problem_id stringlengths 16 19 | problem stringlengths 69 2.04k | solution stringlengths 60 23.9k |
|---|---|---|
sols-TST-IMO-2014_2 | Let $a_1$, $a_2$, $a_3$, \dots be a sequence of integers,
with the property that every consecutive group of $a_i$'s
averages to a perfect square.
More precisely, for all positive integers $n$ and $k$, the quantity
\[ \frac{a_n + a_{n+1} + \dots + a_{n+k-1}}{k} \]
is always the square of an integer.
Prove that the sequ... | Let $\nu_p(n)$ denote the largest exponent of $p$ dividing $n$.
The problem follows from the following proposition.
\begin{proposition*}
Let $(a_n)$ be a sequence of integers and let $p$ be a prime.
Suppose that every consecutive group of $a_i$'s
with length at most $p$ averages to a perfect square.
Then $\nu_... |
sols-TST-IMO-2014_3 | Let $n$ be an even positive integer,
and let $G$ be an $n$-vertex (simple) graph
with exactly $\frac{n^2}{4}$ edges.
An unordered pair of distinct vertices $\{x,y\}$
is said to be \emph{amicable} if they have a common neighbor
(there is a vertex $z$ such that $xz$ and $yz$ are both edges).
Prove that $G$ has at least $... | First, we prove the following lemma.
(\url{https://en.wikipedia.org/wiki/Friendship_paradox}).
\begin{lemma*}
[On average, your friends are more popular than you]
For a vertex $v$, let $a(v)$ denote
the average degree of the neighbors of $v$
(setting $a(v) = 0$ if $\deg v = 0$).
Then
\[ \sum_v a(v) \ge \sum... |
sols-TST-IMO-2014_4 | Let $n$ be a positive even integer,
and let $c_1$, $c_2$, \dots, $c_{n-1}$ be real numbers satisfying
\[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \]
Prove that
\[ 2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2 \]
has no real roots. | We will prove the polynomial is positive for all $x \in \RR$.
As $c_i > 0$, the result is vacuous for $x \le 0$,
so we restrict attention to $x > 0$.
Then letting $c_i = 1 - d_i$ for each $i$,
the inequality we want to prove becomes
\[ x^n + 1 + \frac{x^{n+1}+1}{x+1} > \sum_1^{n-1} d_i x^i
\qquad\text{given } \sum |... |
sols-TST-IMO-2014_5 | Let $ABCD$ be a cyclic quadrilateral,
and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively.
Let $W$, $X$, $Y$ and $Z$ be the orthocenters of
triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively.
Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area. | The following solution is due to Grace Wang.
We begin with:
\begin{claim*}
Point $W$ has coordinates $\half(2a+b+d)$.
\end{claim*}
\begin{proof}
The orthocenter of $\triangle DAB$ is $d+a+b$,
and $\triangle AHE$ is homothetic to $\triangle DAB$
through $A$ with ratio $1/2$.
Hence $w = \half(a+(d+a+b))$ as ne... |
sols-TST-IMO-2014_6 | For a prime $p$, a subset $S$ of residues modulo $p$
is called a \emph{sum-free multiplicative subgroup}
of $\FF_p$ if
\begin{itemize}
\ii there is a nonzero residue $\alpha$ modulo $p$
such that $S = \left\{ 1, \alpha^1, \alpha^2, \dots \right\}$
(all considered mod $p$), and
\ii there are no $a,b,c \in S$
(... | We first prove the following general lemma.
\begin{lemma*}
If $f,g \in \ZZ[X]$ are relatively prime nonconstant polynomials,
then for sufficiently large primes $p$,
they have no common root modulo $p$.
\end{lemma*}
\begin{proof}
By B\'{e}zout Lemma,
there exist polynomials $a(X)$ and $b(X)$ in $\ZZ[X]$
and... |
sols-TST-IMO-2015_1 | Let $ABC$ be a scalene triangle with incenter $I$ whose incircle is
tangent to $\ol{BC}$, $\ol{CA}$, $\ol{AB}$ at $D$, $E$, $F$,
respectively. Denote by $M$ the midpoint of $\ol{BC}$ and
let $P$ be a point in the interior of $\triangle ABC$
so that $MD = MP$ and $\angle PAB = \angle PAC$.
Let $Q$ be a point on the inc... | We present two solutions.
\paragraph{Official solution.}
Assume without loss of generality that $AB < AC$; we show $\angle PQE=90^{\circ}$.
\begin{center}
\begin{asy}
size(9cm);
pair A = dir(130);
pair B = dir(210);
pair C = dir(330);
filldraw(A--B--C--cycle, opacity(0.2)+palecyan, blue);
pa... |
sols-TST-IMO-2015_2 | Prove that for every positive integer $n$, there exists a set $S$ of $n$ positive integers
such that for any two distinct $a,b \in S$, $a-b$ divides $a$ and $b$
but none of the other elements of $S$. | The idea is to look for a
sequence $d_1, \dots, d_{n-1}$ of ``differences''
such that the following two conditions hold.
Let $s_i = d_1 + \dots + d_{i-1}$,
and $t_{i,j} = d_i + \dots + d_{j-1}$ for $i \le j$.
\begin{enumerate}
\item[(i)] No two of the $t_{i,j}$ divide each other.
\item[(ii)] There exists an integer $a$... |
sols-TST-IMO-2015_3 | A physicist encounters $2015$ atoms called usamons.
Each usamon either has one electron or zero electrons,
and the physicist can't tell the difference.
The physicist's only tool is a diode.
The physicist may connect the diode from any usamon $A$
to any other usamon $B$. (This connection is directed.)
When she does so,... | The answer is no.
Call the usamons $U_1, \dots, U_m$ (here $m=2015$).
Consider models $M_k$ of the following form:
$U_1, \dots, U_k$ are all charged for some $0 \le k \le m$
and the other usamons are not charged.
