OSN 2019

Given that $n$ and $r$ are positive integers. Suppose that $$1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r)$$ Prove that $n$ is a composite number.

Given $19$ red boxes and $200$ blue boxes filled with balls. None of which is empty. Suppose that every red boxes have a maximum of $200$ balls and every blue boxes have a maximum of $19$ balls. Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes. Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.

Given that $ABCD$ is a rectangle such that $AD > AB$ , where $E$ is on $AD$ such that $BE \perp AC$ . Let $M$ be the intersection of $AC$ and $BE$ . Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$ . Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$ . Suppose $FG$ intersects $AB$ at $P$ . Prove that $PM = PN$ .

Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown $a+b=c$ $d+e+f=g+h$ $i+j+k+l=m+n+o$ and so on...

Where at the $j$ -th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms.

Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$ .

Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$ .

For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown. $1+3=4$ $2+5+8=6+9$ $7+11+12+14=13+15+16$

Given that $a$ and $b$ are real numbers such that for infinitely many positive integers $m$ and $n$ , $$\lfloor an + b \rfloor \ge \lfloor a + bn \rfloor$$ $$\lfloor a + bm \rfloor \ge \lfloor am + b \rfloor$$ Prove that $a = b$ .

Given a circle with center $O$ , such that $A$ is not on the circumcircle. Let $B$ be the reflection of $A$ with respect to $O$ . Now let $P$ be a point on the circumcircle. The line perpendicular to $AP$ through $P$ intersects the circle at $Q$ . Prove that $AP \times BQ$ remains constant as $P$ varies.

Determine all solutions of $$x + y^2 = p^m$$ $$x^2 + y = p^n$$ For $x,y,m,n$ positive integers and $p$ being a prime.

Let $n > 1$ be a positive integer and $a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}$ . Suppose $$a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1$$ Prove that some of $a_1, a_2, \dots, a_{2n}$ have sum 0.