Found problems: 85335
2017 China Team Selection Test, 3
Find the numbers of ordered array $(x_1,...,x_{100})$ that satisfies the following conditions:
($i$)$x_1,...,x_{100}\in\{1,2,..,2017\}$;
($ii$)$2017|x_1+...+x_{100}$;
($iii$)$2017|x_1^2+...+x_{100}^2$.
JOM 2025, 4
For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$.
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
2006 Indonesia MO, 1
Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.
2017 Sharygin Geometry Olympiad, P9
Let $C_0$ be the midpoint of hypotenuse $AB$ of triangle $ABC$; $AA_1, BB_1$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_0I$ and $A_1B_1$ meet on the altitude from $C$.
[i]Proposed by A.Zaslavsky[/i]
2021 Tuymaada Olympiad, 5
In a $100\times 100$ table $110$ unit squares are marked. Is it always possible to rearrange rows and columns so that all the marked unit squares
are above the main diagonal or on it?
2005 Singapore MO Open, 4
Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.
1955 Putnam, B5
Given an infinite sequence of $0$'s and $1$'s and a fixed integer $k,$ suppose that there are no more than $k$ distinct blocks of $k$ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence $11011010101$ followed by alternating $0$'s and $1$'s indefinitely, which is periodic beginning with the fifth term.)
2017 Saudi Arabia BMO TST, 2
Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.
2014 IPhOO, 2
An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)?
$ \textbf {(A) } \dfrac {7}{10} \qquad \textbf {(B) } \dfrac {49}{100} \qquad \textbf {(C) } 1 \qquad \textbf {(C) } \dfrac {100}{49} \qquad \textbf {(E) } \dfrac {10}{7} $
[i]Problem proposed by Ahaan Rungta[/i]
Russian TST 2014, P2
A circle centered at $O{}$ passes through the vertices $B{}$ and $C{}$ of the acute-angles triangle $ABC$ and intersects the sides $AC{}$ and $AB{}$ at $D{}$ and $E{}$ respectively. The segments $CE$ and $BD$ intersect at $U{}.$ The ray $OU$ intersects the circumcircle of $ABC$ at $P{}.$ Prove that the incenters of the triangles $PEC$ and $PBD$ coincide.
2020 Iranian Combinatorics Olympiad, 2
Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$?
[i]Proposed by Morteza Saghafian[/i]
2022 Bolivia IMO TST, P4
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2010 LMT, 8
How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$
1998 Iran MO (3rd Round), 4
Let be given $r_1,r_2,\ldots,r_n \in \mathbb R$. Show that there exists a subset $I$ of $\{1,2,\ldots,n \}$ which which has one or two elements in common with the sets $\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)$ such that
\[\left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.\]
2009 Peru MO (ONEM), 3
a) On a circumference $8$ points are marked. We say that Juliana does an “T-operation ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do?
b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?
2013 239 Open Mathematical Olympiad, 1
Among the divisors of a natural number $n$, we have numbers such that when they are devided by $2013$, give us remainders $1001, 1002, \ldots, 2012$. Prove that among the divisors of the number $n^2$, there exist numbers such that when they are divided by $2013$, give us reminders $1, 2, 3, \ldots, 2012$.
1968 IMO Shortlist, 15
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
2002 Moldova Team Selection Test, 2
Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.
2008 Brazil Team Selection Test, 2
Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$.
2010 Contests, 4
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
2007 IMS, 5
Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]
2020 Online Math Open Problems, 14
Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM=2$, $[ABC]=2020$, and $BC=10$. Then $AT=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
[i]Proposed by Ankit Bisain[/i]
2022 IFYM, Sozopol, 5
Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called [i]sozopolian[/i], if it [u]can’t[/u] be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of [i]sozopolian[/i] numbers as a function of $a$, $b$ and $c$.
2014 BMT Spring, 14
Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0$. Find the sum of all possible values of $x$.
2003 Hong kong National Olympiad, 4
Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.