Found problems: 85335
KoMaL A Problems 2023/2024, A. 861
Let $f(x)=x^2-2$ and let $f^{(n)}(x)$ denote the $n$-th iteration of $f$. Let $H=\{x:f^{(100)}(x)\leq -1\}$. Find the length of $H$ (the sum of the lengths of the intervals of $H$).
2018 Regional Olympiad of Mexico West, 1
You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions?
[img]https://cdn.artofproblemsolving.com/attachments/4/9/879d1e144acdbc63ee2ffe34cf13a920d5d836.png[/img]
2013 AIME Problems, 8
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.
2007 China Team Selection Test, 1
Find all functions $ f: \mathbb{Q}^{\plus{}} \mapsto \mathbb{Q}^{\plus{}}$ such that:
\[ f(x) \plus{} f(y) \plus{} 2xy f(xy) \equal{} \frac {f(xy)}{f(x\plus{}y)}.\]
2000 Moldova National Olympiad, Problem 4
Find all polynomials $P(x)$ with real coefficients that satisfy the relation
$$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$
2020 LMT Spring, 20
Let $c_1<c_2<c_3$ be the three smallest positive integer values of $c$ such that the distance between the parabola $y=x^2+2020$ and the line $y=cx$ is a rational multiple of $\sqrt{2}$. Compute $c_1+c_2+c_3$.
2005 Mid-Michigan MO, 7-9
[b]p1.[/b] Prove that no matter what digits are placed in the four empty boxes, the eight-digit number $9999\Box\Box\Box\Box$ is not a perfect square.
[b]p2.[/b] Prove that the number $m/3+m^2/2+m^3/6$ is integral for all integral values of $m$.
[b]p3.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor?
[b]p4.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.)
[img]https://cdn.artofproblemsolving.com/attachments/4/b/ca707bf274ed54c1b22c4f65d3d0b0a5cfdc56.png[/img]
[b]p5.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $7$ rocks in the first pile and $9$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game?
[b]p6.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits.
$\begin{tabular}{ccccc}
& & & a & b \\
* & & & c & d \\
\hline
& & c & e & f \\
+ & & a & b & \\
\hline
& c & f & d & f \\
\end{tabular}$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2015 District Olympiad, 4
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent.
Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $
2016 JBMO Shortlist, 4
Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.
MOAA Team Rounds, 2019.9
Jonathan finds all ordered triples $(a, b, c)$ of positive integers such that $abc = 720$. For each ordered triple, he writes their sum $a + b + c$ on the board. (Numbers may appear more than once.) What is the sum of all the numbers written on the board?
2013 Stanford Mathematics Tournament, 11
What is the smalles positive integer with exactly $768$ divisors? Your answer may be written in its prime factorization.
2015 Azerbaijan JBMO TST, 2
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]
2025 AIME, 10
Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.
2018 Rio de Janeiro Mathematical Olympiad, 4
Find every real values that $a$ can assume such that
$$\begin{cases}
x^3 + y^2 + z^2 = a\\
x^2 + y^3 + z^2 = a\\
x^2 + y^2 + z^3 = a
\end{cases}$$
has a solution with $x, y, z$ distinct real numbers.
2005 Bulgaria National Olympiad, 4
Let $ABC$ be a triangle with $AC\neq BC$, and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$. Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$, find $\widehat{EMF}$.
2021 Balkan MO Shortlist, N2
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
MMPC Part II 1996 - 2019, 2010
[b]p1.[/b] Let $x_1 = 0$, $x_2 = 1/2$ and for $n >2$, let $x_n$ be the average of $x_{n-1}$ and $x_{n-2}$. Find a formula for $a_n = x_{n+1} - x_{n}$, $n = 1, 2, 3, \dots$. Justify your answer.
[b]p2.[/b] Given a triangle $ABC$. Let $h_a, h_b, h_c$ be the altitudes to its sides $a, b, c,$ respectively. Prove: $\frac{1}{h_a}+\frac{1}{h_b}>\frac{1}{h_c}$ Is it possible to construct a triangle with altitudes $7$, $11$, and $20$? Justify your answer.
[b]p3.[/b] Does there exist a polynomial $P(x)$ with integer coefficients such that $P(0) = 1$, $P(2) = 3$ and $P(4) = 9$? Justify your answer.
[b]p4.[/b] Prove that if $\cos \theta$ is rational and $n$ is an integer, then $\cos n\theta$ is rational. Let $\alpha=\frac{1}{2010}$. Is $\cos \alpha $ rational ? Justify your answer.
[b]p5.[/b] Let function $f(x)$ be defined as $f(x) = x^2 + bx + c$, where $b, c$ are real numbers.
(A) Evaluate $f(1) -2f(5) + f(9)$ .
(B) Determine all pairs $(b, c)$ such that $|f(x)| \le 8$ for all $x$ in the interval $[1, 9]$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1987 AMC 12/AHSME, 20
Evaluate
\[ \log_{10}(\tan 1^{\circ})+ \log_{10}(\tan 2^{\circ})+ \log_{10}(\tan 3^{\circ})+ \cdots + \log_{10}(\tan 88^{\circ})+\log_{10}(\tan 89^{\circ}). \]
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac{1}{2}\log_{10}(\frac{\sqrt{3}}{2}) \qquad\textbf{(C)}\ \frac{1}{2}\log_{10}2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{none of these} $
2006 International Zhautykov Olympiad, 1
Solve in positive integers the equation
\[ n \equal{} \varphi(n) \plus{} 402 ,
\]
where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.
1971 Miklós Schweitzer, 11
Let $ C$ be a simple arc with monotone curvature such that $ C$ is congruent to its evolute. Show that under appropriate differentiability conditions, $ C$ is a part of a cycloid or a logarithmic spiral with polar equation $ r\equal{}ae^{\vartheta}$.
[i]J. Szenthe[/i]
2017 China Team Selection Test, 3
For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.
2005 Austria Beginners' Competition, 2
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .
1997 IMC, 4
Let $\alpha$ be a real number, $1<\alpha<2$.
(a) Show that $\alpha$ can uniquely be represented as the infinte product \[ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots \] with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$.
(b) Show that $\alpha\in\mathbb{Q}$ iff from some $k$ onwards we have $n_{k+1}=n_k^2$.
2019 Canada National Olympiad, 5
A 2-player game is played on $n\geq 3$ points, where no 3 points are collinear. Each move consists of selecting 2 of the points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. (An odd cycle must have all its vertices among the $n$ points from the start, so the vertices of the cycle cannot be the intersections of the lines drawn.) Find all $n$ such that the player to move first wins.
1986 IMO Longlists, 39
Let $S$ be a $k$-element set.
[i](a)[/i] Find the number of mappings $f : S \to S$ such that
\[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\]
[i](b)[/i] The same with the condition $\text{(i)}$ left out.