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

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Let $T$ be the set of all positive integer divisors of $2004^{100}$. Size of largest subset $S$ of $T$ such that no element in $S$ divides another?

I am getting an answer slightly over $100^2$. Is this right (working below), or is there a better way of selecting elements of S? The following question appeared on the 2004 Canada National Olympiad: ...
3
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1answer
41 views

Let $S$ be a set of $n$ points in the plane with min spacing of 1. Prove $S$ has a subset of $\ge n/7$ points with min spacing of $\sqrt{3}$.

I believe I have proven the case $n=8,|T|=2$, but welcome feedback. I need help proving the case for general $|T|>2$. From the 2003 Canada National Olympiad: Let $S$ be a set of $n$ points in ...
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3answers
69 views

Show that $(1+p/n)^n$ is a Cauchy sequence for arbitrary $p$

It is a generalization of this question. I am looking for a similar derivation as in here. Can we prove that $(1+p/n)^n$ is a Cauchy sequence for any $p \in [a, b]$ by showing that $$ \Bigg| \left( ...
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2answers
29 views

How to show that this interesting difference of products is $O \left( \frac{1}{n^2} \right) $

Let $k \leq n$. Consider the following difference of products: $$ \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n+1} \right) - \prod_{i=1}^{k-1} \left( 1 - \frac{i}{n} \right)$$ For $n=1,2,3$, this is ...
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2answers
33 views

Given 4 points with 2 on different radius. Obtain the center of the circle.

I'm struggle on a math question that states the following: Black holes have an overwhelming gravity, such that the nearest stars begin spinning around them (Example). Every affected star keeps ...
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0answers
35 views

Can you verify the combinatoric recurrence?

There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical. ...
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1answer
47 views

AMT - Three whole numbers add up to 149 and multiply to give 987. What is the largest of the three number

So about this question I'm not too sure... Can't find out what I should start off with. If anyone can help me I'll be very greatly appreciated. The question is: Three whole numbers add up to 149 ...
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0answers
33 views

Additional explanation needed for the solution to a spesific sequence

Some of my attempts include trying to use the the term formula for geometric sequences and some other manipulations in hope of getting a more clearer, workable expression, though without success. ...
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3answers
52 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
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2answers
183 views

Finding all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$

Question: Find all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$. The notation $\lceil x \rceil$ means: The least integer which is not less than $x$. My ...
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2answers
63 views

Find the maximum value of the fraction

Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime ...
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2answers
20 views

ind $\tan \alpha$ in the square

let say the square has sides of 2 units, $DM = DN = AN = AP = 1$, $NP = \sqrt 2$, $NQ = QP = \frac{\sqrt 2}{2}$, and $AR \ne AP$ (?) we have know that $\tan \alpha = \frac 2{RP}$, but what's the ...
2
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2answers
78 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
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3answers
39 views

Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. The greatest possible value of $ a^3 + b^3 + c^3$ can be written in the form $ ...
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6answers
78 views

find x in $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$

Which one satisfies the equation $\sqrt[3]{6+\sqrt x} + \sqrt[3]{6-\sqrt x} = \sqrt[3] {3}$ (A)$27$ (B)$32$ (C)$45$ (D)$52$ (E)$63$ let $a = 6+\sqrt x , b=6-\sqrt x$ cube each side ...
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4answers
70 views

Find $x$ in the triangle

the triangle without point F is drawn on scale, while I made the point F is explained below So, I have used $\sin, \cos, \tan$ to calculate it Let $\angle ACB = \theta$, $\angle DFC = \angle ...
2
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3answers
70 views

Values of the sums $\sum\limits_{k=1}^{n}\cos^4(πk/(2n+1))$

I have been given a question which asks you to prove that $$ \sum_{k=1}^{n}\cos^4\left(\frac{πk}{2n+1}\right)=\frac{6n-5}{16} $$ The main problem I have with solving this is that since the summands ...
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4answers
58 views

remainder of $a^2+3a+4$ divided by 7

If the remainder of $a$ is divided by $7$ is $6$, find the remainder when $a^2+3a+4$ is divided by 7 (A)$2$ (B)$3$ (C)$4$ (D)$5$ (E)$6$ if $a = 6$, then $6^2 + 3(6) + 4 = 58$, and $a^2+3a+4 ...
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2answers
88 views

