Algebra
Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.
Algebraic Techniques Equations Inequalities Word Problems Sequences-
Average of Averages 3
Consider the following diagram:
Each number in the diagram, connected to 4 other numbers, must be equal to their average:
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What is the number in the circle marked with a question mark? -
Numbers Squared
Ayala took all the numbers divisible by 3 from 1 to 99, inclusive, squared each of them, summed the results, and multiplied the resulting number by 2.
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The duck took all the numbers not divisible by 3 from 1 to 100, inclusive, squared each of them, and summed the results.
What is the difference between the number the duck obtained and the number Ayala obtained? -
How many solutions does the equation have?
Given the equation:
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`x^2+2xy+y^2-200x-200y+1900=0`
How many solutions (x,y) are there, where x and y
are integers from 1 to 100 (inclusive)? -
Mathematical Conference
202 participants from three countries attended a mathematical conference: Israel, Greece, and Japan.
On the first day, every pair of participants from the same country shook hands. On the second day, every pair of participants
where one was Israeli and the other was not Israeli shook hands. On the third day, every pair of participants where one
was Israeli and the other was Greek shook hands. In total, 20200 handshakes occurred. How many
Israeli participants were at the conference?Sources:Topics:Number Theory Combinatorics Algebra -> Word Problems Algebra -> Equations -> Diophantine Equations- Gillis Mathematical Olympiad, 2019-2020 Question 2
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Numbers on a Board
At the beginning of the day, four integers are written on the board (`a_0,b_0,c_0,d_0`). Every minute, Danny replaces the four numbers on the board with a new set of four numbers according to the following rule: If the numbers written on the board are (a,b,c,d), Danny first generates the numbers
`a'=a+4b+16c+64d`
`b'=b+4c+16d+64a`
`c'=c+4d+16a+64b`
`d'=d+4a+16b+64c`
Then he erases the numbers (a,b,c,d) and writes in their place the numbers (a',d',c',b'). For which initial sets (`a_0,b_0,c_0,d_0`) will Danny eventually write a set of four numbers that are all divisible by `5780^5780`Sources:Topics:Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences- Gillis Mathematical Olympiad, 2019-2020 Question 4
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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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The Number Circle
The numbers 1 through 6 are written in a circle in order, as shown in the figure.
In each step, Lior chooses a number a in the circle whose neighbors are b and c, and replaces
it with the number `(bc)/a`.
Can Lior reach a state where the product of the numbers in the circle is greater than `10^100`
(a) in 100 steps.
(b) in 110 steps
Sources:- Gillis Mathematical Olympiad, 2019-2020 Question 6
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Inscribed Circle in a Triangle
Inside a triangle there is a point P, whose distances from the lines containing the sides of the triangle are `d_a,d_b,d_c`. Let R denote the radius of the circumscribed circle of the triangle and r the radius of the inscribed circle in the triangle. Show that `sqrt(d_a)+sqrt(d_b)+sqrt(d_3)<= sqrt (2R+5r) `.
Sources:Topics:Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Circles Algebra -> Inequalities- Gillis Mathematical Olympiad, 2019-2020 Question 7
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Children's Clubs
In a kindergarten, there are three clubs: Judo, Agriculture, and Mathematics. Each child participates in exactly one club, and each club has at least one participant. The total number of children in the kindergarten is 32. On Friday, the kindergarten teacher gathered 6 children to tidy up the classroom. The teacher counted and found that exactly half of the Judo club members, a quarter of the Agriculture club members, and an eighth of the Mathematics club members volunteered for the task. How many students are in each club?
Sources:Topics:Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division- Gillis Mathematical Olympiad, 2018-2019 Question 1
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Shuki's Walk
Shuki walked for 3.5 hours. In every one-hour period, he walked 5 km. Does this mean that Shuki's average speed during the time he walked is 5 km/h?
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