Note that for any pair there's a model
where they are different states, by construction.
We can consider t... |
sols-TST-IMO-2015_4 | Let $f \colon \QQ \to \QQ$ be a function such that for any $x,y \in \QQ$,
the number $f(x+y)-f(x)-f(y)$ is an integer.
Decide whether there must exist a constant $c$
such that $f(x) - cx$ is an integer for every rational number $x$. | No, such a constant need not exist.
One possible solution is as follows:
define a sequence by $x_0 = 1$ and
\begin{align*}
2x_1 &= x_0 \\
2x_2 &= x_1 + 1 \\
2x_3 &= x_2 \\
2x_4 &= x_3 + 1 \\
2x_5 &= x_4 \\
2x_6 &= x_5 + 1 \\
&\vdotswithin=
\end{align*}
Set $f(2^{-k}) = x_k$ and $f(2^k)=2^k$ for $k=0,1,\d... |
sols-TST-IMO-2015_5 | Fix a positive integer $n$.
Find the smallest positive integer $\chi$ for which
there exists a tournament on $n$ vertices,
and a coloring of each of the tournament's edges by one of $\chi$ colors, such that:
any two directed edges $u \to v$ and $v \to w$ have different colors. | The answer is
\[ \chi = \left\lceil \log_2 n \right\rceil. \]
First, we prove by induction on $n$
that $\chi \ge \log_2 n$ for any coloring and any tournament.
The base case $n = 1$ is obvious.
Now given any tournament, consider any used color $c$.
Then it should be possible to divide the tournament
into two subsets $... |
sols-TST-IMO-2015_6 | Let $ABC$ be a non-equilateral triangle
and let $M_a$, $M_b$, $M_c$ be the midpoints
of the sides $BC$, $CA$, $AB$, respectively.
Let $S$ be a point lying on the Euler line.
Denote by $X$, $Y$, $Z$ the second intersections
of $M_a S$, $M_b S$, $M_c S$ with the nine-point circle.
Prove that $AX$, $BY$, $CZ$ are concurre... | We assume now and forever that $ABC$ is scalene since the problem
follows by symmetry in the isosceles case.
We present four solutions.
\paragraph{First solution by barycentric coordinates (Evan Chen).}
Let $AX$ meet $M_b M_c$ at $D$,
and let $X$ reflected over $M_b M_c$'s midpoint be $X'$.
Let $Y'$, $Z'$, $E$, $F$ be... |
sols-TST-IMO-2016_1 | Let $S = \{1, \dots, n\}$.
Given a bijection $f \colon S \to S$ an \emph{orbit} of $f$ is a
set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$.
We denote by $c(f)$ the number of distinct orbits of $f$.
For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$,
the two orbits are $\{1,2\}$ and $\{3\}$, hence... | Most motivated solution is to consider $n - c(f)$ and show this is the transposition distance.
Dumb graph theory works as well. |
sols-TST-IMO-2016_2 | Let $ABC$ be a scalene triangle with circumcircle $\Omega$,
and suppose the incircle of $ABC$ touches $BC$ at $D$.
The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $K$ and $M$.
The circumcircle of $\triangle DKM$ intersects the $A$-excircle
at $S_1$, $S_2$, and $\Omega$ at $T \neq M$.
Prove that line $AT$ pa... | We present an angle-chasing solution,
and then a more advanced alternative finish.
\paragraph{First solution (angle chasing).}
Assume for simplicity $AB < AC$.
Let $E$ be the contact point of the $A$-excircle on $BC$;
also let ray $TD$ meet $\Omega$ again at $L$.
From the fact that $\angle MTL = \angle MTD = 180^{\cir... |
sols-TST-IMO-2016_3 | Let $p$ be a prime number. Let $\FF_p$ denote the integers modulo $p$,
and let $\FF_p[x]$ be the set of polynomials with coefficients in $\FF_p$.
Define $\Psi \colon \FF_p[x] \to \FF_p[x]$ by
\[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \]
Prove that for nonzero polynomials $F,G \in \FF_p[x]$,... | Observe that $\Psi$ is also a linear map of $\FF_p$ vector spaces,
and that $\Psi(xP) = \Psi(P)^p$ for any $P \in \FF_p[x]$.
(In particular, $\Psi(1) = x$, not $1$, take caution!)
\paragraph{First solution (Ankan Bhattacharya).}
We start with:
\begin{claim*}
If $P \mid Q$ then $\Psi(P) \mid \Psi(Q)$.
\end{claim*}
\b... |
sols-TST-IMO-2016_4 | Let $\sqrt{3}=1.b_1b_2b_3\dots_{(2)}$ be the binary representation
of $\sqrt 3$. Prove that for any positive integer $n$, at least one of
the digits $b_n, b_{n+1}, \dots, b_{2n}$ equals 1. | Assume the contrary, so that for some integer $k$ we have
\[k < 2^{n-1} \sqrt 3 < k + \frac{1}{2^{n+1}}. \]
Squaring gives
\begin{align*}
k^2 < 3 \cdot 2^{2n-2} &< k^2 + \frac{k}{2^n} + \frac{1}{2^{2n+2}} \\
&\le k^2 + \frac{2^{n-1} \sqrt 3}{2^n} + \frac{1}{2^{2n+2}} \\
&= k^2 + \frac{\sqrt3}{2} + \frac{1}{2^{2n+2}} \\... |
sols-TST-IMO-2016_5 | Let $n \ge 4$ be an integer.
Find all functions $W \colon \{1, \dots, n\}^2 \to \RR$ such that
for every partition $[n] = A \cup B \cup C$ into disjoint sets,
\[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c)
= |A| |B| |C|. \] | Of course, $W(k,k)$ is arbitrary for $k \in [n]$.
We claim that $W(a,b) = \pm 1$ for any $a \neq b$, with the sign fixed.
(These evidently work.)
First, let $X_{abc} = W(a,b)W(b,c)$ for all distinct $a$, $b$, $c$,
so the given condition is
\[ \sum_{a,b,c \in A \times B \times C} X_{abc} = |A| |B| |C|. \]
Consider the ... |
sols-TST-IMO-2016_6 | Let $ABC$ be an acute scalene triangle
and let $P$ be a point in its interior.
Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto
triangle sides $BC$, $CA$, $AB$, respectively.
Find the locus of points $P$ such that
$AA_1$, $BB_1$, $CC_1$ are concurrent
and $\angle PAB + \angle PBC + \angle PCA = 90\dg$.
\end{enumerat... | In complex numbers with $ABC$ the unit circle,
it is equivalent to solving the following two cubic equations
in $p$ and $q = \ol p$:
\begin{align*}
(p-a)(p-b)(p-c) &= (abc)^2 (q -1/a)(q - 1/b)(q - 1/c) \\
0 &= \prod_{\text{cyc}} (p+c-b-bcq) + \prod_{\text{cyc}} (p+b-c-bcq).
\end{align*}
Viewing this as two cubic curv... |
sols-TST-IMO-2017_1 | In a sports league, each team uses a set of at most $t$ signature colors.
A set $S$ of teams is \emph{color-identifiable} if one can assign
each team in $S$ one of their signature colors,
such that no team in $S$ is assigned
\emph{any} signature color of a different team in $S$.
For all positive integers $n$ and $t$,
d... | Answer: $\left\lceil n/t \right\rceil$.
To see this is an upper bound, note that one can easily construct
a sports league with that many teams anyways.
A quick warning:
\begin{remark*}
[Misreading the problem]
It is common to misread the problem by ignoring the word ``any''.
Here is an illustration.
Suppose ... |
sols-TST-IMO-2017_2 | Let $ABC$ be an acute scalene triangle with circumcenter $O$,
and let $T$ be on line $BC$ such that $\angle TAO = 90\dg$.
The circle with diameter $\ol{AT}$
intersects the circumcircle of $\triangle BOC$ at two points
$A_1$ and $A_2$, where $OA_1 < OA_2$.
Points $B_1$, $B_2$, $C_1$, $C_2$ are defined analogously.
\begi... | Let triangle $ABC$ have circumcircle $\Gamma$.
Let $\triangle XYZ$ be the tangential triangle of $\triangle ABC$
(hence $\Gamma$ is the incircle of $\triangle XYZ$),
and denote by $\Omega$ its circumcircle.
Suppose the symmedian $\ol{AX}$ meets $\Gamma$ again at $D$,
and let $M$ be the midpoint of $\ol{AD}$.
Finally, l... |
sols-TST-IMO-2017_3 | Let $P, Q \in \RR[x]$ be relatively prime nonconstant polynomials.
Show that there can be at most three real numbers $\lambda$
such that $P + \lambda Q$ is the square of a polynomial. | This is true even with $\RR$ replaced by $\CC$,
and it will be necessary to work in this generality.
\paragraph{First solution using transformations.}
We will prove the claim in the following form:
\begin{claim*}
Assume $P, Q \in \CC[x]$ are relatively prime.
If $\alpha P + \beta Q$ is a square for four different
cho... |
sols-TST-IMO-2017_4 | You are cheating at a trivia contest.
For each question, you can peek at each of the
$n > 1$ other contestant's guesses before writing your own.
For each question, after all guesses are submitted, the emcee announces the correct answer.
A correct guess is worth $0$ points.
An incorrect guess is worth $-2$ points for ot... | We will prove the result with $2^{n-1}$ replaced
even by $2^{n-2}+1$.
We first make the following reductions.
First, change the weights to be $+1$, $-1$, $0$ respectively
(rather than $0$, $-2$, $-1$); this clearly has no effect.
Also, WLOG that all contestants except you initially have score zero
(and that your score... |
sols-TST-IMO-2017_5 | Let $ABC$ be a triangle with altitude $\ol{AE}$.
The $A$-excircle touches $\ol{BC}$ at $D$,
and intersects the circumcircle at two points $F$ and $G$.
Prove that one can select points $V$ and $N$
on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus. | Let $I$ denote the incenter, $J$ the $A$-excenter,
and $L$ the midpoint of $\ol{AE}$.
Denote by $\ol{IY}$, $\ol{IZ}$ the tangents
from $I$ to the $A$-excircle.
Note that lines $\ol{BC}$, $\ol{GF}$, $\ol{YZ}$ then concur at $H$
(unless $AB=AC$, but this case is obvious),
as it's the radical center of cyclic hexagon $BIC... |
sols-TST-IMO-2017_6 | Prove that there are infinitely many triples $(a,b,p)$ of integers,
with $p$ prime and $0 < a \le b < p$,
for which $p^5$ divides $(a+b)^p - a^p - b^p$.
\end{enumerate} | The key claim is that if $p \equiv 1 \pmod 3$,
then
\[ p(x^2+xy+y^2)^2 \text{ divides } (x+y)^p - x^p - y^p \]
as polynomials in $x$ and $y$.
Since it's known that one can select $a$ and $b$ such that
$p^2 \mid a^2 + ab + b^2$, the conclusion follows.
(The theory of quadratic forms tells us we can do it with $p^2 = a^2... |
sols-TST-IMO-2018_1 | Let $n \ge 2$ be a positive integer,
and let $\sigma(n)$ denote the sum of the positive divisors of $n$.
Prove that the $n$\textsuperscript{th} smallest positive integer
relatively prime to $n$ is at least $\sigma(n)$,
and determine for which $n$ equality holds. | The equality case is $n = p^e$ for
$p$ prime and a positive integer $e$.
It is easy to check that this works.
\bigskip
\paragraph{First solution.}
In what follows, by $[a,b]$ we mean $\{a,a+1,\dots,b\}$.
