Found problems: 15925
PEN P Problems, 21
Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.
2012 Kosovo National Mathematical Olympiad, 4
Let $x,y$ be positive real numbers such that $x+y+xy=3$. Prove that $x+y\geq 2$. For what values of $x$ and $y$ do we have $x+y=2$?
2007 China Girls Math Olympiad, 7
Let $ a$, $ b$, $ c$ be integers each with absolute value less than or equal to $ 10$. The cubic polynomial $ f(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ satisfies the property
\[ \Big|f\left(2 \plus{} \sqrt 3\right)\Big| < 0.0001.
\]
Determine if $ 2 \plus{} \sqrt 3$ is a root of $ f$.
2008 Iran MO (3rd Round), 1
Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3\plus{}1)\neq0$.
2019 Saudi Arabia JBMO TST, 2
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$
2002 Turkey MO (2nd round), 3
Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition:
For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$
\[a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i| \leq d, \quad \text{and} \quad |a_i - c_i| \leq d.\]
2010 CHMMC Fall, 9
Let $a_0, a_1, . . . ,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{342} (1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{341} =\sum^{n}_{i=0}a_ix^i.$$ Compute the number of odd terms in the sequence $a_0, a_1, . . . ,a_n$.
2015 Iran MO (2nd Round), 3
Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.
2020-2021 OMMC, 1
A man rows at a speed of $2$ mph in still water. He set out on a trip towards a spot $2$ miles downstream. He rowed with the current until he was halfway there, then turned back and rowed against the current for $15$ minutes. Then, he turned around again and rowed with the current until he reached his destination. The entire trip took him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph where $p,q$ are relatively prime positive integers. Find $10p+q$.
1989 IberoAmerican, 1
Determine all triples of real numbers that satisfy the following system of equations:
\[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]
2018 European Mathematical Cup, 3
For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at
least $3$ elements such that
$$k(a - b)\in S$$
for all $a,b\in S $ with $a > b?$
Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that
$x < M$ for all $x \in S.$
2017 Middle European Mathematical Olympiad, 1
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$$f(x^2 + f(x)f(y)) = xf(x + y)$$
for all real numbers $x$ and $y$.
2013 District Olympiad, 4
At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2, 3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?
Albania Round 2, 2
Sides of a triangle form an arithmetic sequence with common difference $2$, and its area is $6 \text{ cm }^2$. Find its
sides.
EGMO 2017, 2
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
1999 Tuymaada Olympiad, 2
Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection?
[i]Proposed by K. Kokhas[/i]
2019 Canadian Mathematical Olympiad Qualification, 1
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$.
Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$
2020 Taiwan TST Round 3, 1
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
2010 Purple Comet Problems, 23
A disk with radius $10$ and a disk with radius $8$ are drawn so that the distance between their centers is $3$. Two congruent small circles lie in the intersection of the two disks so that they are tangent to each other and to each of the larger circles as shown. The radii of the smaller circles are both $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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2000 Iran MO (3rd Round), 3
Suppose $f : \mathbb{N} \longrightarrow \mathbb{N}$ is a function that satisfies $f(1) = 1$ and
$f(n + 1) =\{\begin{array}{cc} f(n)+2&\mbox{if}\ n=f(f(n)-n+1),\\f(n)+1& \mbox{Otherwise}\end {array}$
$(a)$ Prove that $f(f(n)-n+1)$ is either $n$ or $n+1$.
$(b)$ Determine$f$.
2013 Israel National Olympiad, 6
Let $x_1,...,x_n$ be positive real numbers, satisfying $x_1+\dots+x_n=n$. Prove that
$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\leq\frac{4}{x_1\cdot x_2\cdot\dots\cdot x_n}+n-4$.
2023 LMT Fall, 21
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$
[i]Proposed by Muztaba Syed[/i]
1963 IMO, 1
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
2016 Azerbaijan JBMO TST, 1
If $ a,b,c $ represent the lengths of the sides of a triangle, prove the inequality:
$$ 3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} . $$
2011 Bogdan Stan, 2
Let be a natural number $ n\ge 3 $ and a number $ \alpha $ from the interval $ (0,1). $
Find all $ \text{n-tuples} $ of real numbers $ \left( x_0,x_1,x_2,\ldots, x_{n-1} \right) $ such that $ x_0=x_n,x_1=x_{n+1} $ and
$$ x_{k+1}\le \left( 1-\alpha \right) x_k+\alpha x_{k-1}, $$
for all $ k $ in the set $ \{ 1,2,\ldots n \} . $
[i]Vasile Pop[/i]