Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions.

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28 views

Proof of inequality by Muirhead

We have to prove: $$\frac{\sqrt{pq}}{p+q+2r}+\frac{\sqrt{pr}}{p+r+2q}+\frac{\sqrt{pr}}{p+r+2q}\leq\frac{3}{4}$$ By multiplying it all out we get the following equivalent: \begin{align*} 4\sum_{cyc}{...
0
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1answer
47 views

Putnam Pigeonhole principle problem regarding a 20 person college and 6 courses

Question: A certain college has 20 students and offers 6 courses. Each student can enroll in any or all of the 6 courses, or none at all. Prove or disprove: there must exist 5 students and 2 courses, ...
5
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7answers
51 views

Polynomial Big List: Find the polynomial whose roots are given by some functions of the roots of given polynomials.

I would like to create a compilation about polynomials for future reference. The aim is to capture some scenarios that appear in many exams and contests. The five scenarios I have thought of are ...
4
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1answer
116 views

IMO $2001$ problem $2$

Let $a,b,c \in \mathbb{R}$. Prove that $$\frac{a}{\sqrt{a^2+8bc}} + \frac{b}{\sqrt{b^2+8ca}}+ \frac{c}{\sqrt{c^2+8ab}} \geqslant 1.$$ I tried to follow the proposed solution for this which depended ...
0
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0answers
43 views

Geometry question from IMO 2009

(IMO 2009/2). Let $ABC$ be a triangle with circumcenter $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$, respectively. Let $K$, $L$, and $M$ be the midpoints of the segments ...
0
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0answers
37 views

Solution verification: chess board all squares white except one

Problem: On a 8x8 chessboard all squares are the color white, except for one square (which is black). Show that you can't get to the situation that all squares are white by only recoloring whole rows ...
1
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0answers
35 views

Find all odd integers $n > 1$, such that for any relatively prime divisors $a$ and $b$ of $n$, the number $a+b-1$ is also a divisor of $n.$ [duplicate]

Find all odd integers $n$ greater than $1$, such that for any relatively prime divisors $a$ and $b$ of $n$, the number $a+b-1$ is also a divisor of $n.$ I found this online on a Russian Olympiad exam ...
1
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0answers
26 views

Proof verification: circle pawn problem

Problem: A circle is divided in 6 sectors by 3 diameters. Each sector contains a pawn. We are allowed to chose two pawns and move each of them to a sector bordering the one it stands on at the moment. ...
2
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1answer
83 views

USAMO 2018 functional equation

Find all functions $f : (0, ∞) → (0, ∞)$ such that $f(x +\frac{1}{y})+ f(y +\frac{1}{z})+ f(z + \frac{1}{x})= 1$ for all $x, y, z > 0$ with $xyz = 1$. Alright so my main question is that i first ...
2
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0answers
91 views

Number theory question from 2017 MO

(Asian-Pacific Olympiad $2017$). Let $a, b, c$ be positive rational numbers with $abc = 1$. Suppose there exist positive integers $x, y, z$ for which $a^x +b^y +c^z$ is an integer. Prove that when $a, ...
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0answers
19 views

Problem: Sharing Software With 1 Person then 2 Person Worth $3000 [closed]

Person A bought an Education Website Application worth $\$3000$. Later, his friend person B wants to share the Web Application with person A. So, person B pays $\$1500$ to person A. Later Person C ...
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1answer
62 views

find f(1) in the following function $f(t^2) = f(t)^2$

If $f(t^2) = f(t)^2$ then find f(1)?. I've tried but it wasn't enough to solve your problem, I couldn't.
2
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2answers
63 views

How to prove that $ABCD$ is a parallelogram?

Let $ABCD$ be a quadrilateral. Let $E$ and $F$ be midpoint of $AB$ and $BC$. The lines $DE$ and $DF$ intersect $AC$ at $M$ and $N$ respectively. Suppose that $AM$ $=$ $MN$ $=$ $MC$. Prove that $ABCD$ ...
3
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3answers
81 views

If $a+b+c=7$ and $\frac1{a+b}+\frac1{b+c}+\frac1{c+a}=\frac7{10}$, then evaluate $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$. [duplicate]

Let $a$, $b$ and $c$ $\in \mathbb{Q}$ such that $a+b+c=7$ and $\cfrac1{a+b}+\cfrac1{b+c}+\cfrac1{c+a}=\cfrac7{10}$ What does $\cfrac a{b+c}+\cfrac b{c+a}+\cfrac c{a+b}$ equal to? The final result is $...
0
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1answer
108 views

A seemingly impossible combinatorics problem!

