Olympiad

Few Problems - 1

Aug 30, 2016 • 3 min read

Given a straight line $l$ and points $A$ and $B$ on the same side of it find the shortest path from $A$ to $B$ which touches $l$ .
Given a function $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that
$f(x+y)=f(x)+f(y)$
Show that $f(x)=xf(1)$ for all $x\in\mathbb{Q}$ . Thus, infer that for such a function knowing its value at one non-zero point is sufficient for finding its values at any point.
Find all integer values for $x$ , $y$ and $z$ satisfying the equation
$x^2+y^2+z^2=2xyz$ .
Given a triangle $ABC$ , let $AP$ , $BQ$ and $CR$ be its altitudes. Show that $AP$ , $BQ$ and $CR$ are concurrent i.e. they meet a single point.
For a given prime number $p$ find the number of quadruples $(a,b,c,d)$ such that $ad-bc\neq 0$ (mod $p$ ).
There are six cities. Any two of them are connected by either a road or a railway track but not both. Show that there are at least three cities among these which are connected by a common mode of transport.

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