE
Seven men, whose names were Adams, Baker, Carter, Dobson, Edwards, Francis, and Gudgeon, were recently engaged in play. The name of the particular game is of no consequence. They had agreed that whenever a player won a game he should double the money of each of the other players鈥攖hat is, he was to give the players just as much money as they had already in their pockets. They played seven games, and, strange to say, each won a game in turn, in the order in which their names are given. But a more curious coincidence is this鈥攖hat when they had finished play each of the seven men had exactly the same amount鈥攖wo shillings and eightpence鈥攊n his pocket. The puzzle is to find out how much money each man had with him before he sat down to play. Sources: