Found problems: 85335
Swiss NMO - geometry, 2010.2
Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$.
Show that $ PMID$ ist cyclic.
2005 AMC 10, 11
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?
$ \textbf{(A)}\ 29\qquad
\textbf{(B)}\ 55\qquad
\textbf{(C)}\ 85\qquad
\textbf{(D)}\ 133\qquad
\textbf{(E)}\ 250$
2014 Taiwan TST Round 2, 1
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 $u_n = v_n.$
2015 Iran Team Selection Test, 4
$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.
1986 AMC 12/AHSME, 7
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is
$ \textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad\textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad\textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad \\ \textbf{(D)}\ \Big\{x\ |\ 2 < x \le 3\Big\}\qquad\textbf{(E)}\ \Big\{x\ |\ 2 < x < 3\Big\} $
2015 ASDAN Math Tournament, 14
A standard deck of $52$ cards is shuffled and randomly arranged in a queue, with each card having a suit $(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)$ and a rank $(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})$. For example, a card with the $\diamondsuit$ suit and the $7$ rank would be denoted as $\diamondsuit7$, and a card with the $\spadesuit$ and the $\text{Ace}$ rank would be denoted as $\spadesuit\text{Ace}$. In the queue, there exists a card with a rank of $\text{Ace}$ that appears for the first time in the queue. Let the card immediately following the above card be denoted as card $C$. Is the probability that $C$ is a $\spadesuit\text{A}$ higher than, equal to, or lower than the probability that $C$ is a $\clubsuit2$?
2007 District Olympiad, 1
Let $a_1\in (0,1)$ and $(a_n)_{n\ge 1}$ a sequence of real numbers defined by $a_{n+1}=a_n(1-a_n^2),\ (\forall)n\ge 1$. Evaluate $\lim_{n\to \infty} a_n\sqrt{n}$.
2011 USAMTS Problems, 5
In the game of Tristack Solitaire, you start with three stacks of cards, each with a different positive integer number of cards. At any time, you can double the number of cards in any one stack of cards by moving cards from exactly one other, larger, stack of cards to the stack you double. You win the game when any two of the three stacks have the same number of cards.
For example, if you start with stacks of $3$, $5$, and $7$ cards, then you have three possible legal moves:
[list]
[*]You may move $3$ cards from the $5$-card stack to the $3$-card stack, leaving stacks of $6$, $2$, and $7$ cards.
[*]You may move $3$ cards from the $7$-card stack to the $3$-card stack, leaving stacks of $6,$ $5$, and $4$ cards.
[*]You may move $5$ cards from the $7$-card stack to the $5$-card stack, leaving stacks of $3$, $10$, and $2$ cards.[/list]
Can you win Tristack Solitaire from any starting position? If so, then give a strategy for winning. If not, then explain why.
2004 China Girls Math Olympiad, 1
We say a positive integer $ n$ is [i]good[/i] if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k \plus{} a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
2018 239 Open Mathematical Olympiad, 10-11.1
Prove that in any convex polygon where all pairwise distances between vertices are distinct, there exists a vertex such that the closest vertex of the polygon is adjacent to it.
[i]Proposed by D. Shiryayev, S. Berlov[/i]
2014 Bosnia Herzegovina Team Selection Test, 1
Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$
1970 IMO Longlists, 13
Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:
$(1)$ $A,B,C$ are assigned $1,2,3$ respectively
$(2)$ Points on $AB$ are assigned $1$ or $2$
$(3)$ Points on $BC$ are assigned $2$ or $3$
$(4)$ Points on $CA$ are assigned $3$ or $1$
Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.
2018 Cyprus IMO TST, 4
Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$.
(a) Prove that
$$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$
(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:
$$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$
Estonia Open Junior - geometry, 2004.2.3
Circles $c_1$ and $c_2$ with centres $O_1$and $O_2$, respectively, intersect at points $A$ and $B$ so that the centre of each circle lies outside the other circle. Line $O_1A$ intersects circle $c_2$ again at point $P_2$ and line $O_2A$ intersects circle $c_1$ again at point $P_1$. Prove that the points $O_1,O_2, P_1, P_2$ and $B$ are concyclic
2012 IFYM, Sozopol, 2
In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.
2023 MOAA, 13
If real numbers $x$, $y$, and $z$ satisfy $x^2-yz = 1$ and $y^2-xz = 4$ such that $|x+y+z|$ is minimized, then $z^2-xy$ can be expressed in the form $\sqrt{a}-b$ where $a$ and $b$ are positive integers. Find $a+b$.
[i]Proposed by Andy Xu[/i]
2016 China Northern MO, 6
Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying:
$$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$
Prove that $FH\perp EG$.
2023 Paraguay Mathematical Olympiad, 2
Aidée draws ten squares of different sizes. The diagonal of the first square measures $1$ cm, the diagonal of the second measures $2$ cm, the diagonal of the third measures $3$ cm, and so on until the diagonal of the tenth square measures $10$ cm. How much are the areas of the ten squares?
2005 Mid-Michigan MO, 5-6
[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ?
[b]p2.[/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]p3.[/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/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img]
[b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ 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]p5.[/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].
1989 Iran MO (2nd round), 1
[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]
[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]
2017 Brazil Team Selection Test, 1
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2023 ELMO Shortlist, C7
A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\).
Let \(n\) be a nonnegative integer and \(S\) be the set of triples \((x,y,z)\) of nonnegative integers such that \(x+y+z=n\). If \(S\) is partitioned into discrete hexagons, show that at least \(n+1\) hexagons are needed.
[i]Proposed by Linus Tang[/i]
1955 Putnam, A2
$A_1 ~A_2~ \ldots ~A_n$ is a regular polygon inscribed in a circle of radius $r$ and center $O.$ $P$ is a point on line $OA_1$ extended beyond $A_1.$ Show that \[ \prod^n_{i=1} ~ \overline{PA}_{~i} = \overline{OP}^{~n} - r^n. \]
Geometry Mathley 2011-12, 8.1
Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$.
Kostas Vittas
1966 Swedish Mathematical Competition, 2
$a_1 + a_2 + ... + a_n = 0$, for some $k$ we have $a_j \le 0$ for $j \le k$ and $a_j \ge 0$ for $j > k$. If ai are not all $0$, show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$.