Found problems: 85335
1991 Arnold's Trivium, 26
Investigate the behaviour as $t\to+\infty$ of solutions of the systems
\[\begin{cases}
\dot{x}=y\\
\dot{y}=2\sin y-y-x\end{cases}\]
\[\begin{cases}
\dot{x}=y\\
\dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}\]
where $\epsilon\ll 1$.
PEN D Problems, 5
Prove that for $n\geq 2$, \[\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.\]
2011 Mathcenter Contest + Longlist, 2 sl2
For natural $n$, define $f_n=[2^n\sqrt{69}]+[2^n\sqrt{96}]$ Prove that there are infinite even integers and infinite odd integers that appear in number $f_1,f_2,\dots$.
[i](tatari/nightmare)[/i]
2009 Greece Team Selection Test, 3
Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$
2002 AMC 10, 13
Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female?
$\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$
2006 Cuba MO, 1
Determine all monic polynomials $P(x)$ of degree $3$ with coefficients integers, which are divisible by $x-1$, when divided by $ x-5$ leave the same remainder as when divided by$ x+5$ and have a root between $2$ and $3$.
1963 Poland - Second Round, 6
From the point $ S $ of space arise $ 3 $ half-lines: $ SA $, $ SB $ and $ SC $, none of which is perpendicular to both others. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the drawn planes intersect along one line $ d $.
2010 National Olympiad First Round, 8
What is the sum of the digits of the first $2010$ positive integers?
$ \textbf{(A)}\ 30516
\qquad\textbf{(B)}\ 28068
\qquad\textbf{(C)}\ 25020
\qquad\textbf{(D)}\ 20100
\qquad\textbf{(E)}\ \text{None}
$
2022 Harvard-MIT Mathematics Tournament, 9
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.
Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.
2024 Turkey MO (2nd Round), 5
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$:
$$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$
Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .
2008 AMC 10, 4
A semipro baseball league has teams with $ 21$ players each. League rules state that a player must be paid at least $ \$15,000$, and that the total of all players' salaries for each team cannot exceed $ \$700,000$. What is the maximum possiblle salary, in dollars, for a single player?
$ \textbf{(A)}\ 270,000 \qquad
\textbf{(B)}\ 385,000 \qquad
\textbf{(C)}\ 400,000 \qquad
\textbf{(D)}\ 430,000 \qquad
\textbf{(E)}\ 700,000$
2012 NIMO Problems, 6
A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
[i]Proposed by Lewis Chen[/i]
2014 Portugal MO, 5
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.
2004 Belarusian National Olympiad, 8
Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise
aunt Polly is right.
2020 CHMMC Winter (2020-21), 1
[i](5 pts)[/i] Let $n$ be a positive integer, $K = \{1, 2, \dots, n\}$, and $\sigma : K \rightarrow K$ be a function with the property that $\sigma(i) = \sigma(j)$ if and only if $i = j$ (in other words, $\sigma$ is a \textit{bijection}). Show that there is a positive integer $m$ such that
\[ \underbrace{\sigma(\sigma( \dots \sigma(i) \dots ))}_\textrm{$m$ times} = i \]
for all $i \in K$.
2014 AMC 10, 7
Suppose $A>B>0$ and A is $x\%$ greater than $B$. What is $x$?
$ \textbf {(A) } 100\left(\frac{A-B}{B}\right) \qquad \textbf {(B) } 100\left(\frac{A+B}{B}\right) \qquad \textbf {(C) } 100\left(\frac{A+B}{A}\right)\qquad \textbf {(D) } 100\left(\frac{A-B}{A}\right) \qquad \textbf {(E) } 100\left(\frac{A}{B}\right)$
1994 Putnam, 4
For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where
\[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\]
Show that $\lim_{n\to \infty}d_n=\infty$.
2019 PUMaC Combinatorics B, 6
Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.
2003 Tournament Of Towns, 2
Prove that every positive integer can be represented in the form
\[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\]
with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.
2016 Postal Coaching, 3
The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.
2022 EGMO, 1
Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
2013 EGMO, 4
Find all positive integers $a$ and $b$ for which there are three consecutive integers at which the polynomial \[ P(n) = \frac{n^5+a}{b} \] takes integer values.
2008 Moldova Team Selection Test, 2
Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of
$ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.
Geometry Mathley 2011-12, 11.4
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.
Nguyễn Văn Linh
2006 National Olympiad First Round, 35
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$?
$
\textbf{(A)}\ -3
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 2\sqrt 3
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \text{None of above}
$