First, we make the following easy observation.
\begin{claim*}
If $a$ and $d$ are positive integers,
then preci... |
sols-TST-IMO-2018_2 | Find all functions $f \colon \ZZ^2 \to [0,1]$
such that for any integers $x$ and $y$,
\[ f(x, y) = \frac{f(x-1, y) + f(x, y-1)}{2}. \] | We claim that the only functions $f$ are constant functions.
(It is easy to see that they work.)
\paragraph{First solution (hands-on).}
First, iterating the functional equation
relation to the $n$th level shows that
\[ f(x, y) = \frac{1}{2^n} \sum_{i=0}^n \binom{n}{i} f(x-i, y-(n-i)). \]
In particular,
\begin{align*}
... |
sols-TST-IMO-2018_4 | Let $n$ be a positive integer and let $S \subseteq \{0,1\}^n$
be a set of binary strings of length $n$.
Given an odd number $x_1, \dots, x_{2k+1} \in S$ of binary strings
(not necessarily distinct), their \emph{majority} is defined as
the binary string $y \in \{0,1\}^n$ for which
the $i$\textsuperscript{th} bit of $y$ ... | Let $M$ denote the majority function (of any length).
\paragraph{First solution (induction).}
We prove all $P_k$ are equivalent by induction on $n \ge 2$,
with the base case $n = 2$ being easy to check by hand.
(The case $n=1$ is also vacuous; however,
the inductive step is not able to go from $n=1$ to $n=2$.)
For th... |
sols-TST-IMO-2018_5 | Let $ABCD$ be a convex cyclic quadrilateral which is not a kite,
but whose diagonals are perpendicular and meet at $H$.
Denote by $M$ and $N$ the midpoints of $\ol{BC}$ and $\ol{CD}$.
Rays $MH$ and $NH$ meet $\ol{AD}$ and $\ol{AB}$ at $S$ and $T$, respectively.
Prove there exists a point $E$, lying outside quadrilatera... | The main claim is that $E$ is the
intersection of $(ABCD)$ with the circle with diameter $\ol{AH}$.
\begin{center}
\begin{asy}
size(10cm);
pair A = dir(100);
pair B = dir(190);
pair D = dir(-10);
pair F = -A;
pair H = foot(A, B, D);
pair E = foot(A, F, H);
pair C = -A+2*foot(origin, A, H);
filldraw(unitcircle, opacity... |
sols-TST-IMO-2018_6 | Alice and Bob play a game.
First, Alice secretly picks a finite set $S$
of lattice points in the Cartesian plane.
Then, for every line $\ell$ in the plane
which is horizontal, vertical, or has slope $+1$ or $-1$,
she tells Bob the number of points of $S$ that lie on $\ell$.
Bob wins if he can then determine the set $S$... | Clearly Bob can compute the number $N$ of points.
The main claim is that:
\begin{claim*}
Fix $m$ and $n$ as in the problem statement.
Among all sets $T \subseteq \ZZ^2$ with $N$ points,
the set $S$ is the \emph{unique} one which maximizes the value of
\[ F(T):=\sum_{(x,y)\in T} (x^2+y^2)(m+n-(x^2+y^2)). \]
\e... |
sols-TST-IMO-2019_1 | Let $ABC$ be a triangle and let $M$ and $N$
denote the midpoints of $\ol{AB}$ and $\ol{AC}$, respectively.
Let $X$ be a point such that $\ol{AX}$
is tangent to the circumcircle of triangle $ABC$.
Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\ol{MX}$,
and by $\omega_C$ the circle through $N$ and $C$ t... | We present four solutions,
the second of which shows that $M$ and $N$
can be replaced by any two points on $AB$ and $AC$
satisfying $AM/AB + AN/AC = 1$.
\paragraph{First solution using symmedians (Merlijn Staps).}
Let $\ol{XY}$ be the other tangent from $X$ to $(AMN)$.
\begin{claim*}
Line $\ol{XM}$ is tangent to $(B... |
sols-TST-IMO-2019_2 | Let $\ZZ/n\ZZ$ denote the set of integers considered modulo $n$
(hence $\ZZ/n\ZZ$ has $n$ elements).
Find all positive integers $n$ for which there exists a bijective function
$g \colon \ZZ/n\ZZ \to \ZZ/n\ZZ$,
such that the $101$ functions
\[ g(x), \quad g(x)+x, \quad g(x)+2x, \quad \dots, \quad g(x)+100x \]
are all bi... | Call a function $g$ \emph{valiant} if it obeys this condition.
We claim the answer is all numbers relatively prime to $101!$.
The construction is to just let $g$ be the identity function.
Before proceeding to the converse solution,
we make a long motivational remark.
\begin{remark*}
[Motivation for both parts]
Th... |
sols-TST-IMO-2019_3 | A \emph{snake of length $k$} is an animal
which occupies an ordered $k$-tuple
$(s_1, \dots, s_k)$ of cells in an $n \times n$ grid of square unit cells.
These cells must be pairwise distinct,
and $s_i$ and $s_{i+1}$ must share a side for $i=1,\dots,k-1$.
If the snake is currently occupying $(s_1, \dots, s_k)$
and $s$ i... | The answer is yes (and $0.9$ is arbitrary).
\paragraph{First grid-based solution.}
The following solution is due to Brian Lawrence.
For illustration reasons, we give below a figure
of a snake of length $89$ turning around in an $11 \times 11$ square
(which generalizes readily to odd $n$).
We will see that a snake of l... |
sols-TST-IMO-2019_4 | We say a function $f \colon \ZZ_{\ge 0} \times \ZZ_{\ge 0} \to \ZZ$
is \emph{great} if for any nonnegative integers $m$ and $n$,
\[ f(m+1, n+1) f(m,n) - f(m+1,n) f(m,n+1) = 1. \]
If $A = (a_0, a_1, \dots)$ and $B = (b_0, b_1, \dots)$
are two sequences of integers,
we write $A \sim B$ if there exists a great function $f... | We present two solutions.