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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1answer
24 views

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 ...
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4answers
93 views

coefficient of $x^{17}$ in the expansion of $(1+x^5+x^7)^{20}$

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. find the coefficient of $x^{17}$ in the expansion of ...
2
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4answers
132 views

find $\left( \frac{x}{x+y} \right)^{2007} + \left( \frac{y}{x+y} \right)^{2007}$

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. if $x, y$ are non-zero numbers satisfying $x^2 + xy + ...
3
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6answers
125 views

evaluate $\frac 1{1+\sqrt2+\sqrt3} + \frac 1{1-\sqrt2+\sqrt3} + \frac 1{1+\sqrt2-\sqrt3} + \frac 1{1-\sqrt2-\sqrt3}$ [on hold]

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?
3
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3answers
95 views

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 ...
2
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1answer
61 views

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 ...
3
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1answer
42 views

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 ...
2
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2answers
87 views

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 ...
2
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2answers
66 views

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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3answers
55 views

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 ...
8
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1answer
106 views

Evaluate $a^2+b^2+c^2$

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. If $a, b, c$ are distinct numbers such that $a^2 - bc = ...
0
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2answers
51 views

given 3 circles, find relation of the regions

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. I had no idea how to find it nor where to start Note ...
5
votes
2answers
106 views

given $2f(x) + f(1-x) = x^2$ find $f(-5)$

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. A function $f$ has property that $2f(x)+ f(1-x) = x^2$ ...
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1answer
64 views

(Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and

The following is a Putnam math competition problem: Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and $ \int_{1}^{3}f(x)dx = 0 $. What is the max value of ...
3
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0answers
26 views

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 ...
0
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1answer
32 views

Number of ways to invite people to dinner

So I have this maths contest problem which goes like this: Alfred has seven friends (we'll call them A,B,C,D,E,F and G). Each night for a week/7 days he can invite any group of three friends over to ...
3
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1answer
71 views

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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5answers
172 views

Find $x$ if $\frac {1} {x} + \frac {1} {y+z} = \frac {1} {2}$ [on hold]

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} ...
2
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2answers
48 views

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 ...
4
votes
4answers
80 views

$\frac {1} {ab} + \frac {1} {ac} + \frac {1} {ad} + \frac {1} {bc} + \frac {1} {bd} + \frac {1} {cd}$

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. given $$ \frac {1} {a} + \frac {1} {b} + \frac {1} {c} + ...
3
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3answers
88 views

$ x^2 + \frac {x^2}{(x-1)^2} = 2010 $

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. Given $$ x^2 + \frac {x^2}{(x-1)^2} = 2010,$$ find ...
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3answers
65 views

Number theory with binary quadratic

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. Given $$ \frac {x^2-y^2+2y-1}{y^2-x^2+2x-1} = 2$$ find ...
3
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2answers
52 views

Maximum and minimum of a sum involving floor functions of rational numbers (contest question)

This question originates from the 1996 Canada National Olympiad. Let $r_1, r_2, \dots, r_m$ be a given set of $m$ positive rational numbers such that $\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$ ...
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2answers
66 views

How to solve “ways of seating around a circular table”

Recently I asked a question about seating, here it is again: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five ...
3
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0answers
76 views

A sequence of polynomials [duplicate]

I posted this question a while back, and I think I may not have made myself clear or shown what I got so far. So let me post this question again with more information and clarification. I have a ...
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2answers
73 views

How many ways to arrange the seating?

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with ...
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0answers
69 views

A Summation Challenge

I am trying to understand the solution of problem from its editorial by djdolls' answer,I am not able to understand a particulare step which is as follows: $$S(n)=\sum_0^D (-1)^i \cdot ...
2
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1answer
27 views

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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2answers
47 views

Inequalities based on square is always positive [closed]

Let $a$ and $b$ be real numbers such that $9a^2 + 8ab + 7b^2 \leq 6$. Prove that $7a + 5b + 12ab \leq 9$.
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2answers
72 views

Olympiad inequality

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 $\frac{1}{2}$ ...
5
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1answer
178 views

$\sum_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge 1$, for all $x_i>0$

Can you help with the following inequality? I found it experimentally. Prove that, for all $x_1,x_2,\ldots,x_n>0$, $$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod _{j=1}^nx_j}} \ge ...