Found problems: 85335
2022 Portugal MO, 6
Given two natural numbers $a < b$, Xavier and Ze play the following game. First, Xavier writes $a$ consecutive numbers of his choice; then, repeat some of them, also of his choice, until he has $b$ numbers, with the condition that the sum of the $b$ numbers written is an even number. Ze wins the game if he manages to separate the numbers into two groups with the same amount. Otherwise, Xavier wins. For example, for $a = 4$ and $b = 7$, if Xavier wrote the numbers $3,4,5,6,3,3,4$, Ze could win, separating these numbers into groups $3,3 ,4,4$ and $3,5,6$. For what values of $a$ and $b$ can Xavier guarantee victory?
2021 Science ON Juniors, 1
Let $a,p,q\in \mathbb{Z}_{\ge 1}$ be such that $a$ is a perfect square, $a=pq$ and
$$2021~|~p^3+q^3+p^2q+pq^2.$$
Prove that $2021$ divides $\sqrt a$.\\ \\
[i](Cosmin Gavrilă)[/i]
2019 Brazil National Olympiad, 6
In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon
they are said to be [b]mutually visible[/b] if the line segment connecting them has no other blue dots. Prove that
There is a set of $ 2019$ blue dots that are mutually visible two by two.
2012 AMC 10, 13
It takes Clea $60$ seconds to walk down an escalator when it is not operating and only $24$ seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it?
$ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 $
2008 Grigore Moisil Intercounty, 1
Let be a sequence of positive real numbers $ \left( a_n\right)_{n\ge 1} $ defined by the recurrence relation $ a_{n+1}=\ln \left(1+a_n\right) . $ Show that:
[b]1)[/b] $ \lim_{n\to\infty } a_n=0 $
[b]2)[/b] $ \lim_{n\to\infty } na_n=2 $
[b]3[/b] $ \lim_{n\to\infty } \frac{n(na_n-2)}{\ln n}=2/3 $
[i]Dorel Duca[/i] and [i]Dorian Popa[/i]
1974 Poland - Second Round, 2
Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if
$$
\sum_{i=1}^n t_i = 0, \text{ to } \sum_{i=1}^n\sum_{j=1}^n t_it_j |s_i-s_j| \leq 0.$$
2019 Peru IMO TST, 6
Let $p$ and $q$ two positive integers. Determine the greatest value of $n$ for which there exists sets $A_1,\ A_2,\ldots,\ A_n$ and $B_1,\ B_2,\ldots,\ B_n$ such that:
[LIST]
[*] The sets $A_1,\ A_2,\ldots,\ A_n$ have $p$ elements each one. [/*]
[*] The sets $B_1,\ B_2,\ldots,\ B_n$ have $q$ elements each one. [/*]
[*] For all $1\leq i,\ j \leq n$, sets $A_i$ and $B_j$ are disjoint if and only if $i=j$.
[/LIST]
1995 AMC 12/AHSME, 22
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13,19,20,25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is
$\textbf{(A)}\ 459 \qquad
\textbf{(B)}\ 600 \qquad
\textbf{(C)}\ 680 \qquad
\textbf{(D)}\ 720\qquad
\textbf{(E)}\ 745$
2013 Romanian Masters In Mathematics, 2
Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation
\[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\]
that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$
2021 USMCA, 10
Find the sum of all positive integers $n \leq 1000$ with the property that for every prime number $p$ dividing $n,$ we have that $2p-1$ also divides $n.$
1951 AMC 12/AHSME, 44
If $ \frac {xy}{x \plus{} y} \equal{} a, \frac {xz}{x \plus{} z} \equal{} b, \frac {yz}{y \plus{} z} \equal{} c$, where $ a,b,c$ are other than zero, then $ x$ equals:
$ \textbf{(A)}\ \frac {abc}{ab \plus{} ac \plus{} bc} \qquad\textbf{(B)}\ \frac {2abc}{ab \plus{} bc \plus{} ac} \qquad\textbf{(C)}\ \frac {2abc}{ab \plus{} ac \minus{} bc}$
$ \textbf{(D)}\ \frac {2abc}{ab \plus{} bc \minus{} ac} \qquad\textbf{(E)}\ \frac {2abc}{ac \plus{} bc \minus{} ab}$
2009 China Team Selection Test, 1
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
2015 Harvard-MIT Mathematics Tournament, 10
Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of [b]negative slope[/b] tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a [b]unique[/b] point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.
