Found problems: 85335
2004 Estonia Team Selection Test, 3
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
2005 IMO Shortlist, 5
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
2023 CCA Math Bonanza, I5
Find the sum of all distinct prime factors of $2023^3 - 2000^3 - 23^3$.
[i]Individual #5[/i]
2014 ELMO Shortlist, 2
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$.
[i]Proposed by Ryan Alweiss[/i]
1989 Bundeswettbewerb Mathematik, 1
Determine the polynomial
$$f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 $$
of smallest degree such that $a_i \in \{-1,0,1\}$ for $0\leq i \leq k-1$ and $f(n)$ is divisible by $30$ for all positive integers $n$.
2011 Akdeniz University MO, 1
Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that
$$p \mid m,n$$
1996 Tournament Of Towns, (488) 1
Prove that if $a, b$ and $c$ are positive numbers such that
$$a^2 + b^2 - ab = c^2,$$
then $(a - c)(b - c) < 0.$
(A Egorov)
Russian TST 2022, P1
In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.
The Golden Digits 2024, P2
Let $n$ be a positive integer. Consider an infinite checkered board. A set $S$ of cells is [i]connected[/i] if one may get from any cell in $S$ to any other cell in $S$ by only traversing edge-adjacent cells in $S$. Find the largest integer $k_n$ with the following property: in any connected set with $n$ cells, one can find $k_n$ disjoint pairs of adjacent cells (that is, $k_n$ disjoint dominoes).
[i]Proposed by David Anghel and Vlad Spătaru[/i]
2000 National Olympiad First Round, 27
How many different permutations $(\alpha_1 \alpha_2\alpha_3\alpha_4\alpha_5)$ of the set $\{1,2,3,4,5\}$ are there such that $(\alpha_1\dots \alpha_k)$ is not a permutation of the set $\{1,\dots ,k\}$, for every $1\leq k \leq 4$?
$ \textbf{(A)}\ 13
\qquad\textbf{(B)}\ 65
\qquad\textbf{(C)}\ 71
\qquad\textbf{(D)}\ 461
\qquad\textbf{(E)}\ \text{None}
$
VI Soros Olympiad 1999 - 2000 (Russia), 11.1
$16$ different natural numbers are written on the board, none of which exceeds $30$. Prove that there must be two coprime numbers among the written numbers.
1965 AMC 12/AHSME, 3
The expression $ (81)^{ \minus{} 2^{ \minus{} 2}}$ has the same value as:
$ \textbf{(A)}\ \frac {1}{81} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 81 \qquad \textbf{(E)}\ 81^4$
2007 Balkan MO Shortlist, N3
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is:
Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$
\[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]
Indonesia MO Shortlist - geometry, g8
Given a circle centered at point $O$, with $AB$ as the diameter. Point $C$ lies on the extension of line $AB$ so that $B$ lies between $A$ and $C$, and the line through $C$ intersects the circle at points $D$ and $E$ (where $D$ lies between $C$ and $E$). $OF$ is the diameter of the circumcircle of triangle $OBD$, and the extension of the line $CF$ intersects the circumcircle of triangle $OBD$ at point $G$. Prove that the points $O, A, E, G$ lie on a circle.
V Soros Olympiad 1998 - 99 (Russia), 10.10
A circle inscribed in triangle $ABC$ touches $BC$ at point $K$, $M$ is the midpoint of the altitude drawn on $BC$. The straight line $KM$ intersects the circle inscribed in $ABC$ for the second time at point $P$. Prove that the circle passing through $B$, $C$ and $P$ touches the circle inscribed in triangle $ABC$.
2021 Saudi Arabia IMO TST, 2
Find all positive integers $n$, such that $n$ is a perfect number and $\varphi (n)$ is power of $2$.
[i]Note:a positive integer $n$, is called perfect if the sum of all its positive divisors is equal to $2n$.[/i]
1968 AMC 12/AHSME, 9
The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:
$\textbf{(A)}\ \dfrac{1}{3} \qquad
\textbf{(B)}\ \dfrac{2}{3} \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 6\dfrac{1}{3} \qquad
\textbf{(E)}\ 6\dfrac{2}{3} $
2016 Kosovo National Mathematical Olympiad, 4
Solve equation in real numbers $\log_{2}(4^x+4)=x+\log_{2}(2^{x+1}-3)$
EMCC Accuracy Rounds, 2014
[b]p1.[/b] Chad lives on the third floor of an apartment building with ten floors. He leaves his room and goes up two floors, goes down four floors, goes back up five floors, and finally goes down one floor, where he finds Jordan's room. On which floor does Jordan live?
[b]p2.[/b] A real number $x$ satisfies the equation $2014x + 1337 = 1337x + 2014$. What is $x$?
[b]p3.[/b] Given two points on the plane, how many distinct regular hexagons include both of these points as vertices?
[b]p4.[/b] Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are $768$, $1024$, $2304$, $2560$, $4096$, and $7680$ kilobytes. Unfortunately, the email server holds a limit of $S$ kilobytes on the total size of the attachments per email, where $S$ is a positive integer. It is additionally given that all of the files are indivisible. What is the maximum value of S for which it will take Jordan at least three emails to transmit all six files to Chad?
[b]p5.[/b] If real numbers $x$ and $y$ satisfy $(x + 2y)^2 + 4(x + 2y + 2 - xy) = 0$, what is $x + 2y$?
[b]p6.[/b] While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How many possible games could they have played? Two games are considered the same if and only if they include the exact same sequence of scoring.
[b]p7.[/b] For a positive integer $m$, we define $m$ as a factorial number if and only if there exists a positive integer $k$ for which $m = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. We define a positive integer $n$ as a Thai number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest Thai numbers?
[b]p8.[/b] Chad and Jordan are in the Exeter Space Station, which is a triangular prism with equilateral bases. Its height has length one decameter and its base has side lengths of three decameters. To protect their station against micrometeorites, they install a force field that contains all points that are within one decameter of any point of the surface of the station. What is the volume of the set of points within the force field and outside the station, in cubic decameters?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1988 IMO Shortlist, 1
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
1979 Chisinau City MO, 179
Prove that the equation $x^2 + y^2 = 1979$ has no integer solutions.
1998 India Regional Mathematical Olympiad, 3
Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]
2024 239 Open Mathematical Olympiad, 8
Let $x_1, x_2, \ldots$ be a sequence of $0,1$, such that it satisfies the following three conditions:
1) $x_2=x_{100}=1$, $x_i=0$ for $1 \leq i \leq 100$ and $i \neq 2,100$;
2) $x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}$ for $51 \leq n \leq 100$;
3) $x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}$ for $n>100$.
Show that the sequence is periodic.
1999 Tuymaada Olympiad, 1
In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$.
[i]St.Petersburg folklore[/i]
2016 NIMO Problems, 4
Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=n$.
[i]Proposed by Michael Ren[/i]