# Tagged Questions

Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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### Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)

The question below is from the 2002 Canada National Olympiad. I have found one family of functions but need help in finding (or proving the non-existence) of others. Suggestions on how to improve the ...
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### Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, ...
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### evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$

Evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$ How to evalute this equation without using calculator?
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### High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least ...
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### Inequality - Cauchy Schwarz

Let $a, b, c, d > 0 \in \mathbb{R}$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Show that: $S = \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq 4$ My approach: I used the Cauchy-...
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### Choose 8 distinct integers from $\{1, 2,\dots,16,17\}$. Show that the eight contain at least three pairs with a common difference for _any_ choice.

This problem is from the 1999 Canada National Olympiad. I am stuck trying to prove the first part using the pigeonhole principle. Is there a refinement that will allow it to be used more sharply, or ...
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### Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
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### Triangle Geometry and Circles Problem

I have discovered something using Geogebra and I am positive it is true. I have tried to prove and my solution works but it is extremley convoluted. I'm hoping someone can provide a simple proof of ...
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### find total integer solutions for $(x-2)(x-10)=3^y$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. How many integer solutions ($x$, $y$) are there of the ...
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### Show that ordered pairs are solutions to an equation if and only if they are consecutive elements of a recursive sequence (contest question)

The following question appeared on the 1998 Canada National Olympiad. I need help proving that the only solutions to the equation are consecutive elements of the recursively-defined sequence. I ...
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### Minimize Value of Function with Constrain

Let $x$ and $y$ be real number with $xy\neq-1$ and $$\frac{x^7y+xy^7}{1+x^5y^5}=4$$ What is the minimum value of $x^2+y^2?$ I've been trying to solve it by Lagrange Multiplier but it's getting ...
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### Find $x$ if $\frac {1} {x} + \frac {1} {y+z} = \frac {1} {2}$ [closed]

I found this question from past year's maths competition in my country. I've tried any possible way to find it, but it is just way too hard. Find $x$ if \begin{align}\frac {1} {x} + \frac {1} {y+z}...
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### What is $k$ so that $\frac {1001\times 1002 \times … \times 2008} {11^k}$ will be an integer?

I found this question from last year's maths competition in my country. I've tried any possible way to find it, but it is just way too hard. What is the largest integer $k$ such that the following ...
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### How find all finite sets $M$ such that $|M|\ge 2$ and $\frac {2a}{3} - b^2\in M$ for all $a,b\in M$

How find all finite sets of real numbers $M$ such that $|M|\ge 2$ and $\frac {2a}{3} - b^2\in M$ for all $a,b\in M$?
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I want to prove that for $a, b, c > 0$ we have $\frac{a}{2a + b} + \frac{b}{2b + c} + \frac{c}{2c + a} \leq 1$. My approach: I know that each of the individual terms is lesser than $\frac{1}{2}$ ...
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### How many possible guesses?

A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $1$ to $9999$ inclusive. The contestant wins the prizes by correctly guessing the price ...
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### Prove that $(\sqrt2 − 1)^n, \forall n \in \mathbb{Z^+}$ can be represented as $\sqrt{m} − \sqrt{m−1}$ for some $m \in \mathbb{Z^+}$ (no induction).

From the 1994 Canada National Olympiad: Prove that $(\sqrt2 − 1)^n, \forall n \in \mathbb{Z^+}$ can be represented as $\sqrt{m} − \sqrt{m−1}$ for some $m \in \mathbb{Z^+}$. I think one ...
$ABCD$ is a rectangle, $P$ is the midpoint of $AB$, and $Q$ is the point on $PD$ such that $CQ$ is perpendicular to $PD$. Prove that the triangle $BQC$ is isosceles. Clearly, we need to prove that $... 1answer 125 views ### How many ways are there to shake hands? In a group of$9$people, each person shakes hands with exactly$2$of the other people from the group. Let$X$be the number of possible ways to perform these handshakes. Take$2$handshake patterns (... 1answer 87 views ### Is it possible to choose$10$distinct numbers from the set$\{0, 1, 2, . . . , 14\}$so that various differences are all distinct? From the 1991 Canada National Olympiad: Can ten distinct numbers$a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$be chosen from$\{0, 1, 2, \dotsc, 14\}$so that the$14$differences $$\... 1answer 190 views ### Cover the grid graph with simple cycles I have a two dimensional n x m grid graph. And I want to find in how many ways this grid can be covered with simple cycles (it can be a one cycle or it can be many ... 1answer 67 views ### Consider the 1000-element subsets Consider all 1000-element subsets of the set A = \{ 1, 2, 3, ... , 2015 \}. From each such subset choose the least element. The arithmetic mean of all of these least elements is \frac{p}{q}, where ... 1answer 72 views ### Infinite number of ways to write 1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k} How can I show that there is an infinite number of ways in which 1 can be written in the form$$1=\frac{1}{n}+\frac{1}{a_1}+\cdots+\frac{1}{a_k},$$where$n>1$is an integer (this number is fixed)... 1answer 357 views ### IMO 2015 #1: “Balanced” and “Centre-Free” sets of points in the plane [closed] International Mathematical Olympiad 2015, Problem 1: We say that a finite set$S$of points in the plane is$\color{\red}{\text{balanced}}$if, for any two different points$A$and$B$in$S$, ... 1answer 43 views ### Unfairish Probability Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability$\frac{2}{3}$and each of the other five sides has probability$\frac{1}{...
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain either exactly two adjacent chairs or no adjacent chairs. Let $1$ be chair, and $0$ be an empty ...