The problem: If n, m equally spaced straight lines, mutually perpendicular to each other, bounds a rectangle, in which another rectangle is chosen and shaded with a minimum side length of $[\frac{n}{2}...
2
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1answer
46 views

Inequality on reciprocal of squares

Please verify. Is this a correct proof for the following inequality on any $m \in \mathbb{N}$: $$\sum_{i=m}^{\infty} \frac{1}{i^2} \leq \frac{2}{2m-1}$$ Proof: Instead, let's consider an equivalent ...
3
votes
1answer
177 views
+100

We color each unit square of a table $10\times 10$ with one color so that …

A table $10\times 10$ is divided in $100$ unit squares. We color each unit square with one color so that no column or row contains more than $5$ colors. How many colors can we use at most? Any idea ...
1
vote
1answer
64 views

For any positive integers $m$ and $n$, show that $\left(\frac{1}{n+1} \right)^{\frac{1}{m}}+\left(\frac{1}{m+1} \right)^{\frac{1}{n}} \ge 1$ [duplicate]

I'm asked to show that for any positive integers $n, m$ $$\left(\frac{1}{n+1} \right)^{\frac{1}{m}}+\left(\frac{1}{m+1} \right)^{\frac{1}{n}} \ge 1$$ I could not think of any way.
3
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1answer
103 views

Determine the number of ways to go from $(1,1)$ to $(n,1)$ on a chessboard

Problem: Let $S$ be a $n \times 3$ chessboard. Let a rook walk on the board, it is allowed to move $1$ step horizontally or vertically every step. Determine the number of ways the rook can go from the ...
3
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4answers
133 views

How do I find the integer solutions that satisfy $xyz = 288$ and $xy + xz + yz = 144$?

Find all integers $x$, $y$, and $z$ such that $$xyz = 288$$ and $$xy + xz + yz = 144\,.$$ I did this using brute force, where $$288 = 12 \times 24 = 12 \times 6 \times 4$$ and found that these set of ...
1
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2answers
49 views

Prove that $n !$ is a divisor of $ \prod_{k=0}^{n-1}\left(2^{n}-2^{k}\right) $

Prove that for any natural number $n, n !$ is a divisor of $ \prod_{k=0}^{n-1}\left(2^{n}-2^{k}\right) $ i have already seen it here $\prod_{i=0}^{n-1}(2^n-2^i)$ can be divided by $n!$ but my doubt ...
2
votes
1answer
31 views

Proof verification of a number theory problem involving sequences.

$\textbf{Question:}$Does there exist an infinite sequence of integers $a_1, a_2, . . . $ such that $gcd(a_m, a_n) = 1 $ if and only if $|m - n| = 1$? $\textbf{My solution:}$Suppose we have a $n$ ...
-1
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0answers
26 views

Please help me with this question. [closed]

There are 21 problems in mathematics competition. The scores of each problem are located in the following ways: three marks for correct answer, minus two marks for incorrect answer and zero marks for ...
2
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2answers
122 views

Handshaking puzzle

Problem: At a party there are $n$ people, some people give each other a hand. After the party everyone writes down on a piece of paper how many people he/she shook hands with. It turns out there are $...
2
votes
1answer
68 views

Range of $f(x)=\frac{(x+a)^2}{(a-b)(a-c)}+\frac{(x+b)^2}{(b-a)(b-c)}+\frac{(x+c)^2}{(c-a)(c-b)}$

Consider $$f(x)=\frac{(x+a)^2}{(a-b)(a-c)}+\frac{(x+b)^2}{(b-a)(b-c)}+\frac{(x+c)^2}{(c-a)(c-b)}$$ (where $a,b,c$ are distinct real numbers). If $p$ denotes the number of natural numbers in the range ...
2
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1answer
70 views

