# Questions tagged [contest-math]

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

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### 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}{...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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(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, ... 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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### 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.
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### 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 $ABCD$ ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n$ ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $1995$ 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 ...
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### 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 $p$ ...
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### $1996$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...
### 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 ...