In what follows, we say $(A, B)$ form a great pair if $A \sim B$.
\paragraph{First solution (Nikolai Beluhov).}
Let $k = a_0 = b_0 = c_0 = d_0$.
We let $f$, $g$, $h$ be great functions for $(A,B)$, $(B,C)$, $(C,D)$
and write the following infinite array:
\[
\begin{bmatrix}
& \vdots & \v... |
sols-TST-IMO-2019_5 | Let $n$ be a positive integer.
Tasty and Stacy are given a circular necklace
with $3n$ sapphire beads and $3n$ turquoise beads,
such that no three consecutive beads have the same color.
They play a cooperative game where they alternate turns
removing three consecutive beads, subject to the following conditions:
\begin{... | In the necklace, we draw a \emph{divider}
between any two beads of the same color.
Unless there are no dividers,
this divides the necklace into several \emph{zigzags}
in which the beads in each zigzag alternate.
Each zigzag has two \emph{endpoints}
(adjacent to dividers).
Observe that the condition about not having
th... |
sols-TST-IMO-2019_6 | Let $ABC$ be a triangle with incenter $I$,
and let $D$ be a point on line $BC$ satisfying $\angle AID = 90\dg$.
Let the excircle of triangle $ABC$ opposite the vertex $A$
be tangent to $\ol{BC}$ at point $A_1$.
Define points $B_1$ on $\ol{CA}$ and $C_1$ on $\ol{AB}$ analogously,
using the excircles opposite $B$ and $C$... | We present two solutions.
\paragraph{First solution using spiral similarity (Ankan Bhattacharya).}
First, we prove the part of the problem
which does not depend on the condition $A B_1 A_1 C_1$ is cyclic.
\begin{lemma*}
Let $ABC$ be a triangle and define $I$, $D$, $B_1$, $C_1$
as in the problem.
Moreover, let $M... |
sols-TST-IMO-2020_1 | Choose positive integers $b_1$, $b_2$, \dots\ satisfying
\[ 1 = \frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb \]
and let $r$ denote the largest real number
satisfying $\frac{b_n}{n^2} \ge r$ for all positive integers $n$.
What are the possible values of $r$ across all
possible choices o... | The answer is $0 \le r \le 1/2$. Obviously $r \ge 0$.
In one direction, we show that
\begin{claim*}
[Greedy bound]
For all integers $n$, we have
\[ \frac{b_n}{n^2} \le \half + \frac{1}{2n}. \]
\end{claim*}
\begin{proof}
This is by induction on $n$. For $n=1$ it is given.
For the inductive step we have
\beg... |
sols-TST-IMO-2020_2 | Two circles $\Gamma_1$ and $\Gamma_2$
have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$.
Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$.
A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$
again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD... | We present four solutions.
\paragraph{First solution, elementary (original).}
We have $\triangle YAC \sim \triangle YBD$, from which it follows
\[ \frac{d(Y,AC)}{d(Y,BD)} = \frac{AC}{BD}. \]
Moreover, if we denote by $r_1$ and $r_2$ the radii of $\Gamma_1$ and $\Gamma_2$, then
\[
\frac{d(T,AC)}{d(T,BD)}
= \frac{TA... |
sols-TST-IMO-2020_3 | Let $\alpha \ge 1$ be a real number.
Hephaestus and Poseidon play a turn-based game
on an infinite grid of unit squares.
Before the game starts, Poseidon chooses a finite
number of cells to be \emph{flooded}.
Hephaestus is building a \emph{levee},
which is a subset of unit edges of the grid, called \emph{walls},
formin... | We show that if $\alpha > 2$
then Hephaestus wins,
but when $\alpha = 2$ (and hence $\alpha \le 2$)
Hephaestus cannot contain even a single-cell flood initially.
\bigskip
\textbf{Strategy for $\alpha > 2$}:
Impose $\ZZ^2$ coordinates on the cells.
Adding more flooded cells does not make our task easier,
so let us ass... |
sols-TST-IMO-2020_4 | For a finite simple graph $G$, we define $G'$ to be the graph
on the same vertex set as $G$, where for any two vertices $u \neq v$,
the pair $\{u,v\}$ is an edge of $G'$ if and only if
$u$ and $v$ have a common neighbor in $G$.
Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$,
then $G$ is also ... | We say a vertex of a graph is \emph{fatal} if it has degree at least $3$,
and some two of its neighbors are not adjacent.
\begin{claim*}
The graph $G'$ has at least as many triangles as $G$,
and has strictly more if $G$ has any fatal vertices.
\end{claim*}
\begin{proof}
Obviously any triangle in $G$ persists in ... |
sols-TST-IMO-2020_5 | Find all integers $n \ge 2$ for which there exists an integer $m$ and
a polynomial $P(x)$ with integer coefficients satisfying the following three conditions:
\begin{itemize}
\item $m > 1$ and $\gcd(m,n) = 1$;
\item the numbers $P(0)$, $P^2(0)$, \dots, $P^{m-1}(0)$ are not divisible by $n$; and
\item $P^m(0)$ is ... | The answer is that this is possible if and only if
there exists primes $p' < p$ such that $p \mid n$ and $p' \nmid n$.
(Equivalently, the radical $\opname{rad}(n)$
must not be the product of the first several primes.)
For a polynomial $P$, and an integer $N$, we introduce the notation
\[ \mathbf{zord}(P \bmod N)
\co... |
sols-TST-IMO-2020_6 | Let $P_1P_2 \dots P_{100}$ be a cyclic $100$-gon,
and let $P_i = P_{i+100}$ for all $i$.
Define $Q_i$ as the intersection of diagonals
$\ol{P_{i-2}P_{i+1}}$ and $\ol{P_{i-1}P_{i+2}}$ for all integers $i$.
Suppose there exists a point $P$
satisfying $\ol{PP_i} \perp \ol{P_{i-1}P_{i+1}}$
for all integers $i$.
