Found problems: 66
2015 Bangladesh Mathematical Olympiad, 3
Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$?
Post no:[size=300]$100$[/size]
2021 Argentina National Olympiad, 2
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$
2018 Bangladesh Mathematical Olympiad, 7
[b]Evaluate[/b]
$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$
2021 Argentina National Olympiad, 4
The sum of several positive integers, not necessarily different, all of them less than or equal to $10$, is equal to $S$. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to $80.$ Determine all values of $S$ for which this is possible.
2015 Bangladesh Mathematical Olympiad, 6
Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?
2021 Argentina National Olympiad, 2
On each OMA lottery ticket there is a $9$-digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ is red, $222222222$ is green, what color is bill $123123123$?
2021 Argentina National Olympiad Level 2, 2
In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$
2000 District Olympiad (Hunedoara), 1
Solve in the set of $ 2\times 2 $ integer matrices the equation
$$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$
2016 Miklós Schweitzer, 2
Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.
the 16th XMO, 4
Given an integer $n$ ,For a sequence of $X$ with the number of $n$ and $Y$ with the number of $100n$ , we call it a [b]spring [/b] . We have two following rules
$\blacksquare$ Choose four adjacent character , if it is $YXXY$ , than it can be changed into $XYYX$
$\blacksquare $ Choose. four adjacent character , if it is $XYYX $ , than it can be changed into $YXXY$
If [b]spring [/b] $A$ can become $B$ using the rules , than we call they are [b][color=#3D85C6]similar [/color][/b]
Thy to find the maximum of $C$ such that there exists $C$ distinct [b]springs[/b] and they are [b][color=#3D85C6]similar[/color][/b]
2018 Czech-Polish-Slovak Match, Source
[url=https://artofproblemsolving.com/community/c678145][b]Czech-Polish-Slovak Match 2018[/b][/url]
[b]Austria, 24 - 27 June 2018[/b]
[url=http://artofproblemsolving.com/community/c6h1667029p10595005][b]Problem 1.[/b][/url] Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$,
$$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$
[i]Proposed by Walther Janous, Austria[/i]
[url=http://artofproblemsolving.com/community/c6h1667030p10595011][b]Problem 2.[/b][/url] Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$.
[i]Proposed by Patrik Bak, Slovakia[/i]
[url=http://artofproblemsolving.com/community/c6h1667031p10595016][b]Problem 3.[/b][/url] There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
[i]Proposed by Peter Novotný, Slovakia[/i]
[url=http://artofproblemsolving.com/community/c6h1667033p10595021][b]Problem 4.[/b][/url] Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles.
[i]Proposed by Josef Tkadlec, Czechia[/i]
[url=http://artofproblemsolving.com/community/c6h1667034p10595023][b]Problem 5.[/b][/url] In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points.
[i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]
[url=http://artofproblemsolving.com/community/c6h1667036p10595032][b]Problem 6.[/b][/url] We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that
$$ n = a + \frac 1a + b + \frac 1b.$$
[b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.
[b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.
[i]Proposed by Walther Janous, Austria[/i]
2015 Bangladesh Mathematical Olympiad, 1
[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b]
A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:
Witness [b]One[/b] said exactly one of the four witnesses is a liar.
Witness [b]Two[/b] said exactly two of the four witnesses is a liar.
Witness [b]Three[/b] said exactly three of the four witnesses is a liar.
Witness [b]Four[/b] said exactly four of the four witnesses is a liar.
Assume that each of the statements is either true or false.How many of the winesses are liars?
2015 District Olympiad, 3
Let $ m, n $ natural numbers with $ m\ge 2,n\ge 3. $ Prove that there exist $ m $ distinct multiples of $ n-1, $ namely, $ a_1,a_2,a_3,...,a_m, $ such that:
$$ \frac{1}{n} =\sum_{i=1}^m \frac{(-1)^{i-1}}{a_i} . $$
2000 District Olympiad (Hunedoara), 3
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that:
$ \text{(i)}\quad f(0)=0 $
$ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $
$ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $
Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$
is primitivable.
2016 Bangladesh Mathematical Olympiad, 5
Suppose there are $m$ Martians and $n$ Earthlings at an intergalactic peace conference. To ensure the Martians stay peaceful at the conference, we must make sure that no two Martians sit together, such that between any two Martians there is always at least one Earthling.
(a) Suppose all $m + n$ Martians and Earthlings are seated in a line. How many ways can the Earthlings and Martians be seated in a line?
(b) Suppose now that the $m+n$ Martians and Earthlings are seated around a circular round-table. How many ways can the Earthlings and Martians be seated around the round-table?
2014 Contests, 2
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.