Olympiad problems

2020 SMO, 3

Tags: S(J)MO , geometry

Let $\triangle ABC$ be an acute scalene triangle with incenter $I$ and incircle $\omega$. Two points $X$ and $Y$ are chosen on minor arcs $AB$ and $AC$, respectively, of the circumcircle of triangle $\triangle ABC$ such that $XY$ is tangent to $\omega$ at $P$ and $\overline{XY}\perp \overline{AI}$. Let $\omega$ be tangent to sides $AC$ and $AB$ at $E$ and $F$, respectively. Denote the intersection of lines $XF$ and $YE$ as $T$. Prove that if the circumcircles of triangles $\triangle TEF$ and $\triangle ABC$ are tangent at some point $Q$, then lines $PQ$, $XE$, and $YF$ are concurrent. [i]Proposed by Andrew Wen[/i]

2020 SJMO, 3

Tags: geometry , S(J)MO

Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$. [i]Proposed by Andrew Wen[/i]

2020 SJMO, 4

Tags: S(J)MO , geometry

Let $B$ and $C$ be points on a semicircle with diameter $AD$ such that $B$ is closer to $A$ than $C$. Diagonals $AC$ and $BD$ intersect at point $E$. Let $P$ and $Q$ be points such that $\overline{PE} \perp \overline{BD}$ and $\overline{PB} \perp \overline{AD}$, while $\overline{QE} \perp \overline{AC}$ and $\overline{QC} \perp \overline{AD}$. If $BQ$ and $CP$ intersect at point $T$, prove that $\overline{TE} \perp \overline{BC}$. [i]Proposed by Andrew Wen[/i]

2020 SMO, 4

Tags: S(J)MO , algebra , combinatorics

Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$, where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$, such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$. [i]Proposed by Grant Yu[/i]

2020 SJMO, 1

Tags: S(J)MO , number theory

Find all positive integers $k \geq 2$ for which there exists some positive integer $n$ such that the last $k$ digits of the decimal representation of $10^{10^n} - 9^{9^n}$ are the same. [i]Proposed by Andrew Wen[/i]

2020 SMO, 5

Tags: S(J)MO , geometry

In triangle $\triangle ABC$, let $E$ and $F$ be points on sides $AC$ and $AB$, respectively, such that $BFEC$ is cyclic. Let lines $BE$ and $CF$ intersect at point $P$, and $M$ and $N$ be the midpoints of $\overline{BF}$ and $\overline{CE}$, respectively. If $U$ is the foot of the perpendicular from $P$ to $BC$, and the circumcircles of triangles $\triangle BMU$ and $\triangle CNU$ intersect at second point $V$ different from $U$, prove that $A, P,$ and $V$ are collinear. [i]Proposed by Andrew Wen and William Yue[/i]

2020 SJMO, 5

Tags: algebra , S(J)MO , inequalities

A nondegenerate triangle with perimeter $1$ has side lengths $a, b,$ and $c$. Prove that \[\left|\frac{a - b}{c + ab}\right| + \left|\frac{b - c}{a + bc}\right| + \left|\frac{c - a}{b + ac}\right| < 2.\] [i]Proposed by Andrew Wen[/i]

2020 SJMO, 6

Tags: S(J)MO , number theory , combinatorics

We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty. [i]Proposed by Anthony Wang[/i]

2020 SMO, 6

Tags: S(J)MO , algebra , number theory

We say that a number is [i]angelic[/i] if it is greater than $10^{100}$ and all of its digits are elements of $\{1,3,5,7,8\}$. Suppose $P$ is a polynomial with nonnegative integer coefficients such that over all positive integers $n$, if $n$ is angelic, then the decimal representation of $P(s(n))$ contains the decimal representation of $s(P(n))$ as a contiguous substring, where $s(n)$ denotes the sum of digits of $n$. Prove that $P$ is linear and its leading coefficient is $1$ or a power of $10$. [i]Proposed by Grant Yu[/i]

2020 SJMO, 2

Tags: combinatorics , S(J)MO , carefully

Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made. [i]Proposed by Andrew Wen[/i]

2020 SMO, 1

Tags: S(J)MO , number theory

The sequence of positive integers $a_0, a_1, a_2, \ldots$ is recursively defined such that $a_0$ is not a power of $2$, and for all nonnegative integers $n$: (i) if $a_n$ is even, then $a_{n+1} $ is the largest odd factor of $a_n$ (ii) if $a_n$ is odd, then $a_{n+1} = a_n + p^2$ where $p$ is the smallest prime factor of $a_n$ Prove that there exists some positive integer $M$ such that $a_{m+2} = a_m $ for all $m \geq M$. [i]Proposed by Andrew Wen[/i]

2020 SMO, 2

Tags: combinatorics , S(J)MO

Adam has a single stack of $3 \cdot 2^n$ rocks, where $n$ is a nonnegative integer. Each move, Adam can either split an existing stack into two new stacks whose sizes differ by $0$ or $1$, or he can combine two existing stacks into one new stack. Adam keeps performing such moves until he eventually gets at least one stack with $2^n$ rocks. Find, with proof, the minimum possible number of times Adam could have combined two stacks. [i]Proposed by Anthony Wang[/i]