Let $ABC$ be a triangle with incircle $(I)$ and circumcircle $(O)$. Let $D,E,F$ be the points where $(I)$ touches $BC,CA,AB$, respectively. Let $\omega_A,\omega_B,\omega_C$ be circles tangent to both $(I)$ and $(O)$, respectively at $D,K$ for $\omega_A$, at $E,M$ for $\omega_B$, and at $F,N$ for $\omega_C$. Prove that:
1. The lines $DK,EM,FN$ concur at $P$.
2. The orthocenter of triangle $DEF$ lies on the segment $OP$.
The original statement is retained, including the word “segment” and the unspecified tangency branches. It is false on the branch used by the supplied proof; see the explicit counterexample.
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
There are $n$ chairs on a circle, numbered $1$ through $n$. Select $k$ chairs. Two selected chairs are called adjacent when they are consecutive among the selected chairs. Find the number of selections for which between any two adjacent selected chairs there are at least three other chairs.
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Find all functions $f:\mathbb Z\to\mathbb Z$ such that, for all integers $x,y,z$,
$$f(x^3+y^3+z^3)=f(x)^3+f(y)^3+f(z)^3.$$
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Prove that for all positive real numbers $a,b,c$,
$$\frac{a^3}{(a+b)^3}+\frac{b^3}{(b+c)^3}+\frac{c^3}{(c+a)^3}\ge\frac38.$$
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
Let $p>3$ be a prime. In each part, $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$, and the congruence condition belongs to that part. Calculate:
(a) If $p\equiv1\pmod4$,
$$S=\sum_{k=1}^{(p-1)/2}\left(\left\lfloor\frac{2k^2}{p}\right\rfloor-2\left\lfloor\frac{k^2}{p}\right\rfloor\right).$$
(b) If $p\equiv1\pmod8$,
$$S=\sum_{k=1}^{(p-1)/2}\left\lfloor\frac{k^2}{p}\right\rfloor.$$
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.
A positive integer is called a “2005-diamond number” if its decimal representation contains 2005 consecutive digits all equal to 9. Let $(a_n)_{n\ge1}$ be a strictly increasing sequence of positive integers satisfying $a_n\lt nC$ for all $n\ge1$, where $C$ is some fixed positive real constant. Prove that the sequence contains infinitely many 2005-diamond numbers.
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; the rubric is editorial. Import multipliers: correct 1, incorrect 0, blank 0.