Geometry solution involving complex numbers from USAMO

Quadrilateral $AP BQ$ is inscribed in circle $ω$ with $∠P = ∠Q = 90◦$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $P Q$. Line $AX$ meets $ω$ again at $S$ (other than $A$). Point $T$ ...
1
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2answers
60 views

Coloring the numbers 1 and to including 10 with constraint

The question: Consider the colors red, green, blue. In how many ways can we color the numbers 1 up to including 10 such that: 2 consecutive numbers dont have the same color odd numbers cant be red. ...
4
votes
0answers
61 views

Show that for all $n$ there exist some $n$-digit number with no $0$ in it whose digit sum divides it.

$\textbf{Question:}$Prove that for each positive integer $n$, there exists a positive integer with the following properties: • it has exactly $n$ digits, • none of the digits is $0$, • it is divisible ...
14
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2answers
509 views

Prove that $\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$ is an integer.

Prove that $$\frac{(3 a+3 b) !(2 a) !(3 b) !(2 b) !}{(2 a+3 b) !(a+2 b) !(a+b) ! a !(b !)^{2}}$$ is an integer for all pairs of positive integers $a, b$ (American Mathematical Monthly) My work - $ v_{...
1
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2answers
184 views

Find all functions which satisfy $f(m^2+n^2)=f(m)^2+f(n)^2$ $\forall\space m,n\in\Bbb{N}$ and $f(1)>0$

Remark. Dear voters, this question is not a duplicate of the following old question. Please refrain from closing it for being a duplicate. QUESTION: Find all functions $f:\Bbb{N}→\Bbb{N}$ which ...
0
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0answers
54 views

Is there a nice way to represent $\prod_{m=1}^{2^{n-1}} \cos\left(2^{m-1} x\right)$?

Does $$\prod_{m=1}^{2^{n-1}} \cos\left(2^{m-1} x\right) \stackrel{?}{=} \frac{\sin\left(2^nx\right)}{2^n\sin x}?$$ I want a more rigorous way to derive this than my approach: Using that $\sin(2x)= 2\...
1
vote
3answers
74 views

Olympiad inequality proof issue

Prove that $(a^2+b^2)^2\geq(a+b+c)(a+b-c)(b+c-a)(c+a-b)\ \forall \ a,b,c\in\mathbb{R^+} $. I, forgetting to consider whether $a_1$ and $a_2$ are strictly non-negative (don't think they are), found a ...
1
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2answers
69 views

Multiplication using pi notation problem

$$ \prod _{n=4}^{\infty}\frac{(n^3+2n)^2}{(n^2-4)(n^4+2n^2+9)} $$ I am not understanding this question.I think the measure is decreasing while $n$ is increasing. So it will be $0$ at last. But that ...
2
votes
1answer
96 views

Find when $\frac{x^5-1}{x-1}$ is a perfect square?

$\textbf{Question:}$Find when $f(x)=\frac{x^5-1}{x-1}$ is a perfect square? where $x \in \mathbb N/ \{1\}$. I tried upto certain number and somewhat convinced that $3$ is the only solution.But I ...
-2
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1answer
8 views

in The rectangle ABCD the ratio DN:NC = 1:4 and BM=MC, find the ratio NS:SR:RB

in Rectangle ABCD, find the length ratio of NS:SR:RB enter image description here
3
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1answer
71 views

Ascending order of $3^a-3^b+3^c-3^d+3^e$.

Let $S$ be set of all numbers of the form $3^a-3^b+3^c-3^d+3^e$ where $a>b>c>d>e>0$ are all natural numbers. If the elements of $S$ are arranged in ascending order, find the $20^{\text{...
2
votes
0answers
69 views

What is the minimum number of doors to a laboratory that would satisfy these conditions?

