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

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0
votes
0answers
30 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. ...
1
vote
2answers
39 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 ...
6
votes
2answers
176 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 ...
4
votes
2answers
62 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 ...
0
votes
2answers
16 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
votes
2answers
39 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. ...
0
votes
3answers
36 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 $ ...
4
votes
6answers
73 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 ...
1
vote
3answers
57 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
votes
3answers
66 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 ...
1
vote
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 ...
4
votes
2answers
86 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 ...
0
votes
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 ...
5
votes
4answers
88 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
votes
4answers
130 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
votes
6answers
124 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?
2
votes
3answers
88 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
votes
1answer
59 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
votes
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
votes
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 ...
1
vote
2answers
62 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 ...
1
vote
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
votes
1answer
105 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
votes
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
105 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 $2f(x)+ f(1-x) = x^2$ ...
1
vote
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
votes
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
votes
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
votes
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 ...
1
vote
5answers
170 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
votes
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
77 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
votes
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 ...
1
vote
3answers
64 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
votes
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 $\sum\limits^{m}_{k=1}{r_k} = 1 \tag{1}$ ...
2
votes
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
votes
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 ...
1
vote
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 ...
1
vote
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
votes
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$?
-2
votes
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 $7a + 5b + 12ab \leq 9$.
0
votes
2answers
71 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
votes
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 ...
4
votes
0answers
46 views

Find all integers $a,b,c$ [duplicate]

This question comes from the 2007 IMO shortlist: Find all integers $a,b,c$ such that $ab-c$, $bc-a$ and $ca-b$ are powers of two (of the form $2^k$ where $k \geq0$). What are some methods of ...
0
votes
0answers
40 views

Interested in Math Olympaids [closed]

I study Advanced Mathematics in University and am interested in Math Olympaids. Where can I start to build the fundamental skills to solve some questions?
1
vote
0answers
61 views

Graph Theory number of handshakes of couples

This is an Olympiad question which I now know the answer to, but I am a bit unsatisfied with it. So maybe someone can shed some light: Question: $5$ couples go to a party. Each person shakes the ...
0
votes
1answer
32 views

Find all real solutions to the following system of equations (involving fixed point iteration)

From the 1996 Canada National Olympiad. I have emphasised the real point of the question. Find all real solutions to the following system of equations. Carefully justify your answer. ...
3
votes
1answer
50 views

Largest sum of compatible triples

A triple $(a,b,c)$ of distinct integers is called compatible if at least one of them, say $b$ has the property that either $n\mid b$ or $b\mid n,$ for each $n\in\{a,c\}.$ Let $X$ be the set of all ...
1
vote
2answers
64 views

How many such polynomial exist?

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. I got: $$P(x) = ax^2 + bx + c \implies P(0) = c = 2010$$ Let $P(r_1, r_2) ...
1
vote
1answer
44 views

Probability of not making a shoe pair.

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The ...