Found problems: 85335
2020 Brazil National Olympiad, 4
A positive integer is [i]brazilian[/i] if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is [i]superbrazilian[/i] if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, but $561=484+77$ is not superbrazilian, because $561$ is not brazilian. How many $4$-digit numbers are superbrazilians?
2005 Postal Coaching, 8
Prove that For all positive integers $m$ and $n$ , one has $| n \sqrt{2005} - m | > \frac{1}{90n}$
2017 Latvia Baltic Way TST, 4
The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.
2023 Canadian Mathematical Olympiad Qualification, 2
How many ways are there to fill a $3 \times 3$ grid with the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$, such that the set of three elements in every row and every column form an arithmetic progression in some order? (Each number must be used exactly once)
LMT Speed Rounds, 2011.4
What is the sum of the first $2011$ integers closest in value to $0,$ including $0$ itself?
2018 ELMO Shortlist, 3
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
2006 Estonia National Olympiad, 5
Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.
2005 Postal Coaching, 18
Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.
1961 Polish MO Finals, 3
Prove that if a plane section of a tetrahedron is a parallelogram, then half of its perimeter is contained between the length of the smallest and the length of the largest edge of the tetrahedron.
2010 Contests, 3
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
2008 Balkan MO Shortlist, A5
Consider an integer $n \geq 1$, $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying
\begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*}
and a function $f: [-1,1] \mapsto \mathbb{R}$ such
\begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*}
for every $x,y \in [-1,1]$. Prove
\begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*}
for every $x$ $\in [-1,1]$. For a given sequence $a_1,a_2, \ldots ,a_n$, Find $f$ and $x$ so hat the equality holds.
1994 Baltic Way, 16
The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.
2023 Brazil Team Selection Test, 4
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
2024 Pan-American Girls’ Mathematical Olympiad, 5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$
for all real numbers $x, y$.
2019 Turkey Team SeIection Test, 2
$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$.
$a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite.
$b)$ Find 3 different prime numbers that do not divide any terms of this sequence.
1975 Miklós Schweitzer, 5
Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$
[i]J. Szucs[/i]
2004 China Team Selection Test, 2
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$
1988 Vietnam National Olympiad, 2
Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$.
2021 CHMMC Winter (2021-22), 5
How many cubics in the form $x^3 -ax^2 + (a+d)x -(a+2d)$ for integers $a,d$ have roots that are all non-negative integers?
2005 Iran MO (3rd Round), 3
Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: \[a^2+b^2+c^2\geq 4(R+r)^2\]
2005 Flanders Math Olympiad, 3
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
2021 China Team Selection Test, 4
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
2015 Israel National Olympiad, 5
Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$.
Prove that $L,T,K$ lie on one line.
2016 CMIMC, 1
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\left\lceil \sqrt{s} \right\rceil \mid s \in S\}$. Find the number of sets $T$ such that $\vert f(T) \vert = 2$ and $f(f(T)) = \{2\}$.
2015 Sharygin Geometry Olympiad, P10
The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral