Found problems: 85335
1991 Tournament Of Towns, (307) 4
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
2014 Contests, 3
The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
[asy]
size(4cm);
dotfactor=3.5;
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,12);
B=(0,24);
C=(16,36);
D=(32,36);
E=(48,24);
F=(48,12);
G=(32,0);
H=(16,0);
W=origin;
X=(0,36);
Y=(48,36);
Z=(48,0);
filldraw(W--A--H--cycle^^B--X--C--cycle^^D--Y--E--cycle^^F--Z--G--cycle,rgb(.76,.76,.76));
draw(W--X--Y--Z--cycle,linewidth(1.2));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
[/asy]
2017 CMIMC Computer Science, 2
We are given the following function $f$, which takes a list of integers and outputs another list of integers. (Note that here the list is zero-indexed.)
\begin{tabular}{l}
1: \textbf{FUNCTION} $f(A)$ \\
2: $\quad$ \textbf{FOR} $i=1,\ldots, \operatorname{length}(A)-1$: \\
3: $\quad\quad$ $A[i]\leftarrow A[A[i]]$ \\
4: $\quad\quad$ $A[0]\leftarrow A[0]-1$ \\
5: $\quad$ \textbf{RETURN} $A$
\end{tabular}
Suppose the list $B$ is equal to $[0,1,2,8,2,0,1,7,0]$. In how many entries do $B$ and $f(B)$ differ?
Kvant 2021, M2670
There are 100 points on the plane so that any 10 of them are vertices of a convex polygon. Does it follow from this that all these points are the vertices of a convex 100-gon?
[i]From the folklore[/i]
2018 IFYM, Sozopol, 3
The number 1 is a solution of the equation
$(x + a)(x + b)(x + c)(x + d) = 16$,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.
2021 Kazakhstan National Olympiad, 5
Find all functions $f : \mathbb{R^{+}}\to \mathbb{R^{+}}$ such that $$f(x)^2=f(xy)+f(x+f(y))-1$$ for all $x, y\in \mathbb{R^{+}}$
2013 IMO Shortlist, A1
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
2019 Durer Math Competition Finals, 9
A cube has been divided into $27$ equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?
2008 Moldova National Olympiad, 11.2
Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $
2010 Hanoi Open Mathematics Competitions, 3
Find $5$ last digits of the number $M = 5^{2010}$ .
(A): $65625$, (B): $45625$, (C): $25625$, (D): $15625$, (E) None of the above.
2008 AIME Problems, 11
Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?
2013 HMNT, 5
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Company $XYZ$ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A$, $B$, and $C$. There are $1$, $5$, and $4$ workers at $A$, $B$, and $C$, respectively. Find the minimum possible total distance Company $XYZ$'s workers have to travel to get to $P$.
2013 Germany Team Selection Test, 1
$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.
2008 Tuymaada Olympiad, 2
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
1994 Abels Math Contest (Norwegian MO), 2b
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.
2019 Sharygin Geometry Olympiad, 22
Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$. A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$
1978 Putnam, A2
Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$. Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$. Show that
$$ \text{det} \begin{pmatrix} p_1 & a& a & \cdots & a \\
b & p_2 & a & \cdots & a\\
b & b & p_3 & \cdots & a\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b & b& b &\cdots &p_n
\end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$
2017 CHMMC (Fall), 6
The country of Claredena has $5$ cities, and is planning to build a road system so that each of its cities has exactly one outgoing (unidirectional) road to another city.
Two road systems are considered equivalent if we can get from one road system the other by just changing the names of the cities. That is, two road systems are considered the same if given a relabeling of the cities, if in the first configuration a road went from city $C$ to city $D$, then in the second configuration there is road that goes from the city now labeled $C$ to the city now labeled $D$.
How many distinct, nonequivalent possibilities are there for the road system Claredena builds?
1980 AMC 12/AHSME, 25
In the non-decreasing sequence of odd integers $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$ each odd positive integer $k$ appears $k$ times. It is a fact that there are integers $b$, $c$, and $d$ such that for all positive integers $n$,
\[ a_n=b\lfloor \sqrt{n+c} \rfloor +d, \]
where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. The sum $b+c+d$ equals
$\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$
2010 QEDMO 7th, 9
Let $p$ be an odd prime number and $c$ an integer for which $2c -1$ is divisible by $p$. Prove that
$$(-1)^{\frac{p+1}{2}}+\sum_{n=0}^{\frac{p-1}{2}} {2n \choose n}c^n$$ is divisible by $p$.
2014 NIMO Problems, 8
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Proposed by Akshaj[/i]
1950 Miklós Schweitzer, 10
Consider an arc of a planar curve such that the total curvature of the arc is less than $ \pi$. Suppose, further, that the curvature and its derivative with respect to the arc length exist at every point of the arc and the latter nowhere equals zero. Let the osculating circles belonging to the endpoints of the arc and one of these points be given. Determine the possible positions of the other endpoint.
2008 China Team Selection Test, 2
For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$
2008 Postal Coaching, 4
Let $n \in N$ and $k$ be such that $1 \le k \le n$. Find the number of ordered $k$-tuples $(a_1, a_2,...,a_k)$ of integers such the $1 \le a_j \le n$, for $1 \le j \le k$ and [u]either [/u] there exist $l,m \in \{1, 2,..., k\}$ such that $l < m$ but $a_l > a_m$ [u]or [/u] there exists $l \in \{1, 2,..., k\}$ such that $a_l - l$ is an odd number.
2014 Harvard-MIT Mathematics Tournament, 6
Let $n$ be a positive integer. A sequence $(a_0,\ldots,a_n)$ of integers is $\textit{acceptable}$ if it satisfies the following conditions:
[list=a]
[*] $0=|a_0|<|a_1|<\cdots<|a_{n-1}|<|a_n|.$
[*]The sets $\{|a_1-a_0|,|a_2-a_1|,\ldots,|a_{n-1}-a_{n-2}|,|a_n-a_{n-1}|\}$ and $\{1,3,9,\ldots,3^{n-1}\}$ are equal.[/list]
Prove that the number of acceptable sequences of integers is $(n+1)!$.