Found problems: 85335
2006 Rioplatense Mathematical Olympiad, Level 3, 3
An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?
2012 India Regional Mathematical Olympiad, 1
Find with proof all nonzero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.
2021 Science ON all problems, 1
Consider the complex numbers $x,y,z$ such that
$|x|=|y|=|z|=1$. Define the number
$$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$
$\textbf{(a)}$ Prove that $a$ is a real number.
$\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$.
[i] (Stefan Bălăucă & Vlad Robu)[/i]
2000 IMO Shortlist, 7
Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?
2012 ELMO Shortlist, 2
For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$.
a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$.
b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$.
[i]Anderson Wang.[/i]
2019 Canadian Mathematical Olympiad Qualification, 4
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
2020 ASDAN Math Tournament, 5
Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac1 2$ , then compute $a^2 + b^2$.
2010 Puerto Rico Team Selection Test, 2
Find two three-digit numbers $x$ and $y$ such that the sum of all other three digit numbers is equal to $600x$.
2012 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with $\angle A = 90^o$, $AB = 1$, and $AC = 2$. Let $\ell$ be a line through $A$ perpendicular to $BC$, and let the perpendicular bisectors of $AB$ and $AC$ meet $\ell$ at $E$ and $F$, respectively. Find the length of segment $EF$.
1985 Kurschak Competition, 3
We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.
2009 Stanford Mathematics Tournament, 5
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes
2005 Junior Balkan Team Selection Tests - Romania, 3
In a country 6 cities are connected two by two with round-trip air routes operated by exactly one of the two air companies in that country.
Prove that there exist 4 cities $A$, $B$, $C$ and $D$ such that each of the routes $A\leftrightarrow B$, $B\leftrightarrow C$, $C\leftrightarrow D$ and $D\leftrightarrow A$ are operated by the same company.
[i]Dan Schwartz[/i]
2019 Korea Junior Math Olympiad., 3
Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition:
There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.
1985 Putnam, B2
Define polynomials $f_{n}(x)$ for $n \geq 0$ by $f_{0}(x)=1, f_{n}(0)=0$ for $n \geq 1,$ and
$$
\frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1)
$$
for $n \geq 0 .$ Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
2011 Korea National Olympiad, 4
Let $k,n$ be positive integers. There are $kn$ points $P_1, P_2, \cdots, P_{kn}$ on a circle. We can color each points with one of color $ c_1, c_2, \cdots , c_k $. In how many ways we can color the points satisfying the following conditions?
(a) Each color is used $ n $ times.
(b) $ \forall i \not = j $, if $ P_a $ and $ P_b $ is colored with color $ c_i $ , and $ P_c $ and $ P_d $ is colored with color $ c_j $, then the segment $ P_a P_b $ and segment $ P_c P_d $ doesn't meet together.
1987 AMC 8, 7
The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is
[asy]
unitsize(36);
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((3,0)--(5.2,1.4)--(5.2,4.4)--(3,3));
draw((0,3)--(2.2,4.4)--(5.2,4.4));
fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle,black);
fill((1,1)--(1,2)--(2,2)--(2,1)--cycle,black);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,black);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle,black);
draw((1,3)--(3.2,4.4));
draw((2,3)--(4.2,4.4));
draw((.733333333,3.4666666666)--(3.73333333333,3.466666666666));
draw((1.466666666,3.9333333333)--(4.466666666,3.9333333333));
fill((1.73333333,3.46666666666)--(2.7333333333,3.46666666666)--(3.46666666666,3.93333333333)--(2.46666666666,3.93333333333)--cycle,black);
fill((3,1)--(3.733333333333,1.466666666666)--(3.73333333333,2.46666666666)--(3,2)--cycle,black);
fill((3.73333333333,.466666666666)--(4.466666666666,.93333333333)--(4.46666666666,1.93333333333)--(3.733333333333,1.46666666666)--cycle,black);
fill((3.73333333333,2.466666666666)--(4.466666666666,2.93333333333)--(4.46666666666,3.93333333333)--(3.733333333333,3.46666666666)--cycle,black);
fill((4.466666666666,1.9333333333333)--(5.2,2.4)--(5.2,3.4)--(4.4666666666666,2.9333333333333)--cycle,black);
[/asy]
$\text{(A)}\ 10 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 22 \qquad \text{(E)}\ 24$
1996 Romania National Olympiad, 2
$ a,b,c,d \in [0,1]$ and $ x,y,z,t \in [0, \frac{1}{2}]$ and $ a+b+c+d=x+y+z+t=1$.prove that:
$ (i)$ $ ax+by+cz+dt$ $ \geq$ $ min( {\frac{a+b}{2} , \frac{b+c}{2} , \frac{c+d}{2} , \frac{d+a}{2} , \frac{a+c}{2} , \frac{b+d}{2} )}$
$ (ii)$ $ ax+by+cz+dt$ $ \geq$ $ 54abcd$
2003 Argentina National Olympiad, 4
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
KoMaL A Problems 2018/2019, A. 754
Let $P$ be a point inside the acute triangle $ABC,$ and let $Q$ be the isogonal conjugate of $P.$ Let $L,M$ and $N$ be the midpoints of the shorter arcs $BC,CA$ and $AB$ of the circumcircle of $ABC,$ respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $(PBC),$ let $X_B$ be the intersection of ray $MQ$ and circle $PCA,$ and let $X_C$ be the intersection of ray $NQ$ and circle $(PAB).$ Prove that $P,X_A,X_B$ and $X_C$ are concyclic or coincide.
[i]Proposed by Gustavo Cruz (São Paulo)[/i]
1960 Kurschak Competition, 3
$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.
2023 239 Open Mathematical Olympiad, 8
Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.
2016 China National Olympiad, 1
Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that
$a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$
Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$
2002 Romania Team Selection Test, 2
Let $P(x)$ and $Q(x)$ be integer polynomials of degree $p$ and $q$ respectively. Assume that $P(x)$ divides $Q(x)$ and all their coefficients are either $1$ or $2002$. Show that $p+1$ is a divisor of $q+1$.
[i]Mihai Cipu[/i]
1996 Tuymaada Olympiad, 5
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$
1994 Bundeswettbewerb Mathematik, 3
Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$