Find all triangles $ABC$ with $\angle BCA$ acute such that the perpendicular bisector of $BC$ meets rays $Ax,Ay$ trisecting $\angle BAC$, with $\angle BAx=\angle xAy=\angle yAC$, at $M,N$ satisfying
$$AB=NP=2HM,$$
where $H$ is the perpendicular projection of $A$ on $BC$ and $M$ is the midpoint of $BC$.
The notation above reproduces the supplied statement literally; see the explicit defect and conditional interpretation in the solution.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
A positive integer $N_0$ is written on a board. Players A and B alternate, A starting, replacing the current positive integer $N$ by either $N-1$ or $\lfloor N/3\rfloor$. The first player to write 0 wins. Determine which player has a guaranteed winning strategy when (1) $N_0=120$; (2) $N_0=(3^{2002}-1)/2$; (3) $N_0=(3^{2002}+1)/2$. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Let $m$ be a positive integer having a prime divisor greater than $\sqrt{2m}+1$. Find the smallest positive integer $M$ for which there is a finite set $T$ of pairwise distinct positive integers such that (i) its smallest and largest elements are respectively $m,M$; (ii) the product of all its elements is a perfect square.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Let $n\ge2$ be an integer. In an $n\times2n$ board ($n$ rows and $2n$ columns), mark an arbitrary $n^2$ unit cells. Prove that for every integer $k$ satisfying $1\lt k\le\lfloor n/2\rfloor+1$, one can select $k$ rows such that the resulting $k\times2n$ board has at least
$$\frac{k!(n-2k+2)}{(n-k+1)(n-k+2)\cdots(n-1)}$$
columns all of whose cells are marked. Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Find all polynomials $P(x)$ with integer coefficients such that
$$Q(x)=(x^2+6x+10)P(x)^2-1$$
is the square of a polynomial with integer coefficients.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Prove that there exist an integer $m\ge2002$ and pairwise distinct positive integers $a_1,a_2,\ldots,a_m$ such that
$$\prod_{i=1}^{m}a_i^2-4\sum_{i=1}^{m}a_i^2$$
is a perfect square.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; this rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Let $ABC$ be an isosceles triangle with $AB=AC$. A point $O$ varies on the line $BC$, subject to the condition that the circle with center $O$ and radius $OA$ has neither $AB$ nor $AC$ as a tangent. The lines $AB,AC$ meet the circle again at $M,N$, respectively. Determine the locus of the orthocenter of triangle $AMN$.
Let $m\lt 2001$ and $n\lt 2002$ be positive integers. Distinct real numbers are placed in a table with $2001$ rows and $2002$ columns. An entry is called bad if it is smaller than at least $m$ entries in its column and at least $n$ entries in its row. What is the largest possible number of bad entries?
Let $n$ be a positive integer. Prove that the equation
$$\frac1{x-1}+\frac1{2^2x-1}+\cdots+\frac1{n^2x-1}=\frac12$$
has a unique solution $x_n\gt 1$. Prove that $x_n$ tends to $4$ as $n$ tends to infinity.