$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

For how many primes $p$ less than $15$, there exists integer triples $(m,n,k)$ such that \[ \begin{array}{rcl} m+n+k &\equiv& 0 \pmod p \\ mn+mk+nk &\equiv& 1 \pmod p \\ mnk &\equiv& 2 \pmod p. \end{array} \] $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

Let $p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0 $ and $q(x) = x^5 + c_4 x^4 + \cdots +c_0$ . If $p(i)=q(i)$ for $i=1,2,3,\cdots,6$ . Show that there exists a negative integer r such that $p(r)=q(r)$ .

Let $n>3$ be an integer. If $x_1<x_2<\ldots<x_{n+2}$ are reals with $x_1=0$, $x_2=1$ and $x_3>2$, what is the maximal value of $$(\frac{x_{n+1}+x_{n+2}-1}{x_{n+1}(x_{n+2}-1)})\cdot (\sum_{i=1}^{n}\frac{(x_{i+2}-x_{i+1})(x_{i+1}-x_i)}{x_{i+2}-x_i})?$$

A trapezium $ ABCD$, in which $ AB$ is parallel to $ CD$, is inscribed in a circle with centre $ O$. Suppose the diagonals $ AC$ and $ BD$ of the trapezium intersect at $ M$, and $ OM \equal{} 2$. [b](a)[/b] If $ \angle AMB$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b](b)[/b] If $ \angle AMD$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides. [b][Weightage 17/100][/b]

How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?

Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that $I_1I_2 \perp $ bisector of $\angle ABC$

Consider the two triangles $ ABC$ and $ PQR$ shown below. In triangle $ ABC, \angle ADB \equal{} \angle BDC \equal{} \angle CDA \equal{} 120^\circ$. Prove that $ x\equal{}u\plus{}v\plus{}w$. [asy]unitsize(7mm); defaultpen(linewidth(.7pt)+fontsize(10pt)); pair C=(0,0), B=4*dir(5); pair A=intersectionpoints(Circle(C,5), Circle(B,6))[0]; pair Oc=scale(sqrt(3)/3)*rotate(30)*(B-A)+A; pair Ob=scale(sqrt(3)/3)*rotate(30)*(A-C)+C; pair D=intersectionpoints(Circle(Ob,length(Ob-C)), Circle(Oc,length(Oc-B)))[1]; real s=length(A-D)+length(B-D)+length(C-D); pair P=(6,0), Q=P+(s,0), R=rotate(60)*(s,0)+P; pair M=intersectionpoints(Circle(P,length(B-C)), Circle(Q,length(A-C)))[0]; draw(A--B--C--A--D--B); draw(D--C); label("$B$",B,SE); label("$C$",C,SW); label("$A$",A,N); label("$D$",D,NE); label("$a$",midpoint(B--C),S); label("$b$",midpoint(A--C),WNW); label("$c$",midpoint(A--B),NE); label("$u$",midpoint(A--D),E); label("$v$",midpoint(B--D),N); label("$w$",midpoint(C--D),NNW); draw(P--Q--R--P--M--Q); draw(M--R); label("$P$",P,SW); label("$Q$",Q,SE); label("$R$",R,N); label("$M$",M,NW); label("$x$",midpoint(P--R),NW); label("$x$",midpoint(P--Q),S); label("$x$",midpoint(Q--R),NE); label("$c$",midpoint(R--M),ESE); label("$a$",midpoint(P--M),NW); label("$b$",midpoint(Q--M),NE);[/asy]

Determine the number of positive integers $n \leq 1000$ such that the sum of the digits of $5n$ and the sum of the digits of $n$ are the same.

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

Let $I$ be the incenter of a scalene triangle $\vartriangle ABC$. In other words, $\overline{AB},\overline{BC},\overline{CA}$ are distinct. Prove that if $D,E$ are two points on rays $\overrightarrow{BA},\overrightarrow{CA}$, satisfying $\overline{BD}=\overline{CA},\overline{CE}=\overline{BA}$ then line $DE$ pass through the orthocenter of $\vartriangle BIC$. [img]http://2.bp.blogspot.com/-aHCD5tL0FuA/XnYM1LoZjWI/AAAAAAAALeE/C6hO9W9FGhcuUP3MQ9aD7SNq5q7g_cY9QCK4BGAYYCw/s1600/imoc2019g1.png[/img]

$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.

Let $n$ be a positive integer and let $d_1, d_2, \dots, d_k$ be its positive divisors. Consider the number $$f(n) = (-1)^{d_1}d_1 + (-1)^{d_2}d_2 + \dots + (-1)^{d_k}d_k$$ Assume $f(n)$ is a power of 2. Show if $m$ is an integer greater than 1, then $m^2$ does not divide $n$.

Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that: i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$; ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$. Find the maximal value of $M$.

The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.

In triangle $\vartriangle ABC$ holds $\angle A= 40^o$ and $\angle B = 20^o$ . The point $P$ lies on the line $AC$ such that $C$ is between $A$ and $P$ and $| CP | = | AB | - | BC |$. Calculate the $\angle CBP$.

The two lines with slopes $2$ and $1/2$ pass through an arbitrary point $T$ on the axis $Oy$ and intersect the hyperbola $y=1/x$ at two points. [b]a)[/b] Prove that these four points lie on a circle. [b]b)[/b] The point $T$ runs through the entire $y$-axis. Find the locus of centers of such circles. [i](I. Gorodnin)[/i]

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

Let $ABCD$ be a square. Let $M$ be the midpoint of $BC$ and $N$ be the point on $AB$ such that $2AN=BN$. If the area of $\triangle DMN$ is 15, find the area of square $ABCD$. [i]Proposed by Harry Kim[/i]

For $n=2^m$ (m is a positive integer) consider the set $M(n) = \{ 1,2,...,n\}$ of natural numbers. Prove that there exists an order $a_1, a_2, ..., a_n$ of the elements of M(n), so that for all $1\leq i < j < k \leq n$ holds: $a_j - a_i \neq a_k - a_j$.

Let $p$ be a prime integer greater than $10^k$. Pete took some multiple of $p$ and inserted a $k-$digit integer $A$ between two of its neighbouring digits. The resulting integer C was again a multiple of $p$. Pete inserted a $k-$digit integer $B$ between two of neighbouring digits of $C$ belonging to the inserted integer $A$, and the result was again a multiple of $p$. Prove that the integer $B$ can be obtained from the integer $A$ by a permutation of its digits. [i](8 points)[/i] [i]Ilya Bogdanov[/i]

Given a $ \triangle ABC $ and a point $ P. $ Let $ O$, $D$, $E$, $F $ be the circumcenter of $ \triangle ABC$, $\triangle BPC$, $\triangle CPA$, $\triangle APB, $ respectively and let $ T $ be the intersection of $ BC $ with $ EF. $ Prove that the reflection of $ O $ in $ EF $ lies on the perpendicular from $ D $ to $ PT. $ [i]Proposed by Telv Cohl[/i]