Prove that... | We show two solutions.
In addition, Luke Robitaille has a reasonable complex numbers solution
posted at \url{https://aops.com/community/p26795631}.
\paragraph{Solution to proposed problem.}
We let $\ol{PP_2}$ and $\ol{P_1P_3}$
intersect (perpendicularly) at point $K_2$,
and define $K_\bullet$ cyclically.
\begin{cente... |
sols-TST-IMO-2021_1 | Determine all integers $s \ge 4$
for which there exist positive integers $a$, $b$, $c$, $d$
such that $s = a+b+c+d$
and $s$ divides $abc+abd+acd+bcd$. | The answer is $s$ composite.
\paragraph{Composite construction.}
Write $s = (w+x)(y+z)$,
where $w$, $x$, $y$, $z$ are positive integers.
Let $a=wy$, $b=wz$, $c=xy$, $d=xz$.
Then
\[ abc+abd+acd+bcd = wxyz(w+x)(y+z) \]
so this works.
\paragraph{Prime proof.}
Choose suitable $a$, $b$, $c$, $d$. Then
\[
(a+b)(a+c)(a+d)... |
sols-TST-IMO-2021_2 | Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$
lie fixed on a circle $\Gamma$, in that order,
and such that $BU_2 > AU_1 > BV_2 > AV_1$.
Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$
not containing $A$ or $B$.
Line $XA$ meets line $U_1 V_1$ at $C$,
while line $XB$ meets line $U_2 V_2$ at $D$.
Prove there ... | For brevity, we let $\ell_i$ denote line $U_iV_i$ for $i=1,2$.
We first give an explicit description of the fixed point $K$.
Let $E$ and $F$ be points on $\Gamma$ such that $\ol{AE} \parallel \ell_1$
and $\ol{BF} \parallel \ell_2$.
The problem conditions imply that $E$ lies between $U_1$ and $A$
while $F$ lies between... |
sols-TST-IMO-2021_3 | Find all functions $f \colon \RR \to \RR$ that satisfy the inequality
\[
f(y) -
\left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq
f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2}
\]
for all real numbers $x < y < z$.
\end{enumerate} | Answer: all functions of the form $f(y) = a y^2 + by + c$, where
$a, b, c$ are constants with $a \leq 0$.
If $I = (x,z)$ is an interval,
we say that a real number $\alpha$ is a
\emph{supergradient} of $f$ at $y \in I$
if we always have
\[ f(t) \le f(y) + \alpha(t-y) \]
for every $t \in I$.
(This inequality may be fami... |
sols-TST-IMO-2023_1 | There are $2022$ equally spaced points
on a circular track $\gamma$ of circumference $2022$.
The points are labeled $A_1$, $A_2$, \dots, $A_{2022}$ in some order, each label used once.
Initially, Bunbun the Bunny begins at $A_1$.
She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$,
until she reaches $... | Replacing $2022$ with $2n$, the answer is $2n^2 - 2n + 2$.
(When $n=1011$, the number is $2042222$.)
\begin{center}
\begin{asy}
unitsize(48);
int n = 5;
int m = 2*n;
int[] a = {};
a.push(0);
for (int i = 1; i < m; ++i)
a.push(a[i-1] + (i % 2 > 0 ? n : n-1));
int[] b = new int[m];
... |
sols-TST-IMO-2023_2 | Let $ABC$ be an acute triangle.
Let $M$ be the midpoint of side $BC$,
and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively.
Suppose that the common external tangents
to the circumcircles of triangles $BME$ and $CMF$ intersect at a point $K$,
and that $K$ lies on the circumcircle of $ABC$.
Pro... | We present several distinct approaches.
\paragraph{Inversion solution submitted by Ankan Bhattacharya and Nikolai Beluhov.}
Let $H$ be the orthocenter of $\triangle ABC$.
We use inversion in the circle with diameter $\ol{BC}$.
We identify a few images:
\begin{itemize}
\ii The circumcircles of $\triangle BME$ and $\t... |
sols-TST-IMO-2023_3 | Consider pairs $(f,g)$ of functions
from the set of nonnegative integers to itself such that
\begin{itemize}
\ii $f(0) \ge f(1) \ge f(2) \ge \dots \ge f(300) \ge 0$;
\ii $f(0) + f(1) + f(2) + \dots + f(300) \leq 300$;
\ii for any $20$ nonnegative integers $n_1$, $n_2$, \dots, $n_{20}$,
not necessarily distinct,... | Replace $300 = \frac{24 \cdot 25}{2}$ with $\frac{s(s+1)}{2}$ where $s=24$, and
$20$ with $k$. The answer is $115440 = \frac{ks(ks+1)}{2}$. Equality is achieved
at $f(n) = \max (s-n, 0)$ and $g(n) = \max (ks-n,0)$. To prove
\[g(n_1+\dotsb+n_k)\leq f(n_1)+\dotsb+f(n_k),\]
write it as
\[\max(x_1+\dotsb+x_k, 0)\leq \max(x... |
sols-TST-IMO-2023_4 | For nonnegative integers $a$ and $b$,
denote their \emph{bitwise xor} by $a \oplus b$.
(For example, $9 \oplus 10 = 1001_2 \oplus 1010_2 = 0011_2 = 3$.)
Find all positive integers $a$ such that
for any integers $x > y \ge 0$, we have
\[ x \oplus ax \neq y \oplus ay. \] | Answer: the function $x \mapsto x \oplus ax$ is injective if and only if
$a$ is an even integer.
\paragraph{Even case.}
First, assume $\nu_2(a) = k > 0$.
We wish to recover $x$ from $c \coloneq x \oplus ax$.
Notice that:
\begin{itemize}
\ii The last $k$ bits of $c$ coincide with the last $k$ bits of $x$.
\ii Now t... |
sols-TST-IMO-2023_5 | Let $m$ and $n$ be fixed positive integers.
