Found problems: 85335
1996 India Regional Mathematical Olympiad, 7
If $A$ is a fifty element subset of the set $1,2,\ldots 100$ such that no two numbers from $A$ add up to $100$, show that $A$ contains a square.
2022 HMNT, 10
A real number $x$ is chosen uniformly at random from the interval $[0, 1000].$ Find the probability that $$\left\lfloor\frac{\lfloor \tfrac{x}{2.5}\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor.$$
2007 Balkan MO Shortlist, G2
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
2016 Hanoi Open Mathematics Competitions, 1
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ :
(A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.
1999 Moldova Team Selection Test, 12
Solve the equation in postive integers $$x^2+y^2+1998=1997x-1999y.$$
1995 Greece National Olympiad, 3
If the equation $ ax^2+(c-b)x+(e-d)=0$ has real roots greater than $1$, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.
2020 USEMO, 1
Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?
2004 National Olympiad First Round, 7
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 8
$
2024 Lusophon Mathematical Olympiad, 5
In a $9\times9$ board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column.
One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side.
How many ways are there to paint the board such that the squares $44$ and $49$ are both black?
2023 BMT, Tie 1
Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.
1989 All Soviet Union Mathematical Olympiad, 497
$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.
2018 Ecuador NMO (OMEC), 5
Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.
2021 Korea - Final Round, P1
An acute triangle $\triangle ABC$ and its incenter $I$, circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$, $AC$ in $D$,$E$. The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$. Denote the circumcircle of $DEF$ as $\omega$, and its center as $K$. $\omega$ and $FI$ intersect in $P$($\neq F$). Prove that $O,K,P$ is collinear.
2017 QEDMO 15th, 11
Calculate $$\frac{(2^1+3^1)(2^2+3^2)(2^4+3^4)(2^8+3^8)...(2^{2048}+3^{2048})+2^{4096}}{3^{4096}}$$
2010 Harvard-MIT Mathematics Tournament, 6
Suppose that a polynomial of the form $p(x)=x^{2010}\pm x^{2009}\pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?
Gheorghe Țițeica 2025, P4
Consider $4n$ points in the plane such that no three of them are collinear ($n\geq 1$). Show that the set of centroids of all the triangles formed by any three of these points contains at least $4n$ elements.
[i]Radu Bumbăcea[/i]
2012 USA TSTST, 5
A rational number $x$ is given. Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties:
(a) $x_0=x$;
(b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$;
(c) $x_n$ is an integer for some $n$.
2005 Miklós Schweitzer, 2
Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic.
Any Hints or Sols for this hard problem?? :help:
1991 Greece Junior Math Olympiad, 2
Given a semicircle of diameter $AB$ and center $O$. Let $CD$ be the chord of the semicircle tangent to two circles of diameters $AO$ and $OB$. If $CD=120$ cm,, caclulate area of the semicircle.
2024 Harvard-MIT Mathematics Tournament, 16
Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$
2017 AMC 8, 15
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
[asy]
fill((0.5, 4.5)--(1.5,4.5)--(1.5,2.5)--(0.5,2.5)--cycle,lightgray);
fill((1.5,3.5)--(2.5,3.5)--(2.5,1.5)--(1.5,1.5)--cycle,lightgray);
label("$8$", (1, 0));
label("$C$", (2, 0));
label("$8$", (3, 0));
label("$8$", (0, 1));
label("$C$", (1, 1));
label("$M$", (2, 1));
label("$C$", (3, 1));
label("$8$", (4, 1));
label("$C$", (0, 2));
label("$M$", (1, 2));
label("$A$", (2, 2));
label("$M$", (3, 2));
label("$C$", (4, 2));
label("$8$", (0, 3));
label("$C$", (1, 3));
label("$M$", (2, 3));
label("$C$", (3, 3));
label("$8$", (4, 3));
label("$8$", (1, 4));
label("$C$", (2, 4));
label("$8$", (3, 4));[/asy]
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }24\qquad\textbf{(E) }36$
1977 IMO Shortlist, 11
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2005 Greece National Olympiad, 3
We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with
$x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that
i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$;
ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.
2017-IMOC, G4
Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear.
[img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]
2021 BMT, Tie 1
Let the sequence $\{a_n\}$ for $n \ge 0$ be defined as $a_0 = c$, and for $n \ge 0$,
$$a_n =\frac{2a_{n-1}}{4a^2_{n-1} -1}.$$
Compute the sum of all values of $c$ such that $a_{2020}$ exists but $a_{2021}$ does not exist.