Found problems: 85335
2024 Cono Sur Olympiad, 5
A permutation of $\{1, 2 \cdots, n \}$ is [i]magic[/i] if each element $k$ of it has at least $\left\lfloor \frac{k}{2} \right\rfloor$ numbers less to it at the left. For each $n$ find the number of [i]magical[/i] permutations.
1986 AMC 12/AHSME, 5
Simplify $\left(\sqrt[6]{27} - \sqrt{6 \frac{3}{4} }\right)^2$
$ \textbf{(A)}\ \frac{3}{4} \qquad
\textbf{(B)}\ \frac{\sqrt 3}{2} \qquad
\textbf{(C)}\ \frac{3 \sqrt 3}{4} \qquad
\textbf{(D)}\ \frac{3}{2} \qquad
\textbf{(E)}\ \frac{3 \sqrt 3}{2} $
2018 Balkan MO, 3
Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile. The game ends if a player cannot move, in which case the other player wins.
Determine all pairs $(a,b)$ of positive integers such that if initially the two piles have $a$ and $b$ coins respectively, then Bob has a winning strategy.
Proposed by Dimitris Christophides, Cyprus
2017 Turkey Team Selection Test, 7
Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.
2017 India IMO Training Camp, 1
Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.
2012 Bogdan Stan, 4
Let be three real positive numbers $ \alpha ,\beta ,\gamma $ and let $ M,N $ be points on the sides $ AB,BC, $ respectively, of a triangle $ ABC, $ such that $ \frac{MA}{MB} =\frac{\alpha }{\beta } $ and $ \frac{NB}{NC} =\frac{\beta }{\gamma } . $ Also, let $ P $ be the intersection of $ CM $ with $ AN. $ Show that:
$$ \frac{1}{\alpha }\overrightarrow{PA} +\frac{1}{\beta }\overrightarrow{PB} +\frac{1}{\gamma }\overrightarrow{PC} =0 $$
2022 Purple Comet Problems, 10
Let $a$ be a positive real number such that $$4a^2+\frac{1}{a^2}=117.$$ Find $$8a^3+\frac{1}{a^3}.$$
2014 Canada National Olympiad, 3
Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if
[list]
[*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and
[*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and
[*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list]
Determine the number of good $p$-tuples.
2007 Olympic Revenge, 4
Let $A_{1}A_{2}B_{1}B_{2}$ be a convex quadrilateral. At adjacent vertices $A_{1}$ and $A_{2}$ there are two Argentinian cities. At adjacent vertices $B_{1}$ and $B_{2}$ there are two Brazilian cities. There are $a$ Argentinian cities and $b$ Brazilian cities in the quadrilateral interior, no three of which collinear. Determine if it's possible, independently from the cities position, to build straight roads, each of which connects two Argentinian cities ou two Brazilian cities, such that:
$\bullet$ Two roads does not intersect in a point which is not a city;
$\bullet$ It's possible to reach any Argentinian city from any Argentinian city using the roads; and
$\bullet$ It's possible to reach any Brazilian city from any Brazilian city using the roads.
If it's always possible, construct an algorithm that builds a possible set of roads.
2019 MIG, 24
Regular hexagon $ABCDEF$ has area $1$. Starting with edge $AB$ and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side $AB$, the new point, $G$, is drawn so $AG = \tfrac12 AB$. This forms hexagon $GHIJKL$, as shown. What is the area of this new hexagon?
[asy]
size(120);
pair A = (-1/2, sqrt(3)/2);
pair B = (1/2, sqrt(3)/2);
pair C = (1,0);
pair D = (1/2, -sqrt(3)/2);
pair EE = (-1/2, -sqrt(3)/2);
pair F = (-1,0);
pair G = (A+B)/2;
pair H = (B+C)/2;
pair I = (C+D)/2;
pair J = (D+EE)/2;
pair K = (EE+F)/2;
pair L = (F+A)/2;
draw(A--B--C--D--EE--F--cycle);
draw(G--H--I--J--K--L--cycle);
dot(A^^B^^C^^D^^EE^^F^^G^^H^^I^^J^^K^^L);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,E);
label("$D$",D,SE);
label("$E$",EE,SW);
label("$F$",F,W);
label("$G$",G,N);
label("$H$",H,NE);
label("$I$",I,SE);
label("$J$",J,S);
label("$K$",K,SW);
label("$L$",L,NW);
[/asy]
$\textbf{(A) }\dfrac35\qquad\textbf{(B) }\dfrac57\qquad\textbf{(C) }\dfrac34\qquad\textbf{(D) }\dfrac79\qquad\textbf{(E) }\dfrac45$
2020 China Girls Math Olympiad, 6
Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$
2014 Contests, 1
In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.
2010 Peru IMO TST, 5
Let $\Bbb{N}$ be the set of positive integers. For each subset $\mathcal{X}$ of $\Bbb{N}$ we define the set $\Delta(\mathcal{X})$ as the set of all numbers $| m - n |,$ where $m$ and $n$ are elements of $\mathcal{X}$, ie: $$\Delta (\mathcal{X}) = \{ |m-n| \ | \ m, n \in \mathcal{X} \}$$ Let $\mathcal A$ and $\mathcal B$ be two infinite, disjoint sets whose union is $\Bbb{N.}$
a) Prove that the set $\Delta (\mathcal A) \cap \Delta (\mathcal B)$ has infinitely many elements.
b) Prove that there exists an infinite subset $\mathcal C$ of $\Bbb{N}$ such that $\Delta (\mathcal C)$ is a subset of $\Delta (\mathcal A) \cap \Delta (\mathcal B).$
2021 Math Prize for Girls Problems, 13
There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch. They all start in the off position when 1011 people walk by. The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on. In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$. After all 1011 people have gone by, how many light bulbs are on?
1977 Spain Mathematical Olympiad, 6
A triangle $ABC$ is considered, and let $D$ be the intersection point of the angle bisector corresponding to angle $A$ with side $BC$. Prove that the circumcircle that passes through $A$ and is tangent to line $BC$ at $D$, it is also tangent to the circle circumscribed around triangle $ABC$.
2021 AMC 12/AHSME Fall, 23
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
\frac{29}{30} \qquad\textbf{(E)}\ 1$
2006 Baltic Way, 20
A $12$-digit positive integer consisting only of digits $1,5$ and $9$ is divisible by $37$. Prove that the sum of its digits is not equal to $76$.
2019 Iran MO (2nd Round), 5
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.
Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!
2005 Polish MO Finals, 2
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
2024 Indonesia TST, 3
Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$.
Determine the smallest number of pieces Paul needs to make in order to accomplish this.
1995 Tournament Of Towns, (474) 2
Do there exist
(a) four
(b) five
distinct positive integers such that the sum of any three of them is a prime number?
(V Senderov)
1954 AMC 12/AHSME, 31
In triangle $ ABC$, $ AB\equal{}AC$, $ \angle A\equal{}40^\circ$. Point $ O$ is within the triangle with $ \angle OBC \cong \angle OCA$. The number of degrees in angle $ BOC$ is:
$ \textbf{(A)}\ 110 \qquad
\textbf{(B)}\ 35 \qquad
\textbf{(C)}\ 140 \qquad
\textbf{(D)}\ 55 \qquad
\textbf{(E)}\ 70$
2008 Postal Coaching, 4
Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.
2013 Junior Balkan Team Selection Tests - Romania, 1
Find all pairs of integers $(x,y)$ satisfying the following condition:
[i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$
[/i]
Tournament of Towns
2001 Turkey MO (2nd round), 1
Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$