Tsvety and Freyja play a game on an infinite grid of unit square cells.
Tsvety has secretly written a real number inside of each cell so that
the sum of the numbers within every rectangle of size
either $m \times n$ or $n \times m$ is zero.
Freyja wants to learn all of these ... | The answer is the following:
\begin{itemize}
\ii If $\gcd(m, n) > 1$, then Freyja cannot win.
\ii If $\gcd(m, n) = 1$, then Freyja
can win in a minimum of $(m-1)^2 + (n-1)^2$ questions.
\end{itemize}
First, we dispose of the case where $\gcd(m, n) > 1$.
Write $d = \gcd(m, n)$.
The idea is that any labeling where... |
sols-TST-IMO-2023_6 | Fix a function $f \colon \NN \to \NN$ and for any $m,n \in \NN$ define
\[
\Delta(m,n) = \underbrace{f(f(\dots f}_{f(n)\text{ times}} (m)\dots)) %chktex 9
- \underbrace{f(f(\dots f}_{f(m)\text{ times}} (n)\dots)). %chktex 9
\]
Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \NN$.
Show that $\Delta$ is unbound... | Suppose for the sake of contradiction that $|\Delta(m,n)|\le N$ for all $m$, $n$.
Note that $f$ is injective, as
\[ f(m)=f(n) \implies \Delta(m,n) = 0 \implies m=n, \]
as desired.
Let $G$ be the ``arrow graph'' of $f$,
which is the directed graph with vertex set $\NN$ and edges $n\to f(n)$.
The first step in the solut... |
sols-TST-IMO-2024_1 | Find the smallest constant $C > 1$ such that the following statement holds:
for every integer $n \ge 2$ and sequence of non-integer
positive real numbers $a_1$, $a_2$, \dots, $a_n$ satisfying
\[ \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} = 1, \]
it's possible to choose positive integers $b_i$ such that
\begi... | \paragraph{Answer.} The answer is $C=\frac{3}{2}$.
\paragraph{Lower bound.}
Note that if $a_1 = \frac{4n-3}{2n-1}$ and $a_i = \frac{4n-3}{2}$ for $i > 1$,
then we must have $b_1 \in \{1,2\}$ and $b_i \in \{2n-2,2n-1\}$ for $i > 1$. If
we take $b_1 = 2$ then we obtain
\[
\frac{1}{b_1} + \frac{1}{b_2} + \dots + \frac{... |
sols-TST-IMO-2024_2 | Let $ABC$ be a triangle with incenter $I$.
Let segment $AI$ intersect the incircle of triangle $ABC$ at point $D$.
Suppose that line $BD$ is perpendicular to line $AC$.
Let $P$ be a point such that $\angle BPA = \angle PAI = 90^\circ$.
Point $Q$ lies on segment $BD$ such that the
circumcircle of triangle $ABQ$ is tange... | We show several approaches.
\paragraph{First solution, by author.}
\begin{center}
\begin{asy}
unitsize(1.0inches);
real eps = 1.15104332322704; /* trololol */
real v = 52 + eps;
real w = 150 + eps;
pair V = dir(v);
pair W = dir(w);
pair A = 2*V*W/(V+W);
pair D = dir((v+w)/2);
pai... |
sols-TST-IMO-2024_3 | Let $n > k \ge 1$ be integers and let $p$ be a prime dividing $\tbinom nk$.
Prove that the $k$-element subsets of $\{1, \dots, n\}$ can be
split into $p$ classes of equal size, such that
any two subsets with the same sum of elements belong to the same class. | Let $\sigma(S)$ denote the sum of the elements of $S$, so that
\[ P(x) \coloneq \sum_{\substack{S\subseteq \{1, \dots, n\} \\ |S|=k}} x^{\sigma(S)} \]
is the generating function for the sums of $k$-element subsets of $\{1, \dots, n\}$.
By Legendre's formula,
\[ \nu_p\left(\binom{n}{k}\right) = \sum_{r=1}^{\infty}
\l... |
sols-TST-IMO-2024_4 | Find all integers $n \ge 2$ for which there exists
a sequence of $2n$ pairwise distinct points $(P_1, \dots, P_n, Q_1, \dots, Q_n)$
in the plane satisfying the following four conditions:
\begin{enumerate}[label={(\roman*)}]
\ii no three of the $2n$ points are collinear;
\ii $P_iP_{i+1} \ge 1$ for all $i=1,2,\dots,n... | \paragraph{Answer.}
Even integers only.
\paragraph{Proof that even $n$ work.}
If we ignore the conditions that the points are pairwise distinct and form no
collinear triples, we may take
\[
P_{2i+1}=(0.51, 0), \quad P_{2i}=(-0.51, 0), \quad
Q_{2i+1}=(0, 0.51), \quad Q_{2i}=(0,-0.51).
\]
The distances $P_iP_{i+1}$ ... |
sols-TST-IMO-2024_5 | Suppose $a_1 < a_2 < \dots < a_{2024}$ is an arithmetic sequence of positive integers,
and $b_1 < b_2 < \dots < b_{2024}$ is a geometric sequence of positive integers.
Find the maximum possible number of integers that could appear in both sequences,
over all possible choices of the two sequences. | \paragraph{Answer.} $11$ terms.
\paragraph{Construction.} Let $a_i = i$ and $b_i = 2^{i-1}$.
\paragraph{Bound.}
We show a $\nu_p$-based approach communicated by Derek Liu,
which seems to be the shortest one.
At first, we completely ignore the geometric sequence $b_i$
and focus only on the arithmetic sequence.
\begin{c... |
sols-TST-IMO-2024_6 | Solve over $\RR$ the functional equation
\[ f(xf(y)) + f(y)=f(x+y)+f(xy). \]
\end{enumerate} | In addition to all constant functions, $f(x) \equiv x + 1$ clearly works too.
