# [Solutions] India Regional Mathematical Olympiad 2018

1. Let $ABC$ be a triangle with integer sides in which $AB < AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB, AC) > 1$.
2. Let $n$ be a natural number. Find all real numbers $x$ satsfying the equation $$\sum_{k=1}^{n}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}{4}.$$
3. For a rational number $r$, its period is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers $r$ of the form $r = 0.\overline{abcdefgh}$ having period $8$, find the sum of all the elements of $S$.
4. Let $E$ denote the set of $25$ points $(m, n)$ in the $xy$-plane, where $m$, $n$ are natural numbers, $1 \leq m \leq 5$, $1 \leq n \leq 5$. Suppose the points of $E$ are arbitrarily coloured using two colours, red and blue. Show that there always exist four points in the set E of the form $(a, b)$, $(a + k, b)$, $(a + k, b + k)$, $(a, b + k)$ for some positive integer k such that at least three of these four points have the same colour. (That is, there always exist four points in the set E which form the vertices of a square and having at least three points of the same colour.)
5. Find all natural numbers n such that $1 + [\sqrt{2n}]$ divides $2n$. (For any real number $x$, $[x]$ denotes the largest integer not exceeding $x$.)
6. Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $I$ be the incentre of triangle $ABC$, and let $D$, $E$, $F$ be the points at which its incircle touches the sides $BC$, $CA$, $AB$, respectively. Let $BI$, $CI$ meet the line $EF$ at $Y$, $X$, respectively. Further assume that both $X$ and $Y$ are outside the triangle $ABC$. Prove that
a) $B$, $C$, $Y$, $X$ are concyclic; and
b) $I$ is also the incentre of triangle $DYX$.
 MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng $\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu: $\TeX$, PDF, WORD, IMG,...