$S$ scientists are working in a lab that they want to keep secure. They want to install $D$ doors to the lab, such that each door has $L$ locks that each require a different key to open. The ...
3
votes
2answers
163 views

solving diophantine problem $a^3+b^3=2019(1+ab)$ for coprime $a$ and $b$

We're interested in solving $$\begin{cases} a^3+b^3=2019(1+ab) \\ \gcd(a,b)=1 \end{cases}$$ I'm stuck with the deduction Say why $a^3 \equiv -b^3 \pmod{2019}$ . (done) Using Fermat, prove that $a^{...
3
votes
1answer
54 views

$\frac{x^2}{by+cz}=\frac{y^2}{cz+ax}=\frac{z^2}{ax+by}=2$

If $$\frac{x^2}{by+cz}=\frac{y^2}{cz+ax}=\frac{z^2}{ax+by}=2$$ then find the value of $$\frac{c}{2c+z}+\frac{b}{2b+y}+\frac{a}{2a+x}.$$ I think all the terms need to be manipulated in some way to get ...
-2
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1answer
54 views

Find LCM $(A_0,A_1,…,A_{1999})$ where $A_n=2^{3n}$+$3^{(6n+2)}$+$5^{(6n+2)}$ ; n=(0,…,1999) [closed]

$\textbf{Question:}$ Find $LCM (A_0,A_1,\cdots,A_{1999})$ where $A_n=2^{3n}$+$3^{(6n+2)}$+$5^{(6n+2)}$ ; $n=0,…,1999$.
5
votes
4answers
99 views

Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3 +\dots+ n^3$ is divided by $n+5$ the remainder is $17.$

Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3 +\dots+ n^3$ is divided by $n+5$ the remainder is $17.$ Letting $k= n+5$ we get that $1^3+2^3+3^3 +\dots+ (k-5)^3 \equiv 17 \text{...
3
votes
0answers
102 views

Finding $t_{2020}$ when $t_n = \frac{5t_{n-1} + 1}{25t_{n-2}}$

Define a sequence recursively by $t_1 = 20, t_2=21$ and $$t_n = \frac{5t_{n-1} + 1}{25t_{n-2}}$$ for all $n \geqslant 3.$ Then $t_{2020}$ can be written as $\frac{p}{q},$ where $p$ and $q$ are ...
0
votes
1answer
91 views

$1995$ Hungarian Olympiad Number theory Problem

Let k,n be positive integers such that $(n+2)^{n+2}, (n+4)^{n+4}, (n+6)^{n+6}, ..., (n+2k)^{n+2k}$ end in the same digit in decimal representation. At most how large is k? i seen same question here ...
2
votes
1answer
91 views

Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$

$\textbf{Question:}$Prove that there exist infinitely many primes $p$ such that $13 \mid p^3+1$ I could easily see that the given is equivalent to showing that there are infinitely many primes $p$ ...
2
votes
2answers
68 views

$1996$ Austrian-Polish Number theory problem

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all ...
3
votes
1answer
57 views

number theory question from USAMO 2010 preparation session

Prove that for any natural number k, there exists a natural number n such that n has exactly k different prime factors and $2^{n^{2}}+ 1$ is divisible by $n^3$. Below i present my attempt. PLease ...
2
votes
2answers
105 views

Geometry question: Find the area of blue-shared area inside this isosceles

See below, looks a bit interesting, but I cannot find a solution. I think a starting point might be the similarity of the lower white triangle and the larger triangle composed of lower blue, pink, and ...
3
votes
2answers
95 views

Functional equation from USAMO 2010 preparation session

Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that $(1 + yf(x))(1 − yf(x + y)) = 1$ for all $x, y \in \mathbb R^+$, where $\mathbb R^+$ is a set of all positive real numbers. Well I don't ...
2
votes
1answer
94 views

If $x+y+z=1$, prove that $9xyz+1\ge 4(xy+yz+zx)$

If $x+y+z=1$, prove that $9xyz+1\ge 4(xy+yz+zx)$ for $x,y,z\in \Bbb R^+$ I tried to solve this by splitting $9xyz$ as $3xyz+3xyz+3xyz$ and taking all the terms to the LHS before factoring, but I ...
0
votes
1answer
64 views

In a math competition with $8$ students and $8$ problems, if each problem is solved by $5$ students, then two students together solve all problems. [closed]

Eight students are entered in a math competition. They all have to solve the same set of $8$ problems. After correction, we see that each problem was correctly resolved by exactly $5$ students. Show ...

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