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-
Question
By what factor is the sum of all numbers from 1 to 99 smaller than the sum of all numbers from 1 to 9999?
Sources:Topics:Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence -
Question
Grandma Hannah has many flowerpots with flowers in her house. One day, her three grandchildren, Avi, Beni, and Gili, came to visit and tried to guess how many flowerpots she has.
Sources:
Avi said: "Grandma has more than 8 flowerpots,"
Beni said: "More than 10, I think,"
And then Gili said: "Grandma Hannah has at least 12 flowerpots."
"Two of you are right, and one of you is wrong," answered the grandmother. So how many flowerpots does she have in the house? -
The Matchstick Root
Move one matchstick to make a correct equality:
Topics:Algebra -> Algebraic Techniques -> Roots / Radicals Puzzles and Rebuses -> Matchstick Puzzles -
Products of Areas
In the figure, there is a rectangle and a point inside it. Two segments are drawn through the point, parallel to the sides of the rectangle, dividing the rectangle into 4 smaller rectangles.
Prove that the product of the areas of the shaded rectangles inside the rectangle is equal to the product of the areas of the unshaded rectangles inside the rectangle.
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Sheep and Camels
Shmuel has a flock of 9 sheep and 5 camels. He wants to divide the flock between his two sons, Yossi and Danny, so that each of them receives an equal share of the value.
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It is known that 7 sheep cost as much as 3 camels. How can Shmuel divide the flock between his two sons equally, without selling any animals? -
Log of Wood
You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
邪. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
斜. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,Assuming you have no other measuring tools? Explain!
Sources:Topics:Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder -
School Supplies
An eraser and two pencils together cost 19 lira, a pencil and two notebooks together cost 23 lira, a notebook and two
erasers together cost 18 lira. How much do an eraser, a pencil, and a notebook cost together?Sources: -
Sum
In the following problem, different digits have been replaced with different letters, and the sum is a three-digit number.
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What is the sum if the number "BA" is as small as possible?
\(\begin{align} AB& \\ \underline{+\quad BA}& \\ ***& \end{align}\) -
Puzzle
You have a puzzle where each piece looks like this:
The following shape is made up of 12 pieces. What is its perimeter?
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Fibonacci Polynomial
The Fibonacci sequence is defined by `F_1 = F_2 = 1` and the recurrence relation `F_n = F_{n-1} + F_{n-2}` for every integer `n >= 3`. Given natural numbers `m,n >= 1`, find the minimum degree `d` such that there exists a polynomial `f(x) = a_d x^d + a_{d-1} x^{d-1} + ... + a_1 x + a_0` satisfying `f(k) = F_{m+k}` for all `k = 0,1,...,n`. Justify your answer.
Sources:Topics:Algebra -> Sequences -> Recurrence Relations- Gillis Mathematical Olympiad, 2015-2016 Question 5