2025 Kosovo National Mathematical Olympiad`, P3
A number is said to be [i]regular[/i] if when a digit $k$ appears in that number, the digit appears exactly $k$ times. For example, the number $3133$ is a regular number because the digit $1$ appears exactly once and the digit $3$ appears exactly three times. How many regular six-digit numbers are there?
2018 Ramnicean Hope, 3
Prove that for any noncollinear points $ A,B,C $ and positive real numbers $ x,y, $ the following inequality is true.
$$ xAB^2- \frac{xy}{x+y}BC^2 +yCA^2\ge 0 $$
[i]Constantin Rusu[/i]
2022 Malaysian IMO Team Selection Test, 4
Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \] What is the minimum degree of $P$?
[i]Proposed by Loke Zhi Kin[/i]
2024 Irish Math Olympiad, P10
Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ with the property that $$f(x+f(y)+f(f(z)))=z+f(y)+f(f(x))$$ for all positive integers $x,y,z$.
2000 Brazil Team Selection Test, Problem 3
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
2021 Israel TST, 3
Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.).
The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.
2010 Olympic Revenge, 4
Let $a_n$ and $b_n$ to be two sequences defined as below:
$i)$ $a_1 = 1$
$ii)$ $a_n + b_n = 6n - 1$
$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.
Determine $a_{2009}$.
2001 Miklós Schweitzer, 8
Let $H$ be a complex Hilbert space. The bounded linear operator $A$ is called [i]positive[/i] if $\langle Ax, x\rangle \geq 0$ for all $x\in H$. Let $\sqrt A$ be the positive square root of $A$, i.e. the uniquely determined positive operator satisfying $(\sqrt{A})^2=A$. On the set of positive operators we introduce the
$$A\circ B=\sqrt A B\sqrt B$$
operation. Prove that for a given pair $A, B$ of positive operators the identity
$$(A\circ B)\circ C=A\circ (B\circ C)$$
holds for all positive operator $C$ if and only if $AB=BA$.
2019 Harvard-MIT Mathematics Tournament, 9
How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7?
2008 AMC 12/AHSME, 13
Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$?
$ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad
\textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad
\textbf{(C)}\ \frac{\sqrt3}{18} \qquad
\textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad
\textbf{(E)}\ \frac{\sqrt3}{12}$
2000 Turkey Team Selection Test, 2
Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus
$ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.
2003 AMC 10, 23
A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt)+fontsize(6pt));
pair C=dir(22.5), B=dir(67.5), A=dir(112.5), H=dir(157.5), G=dir(202.5), F=dir(247.5), E=dir(292.5), D=dir(337.5);
draw(A--B--C--D--E--F--G--H--cycle);
label("$A$",A,NNW);
label("$B$",B,NNE);
label("$C$",C,ENE);
label("$D$",D,ESE);
label("$E$",E,SSE);
label("$F$",F,SSW);
label("$G$",G,WSW);
label("$H$",H,WNW);[/asy]$ \textbf{(A)}\ 1\minus{}\frac{\sqrt2}{2} \qquad
\textbf{(B)}\ \frac{\sqrt2}{4} \qquad
\textbf{(C)}\ \sqrt2\minus{}1 \qquad
\textbf{(D)}\ \frac12 \qquad
\textbf{(E)}\ \frac{1\plus{}\sqrt2}{4}$