Found problems: 85335
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 $(n+1)!$.
2011 Singapore Junior Math Olympiad, 5
Initially, the number $10$ is written on the board. In each subsequent moves, you can either
(i) erase the number $1$ and replace it with a $10$, or
(ii) erase the number $10$ and replace it with a $1$ and a $25$ or
(iii) erase a $25$ and replace it with two $10$.
After sometime, you notice that there are exactly one hundred copies of $1$ on the board. What is the least possible sum of all the numbers on the board at that moment?
2024 Korea Winter Program Practice Test, Q8
Let $\omega$ be the incircle of triangle $ABC$. For any positive real number $\lambda$, let $\omega_{\lambda}$ be the circle concentric with $\omega$ that has radius $\lambda$ times that of $\omega$. Let $X$ be the intersection between a trisector of $\angle B$ closer to $BC$ and a trisector of $\angle C$ closer to $BC$. Similarly define $Y$ and $Z$. Let $\epsilon = \frac{1}{2024}$. Show that the circumcircle of triangle $XYZ$ lies inside $\omega_{1-\epsilon}$.
[i]Note. Weaker results with smaller $\epsilon$ may be awarded points depending on the value of the constant $\epsilon <\frac{1}{2024}$.[/i]
2020 Iranian Geometry Olympiad, 1
By a [i]fold[/i] of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper.
Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number
of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet.
(Note that the folding lines do not have to coincide with the grid lines of the shape.)
[i]Proposed by Mahdi Etesamifard[/i]
2000 Harvard-MIT Mathematics Tournament, 38
What is the largest number you can write with three $3$’s and three $8$’s, using only symbols $+,-,/,\times$ and exponentiation?
2019 LIMIT Category C, Problem 11
Let
$$x=\frac1{1\cdot2}-\frac1{2\cdot3}+\frac1{3\cdot4}-\ldots$$Then $e^{x+1}$ is
2024 Yasinsky Geometry Olympiad, 2
Let \( O \) and \( H \) be the circumcenter and orthocenter of the acute triangle \( ABC \). On sides \( AC \) and \( AB \), points \( D \) and \( E \) are chosen respectively such that segment \( DE \) passes through point \( O \) and \( DE \parallel BC \). On side \( BC \), points \( X \) and \( Y \) are chosen such that \( BX = OD \) and \( CY = OE \). Prove that \( \angle XHY + 2\angle BAC = 180^\circ \).
[i]Proposed by Matthew Kurskyi[/i]
2006 Cuba MO, 6
Two concentric circles of radii $1$ and $2$ have centere the point $O$. The vertex $A$ of the equilateral triangle $ABC$ lies at the largest circle, while the midpoint of side $BC$ lies on the smaller circle. If$ B$,$O$ and $C$ are not collinear, what measure can the angle $\angle BOC$ have?