Found problems: 31
2017 India National Olympiad, 2
Suppose $n \ge 0$ is an integer and all the roots of $x^3 +
\alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$.
2024 India IMOTC, 11
There are $n\ge 3$ particles on a circle situated at the vertices of a regular $n$-gon. All these particles move on the circle with the same constant speed. One of the particles moves in the clockwise direction while all others move in the anti-clockwise direction. When particles collide, that is, they are all at the same point, they all reverse the direction of their motion and continue with the same speed as before.
Let $s$ be the smallest number of collisions after which all particles return to their original positions. Find $s$.
[i]Proposed by N.V. Tejaswi[/i]
2017 India National Olympiad, 1
In the given figure, $ABCD$ is a square sheet of paper. It is folded along $EF$ such that $A$ goes to a point $A'$ different from $B$ and $C$, on the side $BC$ and $D$ goes to $D'$. The line $A'D'$ cuts $CD$ in $G$. Show that the inradius of the triangle $GCA'$ is the sum of the inradii of the triangles $GD'F$ and $A'BE$.
[asy]
size(5cm);
pair A=(0,0),B=(1,0),C=(1,1),D=(0,1),Ap=(1,0.333),Dp,Ee,F,G;
Ee=extension(A,B,(A+Ap)/2,bisectorpoint(A,Ap));
F=extension(C,D,(A+Ap)/2,bisectorpoint(A,Ap));
Dp=reflect(Ee,F)*D;
G=extension(C,D,Ap,Dp);
D(MP("A",A,W)--MP("E",Ee,S)--MP("B",B,E)--MP("A^{\prime}",Ap,E)--MP("C",C,E)--MP("G",G,NE)--MP("D^{\prime}",Dp,N)--MP("F",F,NNW)--MP("D",D,W)--cycle,black);
draw(Ee--Ap--G--F);
dot(A);dot(B);dot(C);dot(D);dot(Ap);dot(Dp);dot(Ee);dot(F);dot(G);
draw(Ee--F,dashed);
[/asy]
2004 India IMO Training Camp, 1
Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.
1996 India Regional Mathematical Olympiad, 4
Suppose $N$ is an $n$ digit positive integer such that
(a) all its digits are distinct;
(b) the sum of any three consecutive digits is divisible by $5$.
Prove that $n \leq 6$. Further, show that starting with any digit, one can find a six digit number with these properties.
1990 India National Olympiad, 5
Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity
\[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\]
must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?