Found problems: 31
2005 India IMO Training Camp, 1
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$
India EGMO 2021 TST, 6
Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$.
Show that $a^2+b^2-ab$ is not a square.
2008 India National Olympiad, 3
Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 \plus{} bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.
2014 Postal Coaching, 2
Let $O$ be the centre of the square $ABCD$. Let $P,Q,R$ be respectively on the segments $OA,OB,OC$ such that $OP=3,OQ=5,OR=4$. Suppose $S$ is on $OD$ such that $X=AB\cap PQ,Y=BC\cap QR$ and $Z=CD\cap RS$ are collinear. Find $OS$.
India EGMO 2022 TST, 2
Let $a,b$ be arbitrary co-prime natural numbers. Alice writes the natural number $t < b$ on a blackboard. Every second she replaces the number on the blackboard, say $x$, with the smallest natural number in $\{x \pm a, x \pm b \}$ that she has not yet ever written. She keeps doing this as long as possible. Prove that this process goes on indefinitely and that Alice will write down every natural number.
[i]~Pranjal Srivastava and Rohan Goyal[/i]
2011 India Regional Mathematical Olympiad, 6
Find the largest real constant $\lambda$ such that
\[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\]
For all positive real numbers $a,b,c.$
2014 India Regional Mathematical Olympiad, 5
Let $a,b,c$ be positive real numbers such that
\[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \]
Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?
2005 India IMO Training Camp, 1
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$
2016 India Regional Mathematical Olympiad, 1
Let $ABC$ be an isosceles triangle with $AB=AC.$ Let $ \Gamma $ be its circumcircle and let $O$ be the centre of $ \Gamma $ . let $CO$ meet $ \Gamma$ in $D .$ Draw a line parallel to $AC$ thrugh $D.$ Let it intersect $AB$ at $E.$ Suppose $AE : EB=2:1$ .Prove that $ABC$ is an equilateral triangle.
1993 India National Olympiad, 9
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
1998 India National Olympiad, 1
In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.
1995 India Regional Mathematical Olympiad, 6
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with centre $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the centre point $O$ in their interior?
2023 India IMO Training Camp, 2
In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to like $DK$ meets the circle with diameter $BC$ at points $X,Y$. Prove that $AX \cdot DY = DX \cdot AY$
2019 India Regional Mathematical Olympiad, 3
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations
$$a^2+ab=c$$
$$b^2+bc=a$$
$$c^2+ca=b$$
1988 India National Olympiad, 4
If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that
\[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]
2024 India IMOTC, 1
A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped.
[i]Proposed by Anant Mudgal and Rohan Goyal[/i]
2002 India National Olympiad, 3
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.
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.
2019 India IMO Training Camp, P1
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that
\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]
Prove that
\[5m+12n\le 581.\]
2022-23 IOQM India, 10
Consider the $10$-digit number $M=9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M=9\underline{87}6 \underline{54} 3210$ by interchanging the $2$ underlined pairs, and keeping the others in their places, we get $M_{1}=9786453210$. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $M$.
2019 India IMO Training Camp, P1
Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that
\[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\]
Prove that
\[5m+12n\le 581.\]
1995 India Regional Mathematical Olympiad, 4
Show that the quadratic equation $x^2 + 7x - 14 (q^2 +1) =0$ , where $q$ is an integer, has no integer root.
1997 India Regional Mathematical Olympiad, 4
In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$
2018 CMI B.Sc. Entrance Exam, 6
Imagine the unit square in the plane to be a [i]carrom board[/i]. Assume the [i]striker[/i] is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x,y)$ and its initial velocity $\overrightarrow{(p,q)}$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its [i]bounce number[/i] to be the number of edges it hits before returning to its initial state for the $1^{\text{st}}$ time.
For example, the trajectory with initial state $[(.5,.5);\overrightarrow{(1,0)}]$ has bounce number $2$ and it returns to its initial state for the $1^{\text{st}}$ time in $2$ time units. And the trajectory with initial state $[(.25,.75);\overrightarrow{(1,1)}]$ has bounce number $4$.
$\textbf{(a)}$ Suppose the striker has initial state $[(.5,.5);\overrightarrow{(p,q)}]$. If $p>q\geqslant 0$ then what is its velocity after it hits an edge for the $1^{\text{st}}$ time ? What if $q>p\geqslant 0$ ?
$\textbf{(b)}$ Draw a trajectory with bounce number $5$ or justify why it is impossible.
$\textbf{(c)}$ Consider the trajectory with initial state $[(x,y);\overrightarrow{(p,0)}]$ where $p$ is a positive integer. In how much time will the striker $1^{\text{st}}$ return to its initial state ?
$\textbf{(d)}$ What is the bounce number for the initial state $[(x,y);\overrightarrow{(p,q)}]$ where $p,q$ are relatively prime positive integers, assuming the striker never hits a corner ?
1999 India National Olympiad, 2
In a village $1998$ persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equla to $3996$ feet. For this purpose, the field was divided into $1998$ equal parts. If each part had an integer area, find the length and breadth of the field.