Found problems: 85335
2020 Brazil National Olympiad, 1
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
2016 Harvard-MIT Mathematics Tournament, 6
Consider a $2 \times n$ grid of points and a path consisting of $2n-1$ straight line segments connecting all these $2n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called $\textit{efficient}$ if each point is only passed through once and no two line segments intersect. How many efficient paths are there when $n = 2016$?
2020 LIMIT Category 1, 2
Prove that any integer has a multiple consisting of all ten digits $\{0,1,2,3,4,5,6,7,8,9\}$.
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[i]Note: Any digit can be repeated any number of times[/i]
1994 Turkey MO (2nd round), 1
For $n\in\mathbb{N}$, let $a_{n}$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty{\frac{1}{a_{n}^{3}}}.\]
1982 IMO Longlists, 19
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
2020 Junior Balkаn MO, 3
Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy.
Proposed by [i]Demetres Christofides, Cyprus[/i]
2021 Princeton University Math Competition, A2 / B4
For a bijective function $g : R \to R$, we say that a function $f : R \to R$ is its superinverse if it satisfies the following identity $(f \circ g)(x) = g^{-1}(x)$, where $g^{-1}$ is the inverse of $g$. Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its superinverse, find $|f(-289)|$.
2003 Putnam, 5
A Dyck $n$-path is a lattice path of $n$ upsteps $(1, 1)$ and $n$ downsteps $(1, -1)$ that starts at the origin $O$ and never dips below the $x$-axis. A return is a maximal sequence of contiguous downsteps that terminates on the $x$-axis. For example, the Dyck $5$-path illustrated has two returns, of length $3$ and $1$ respectively. Show that there is a one-to-one correspondence between the Dyck $n$-paths with no return of even length and the Dyck $(n - 1)$ paths.
\[\begin{picture}(165,70)
\put(-5,0){O}
\put(0,10){\line(1,0){150}}
\put(0,10){\line(1,1){30}}
\put(30,40){\line(1,-1){15}}
\put(45,25){\line(1,1){30}}
\put(75,55){\line(1,-1){45}}
\put(120,10){\line(1,1){15}}
\put(135,25){\line(1,-1){15}}
\put(0,10){\circle{1}}\put(0,10){\circle{2}}\put(0,10){\circle{3}}\put(0,10){\circle{4}}
\put(15,25){\circle{1}}\put(15,25){\circle{2}}\put(15,25){\circle{3}}\put(15,25){\circle{4}}
\put(30,40){\circle{1}}\put(30,40){\circle{2}}\put(30,40){\circle{3}}\put(30,40){\circle{4}}
\put(45,25){\circle{1}}\put(45,25){\circle{2}}\put(45,25){\circle{3}}\put(45,25){\circle{4}}
\put(60,40){\circle{1}}\put(60,40){\circle{2}}\put(60,40){\circle{3}}\put(60,40){\circle{4}}
\put(75,55){\circle{1}}\put(75,55){\circle{2}}\put(75,55){\circle{3}}\put(75,55){\circle{4}}
\put(90,40){\circle{1}}\put(90,40){\circle{2}}\put(90,40){\circle{3}}\put(90,40){\circle{4}}
\put(105,25){\circle{1}}\put(105,25){\circle{2}}\put(105,25){\circle{3}}\put(105,25){\circle{4}}
\put(120,10){\circle{1}}\put(120,10){\circle{2}}\put(120,10){\circle{3}}\put(120,10){\circle{4}}
\put(135,25){\circle{1}}\put(135,25){\circle{2}}\put(135,25){\circle{3}}\put(135,25){\circle{4}}
\put(150,10){\circle{1}}\put(150,10){\circle{2}}\put(150,10){\circle{3}}\put(150,10){\circle{4}}
\end{picture}\]
2013 Portugal MO, 4
Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?
2015 China Team Selection Test, 2
Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$, and a pair of vertices $(u,v)$, define \[ L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}\]Prove that, for any choice of $V$, there exist at least $120$ distinct values of $L(u,v)$.
2010 Contests, 3
Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.
1971 Poland - Second Round, 4
On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.
2014 Contests, 1
In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?
Kyiv City MO Seniors 2003+ geometry, 2015.10.5.1
The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
2006 China Team Selection Test, 2
Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.
2010 IFYM, Sozopol, 8
Let $k$ be a circle and $l$–line that is tangent to $k$ in point $P$. On $l$ from the two sides of $P$ are chosen arbitrary points $A$ and $B$. The tangents through $A$ and $B$ to $k$, different than $l$, intersect in point $C$. Find the geometric place of points $C$, when $A$ and $B$ change in such way so that $AP.BP$ is a constant.
2018 IOM, 4
Let $1 = d_0 < d_1 < \dots < d_m = 4k$ be all positive divisors of $4k$, where $k$ is a positive integer. Prove that there exists $i \in \{1, \dots, m\}$ such that $d_i - d_{i-1} = 2$.
[i]Ivan Mitrofanov[/i]
2009 Brazil Team Selection Test, 2
The cities of Terra Brasilis are connected by some roads. There are no two cities directly connected by more than one road. It is known that it is possible to go from one city to any other using one or more roads. We call [i]role[/i] any closed road route (ie, it starts in a city and ends in the same city) that does not pass through a city more than once. At Terra Brasilis, all roles go through an odd number of cities.
The government of Terra Brasilis decided to close some roles for reform. When you close a role, all it;s roads are closed, so traffic is not allowed on these roads. By doing this, Terra Brasilis was divided into several regions such that from any city in each region it is possible to reach any other in the same region by road, but it is not possible to reach cities in other regions.
Prove that the number of regions is odd
[hide=original wording]As cidades da Terra Brasilis sao conectadas por algumas estradas. Nao ha duas cidades conectadas diretamente por mais de uma estrada. Sabe-se que, e possivel ir de uma cidade para qualquer outra utilizando uma ou mais estradas. Chamamos de rol^e qualquer rota fechada de estradas (isto e, comeca em uma cidade e termina na mesma cidade) que nao passa por uma cidade mais de uma vez. Na Terra Brasilis, todos os roles passam por quantidades impares de cidades.
O governo da Terra Brasilis decidiu fechar alguns roles para reforma. Ao fechar um role, todas as suas estradas sao
interditadas, de modo que nao e permitido o trafego nessas estradas. Ao fazer isso, a Terra Brasilis ficou dividida em varias regioes de modo que, de qualquer cidade de cada regiao e possivel hegar a qualquer outra da mesma regiao atraves de estradas, mas nao e possivel hegar a cidades de outras regioes.
Prove que o numero de regioes e impar.[/hide]
2019 AMC 8, 3
Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest?
$\textbf{(A) } \frac{15}{11} < \frac{17}{13} < \frac{19}{15} \qquad\textbf{(B) } \frac{15}{11} < \frac{19}{15} < \frac{17}{13} \qquad\textbf{(C) } \frac{17}{13} < \frac{19}{15} < \frac{15}{11}
\newline\newline
\qquad\textbf{(D) } \frac{19}{15} < \frac{15}{11} < \frac{17}{13} \qquad\textbf{(E) } \frac{19}{15} < \frac{17}{13} < \frac{15}{11}$
2013 Tuymaada Olympiad, 3
For every positive real numbers $a$ and $b$ prove the inequality
\[\displaystyle \sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.\]
[i]A. Khabrov[/i]
Geometry Mathley 2011-12, 15.4
Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$.
Kostas Vittas
2020 Junior Balkаn MO, 1
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2005 AMC 12/AHSME, 10
A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
2010 Argentina Team Selection Test, 6
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.
Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.
Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).