We prove these are the only solutions.
The solution that follows is by the original proposer.
Let $P(x,y)$ denote the given assertion.
\begin{claim}\label{periodic}
If $f$ is periodic, then $f$ is constant.
\end{claim}
\begin{proof}
Let... |
sols-TSTST-2011_1 | Find all real-valued functions $f$ defined on pairs of real numbers,
having the following property: for all real numbers $a, b, c$,
the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The \emph{median} of three real numbers, not necessarily distinct,
is the number that is in the middle when the thr... | The following solution is joint with Andrew He.
We prove the following main claim,
from which repeated applications can deduce the problem.
\begin{claim*}
Let $a < b < c$ be arbitrary. On $\{a,b,c\}^2$,
$f$ takes one of the following two forms,
where the column indicates the $x$-value
and the row indicates th... |
sols-TSTST-2011_2 | Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$.
Line $\ell$ is tangent to $\omega_1$ at $P$
and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$.
Let $X$ and $Y$ be points on major arcs $\arc{PA}$
(on $\omega_1$) and $\arc{AQ}$ (on $\omega_2$), respectively,
such that $AX/PX = AY/QY ... | We begin as follows:
\begin{claim*}
There is a spiral similarity centered at $X$ mapping $AR$ to $PQ$.
Similarly there is a spiral similarity centered at $Y$ mapping $SA$ to $PQ$.
\end{claim*}
\begin{proof}
Since $\dang XAR = \dang XAP = \dang XPQ$,
and $AR/AX = PQ/PX$ is given.
\end{proof}
Now the composition ... |
sols-TSTST-2011_3 | Prove that there exists a real constant $c$
such that for any pair $(x,y)$ of real numbers,
there exist relatively prime integers $m$ and $n$
satisfying the relation
\[ \sqrt{(x-m)^2 + (y-n)^2} < c \log(x^2+y^2+2). \] | This is actually the same problem as USAMO 2014/6. Surprise! |
sols-TSTST-2011_4 | Acute triangle $ABC$ is inscribed in circle $\omega$.
Let $H$ and $O$ denote its orthocenter and circumcenter, respectively.
Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively.
Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively.
Lines $MN$ and $PQ$ meet at $R$.
Prove that $\ol{OA} \perp \o... | Let $MH$ and $NH$ meet the nine-point circle again at $P'$ and $Q'$, respectively.
Recall that $H$ is the center of the homothety between
the circumcircle and the nine-point circle.
From this we can see that $P$ and $Q$ are the images of this homothety, meaning that
\[ HQ = 2HQ' \quad\text{and}\quad HP = 2HP'. \]
Since... |
sols-TSTST-2011_5 | At a certain orphanage, every pair of orphans are either friends or enemies.
For every three of an orphan's friends,
an even number of pairs of them are enemies.
Prove that it's possible to assign each orphan two parents
such that every pair of friends shares exactly one parent,
no pair of enemies shares any parents,
a... | Of course, we consider the graph with vertices as children and edges as friendships.
Consider all the maximal cliques in the graph.
\begin{claim*}
Every vertex is in at most two maximal cliques.
\end{claim*}
\begin{proof}
Indeed, consider a vertex $v$ adjacent to $w_1$ and $w_2$,
but with $w_1$ not adjacent to $w... |
sols-TSTST-2011_6 | Let $a,b,c$ be real numbers in the interval $[0,1]$
with $a+b,b+c,c+a \ge 1$. Prove that
\[ 1 \le (1-a)^2 + (1-b)^2 + (1-c)^2
+ \frac{2\sqrt2 abc}{\sqrt{a^2+b^2+c^2}}. \] | The following approach is due to Ashwin Sah.
We will prove the inequality for any $a$, $b$, $c$
the sides of a possibly degenerate triangle
(which is implied by the condition),
ignoring the particular constant $1$.
Homogenizing, we instead prove the problem
in the following form:
\begin{claim*}
We have
\[ k^2 \le ... |
sols-TSTST-2011_7 | Let $ABC$ be a triangle.
Its excircles touch sides $BC$, $CA$, $AB$ at $D$, $E$, $F$.
Prove that the perimeter of triangle $ABC$ is
at most twice that of triangle $DEF$. | Solution by August Chen:
It turns out that it is enough to take the
orthogonal projection of $EF$ onto side $BC$
(which has length $a-(s-a)(\cos B + \cos C)$)
and sum cyclically:
\begin{align*}
-s + \sum_{\text{cyc}} EF &\ge
-s +
\sum_{\text{cyc}}
\left[ a - (s-a)\left( \cos B + \cos C \right) \right] \\
&= s... |
sols-TSTST-2011_8 | Let $x_0$, $x_1$, \dots, $x_{n_0-1}$ be integers,
and let $d_1$, $d_2$, \dots, $d_k$ be positive integers
with $n_0 = d_1 > d_2 > \dotsb > d_k$ and
$\gcd(d_1, d_2, \dots, d_k) = 1$.
For every integer $n \geq n_0$, define
\[ x_n = \left\lfloor \frac{x_{n-d_1} + x_{n-d_2}
+ \dots + x_{n-d_k}}{k} \right\rfloor. \]
Show ... | Note that if the initial terms are contained
in some interval $[A,B]$ then they will remain in that interval.
Thus the sequence is eventually periodic.
Discard initial terms and let the period be $T$;
we will consider all indices modulo $T$ from now on.
Let $M$ be the maximal term in the sequence
(which makes sense si... |
sols-TSTST-2011_9 | Let $n$ be a positive integer.
Suppose we are given $2^n+1$ distinct sets,
each containing finitely many objects.
Place each set into one of two categories, the red sets and the blue sets,
so that there is at least one set in each category.
We define the \textit{symmetric difference} of two sets as
the set of objects b... | We can interpret the problem as working with
binary strings of length $\ell \ge n+1$,
with $\ell$ the number of elements across all sets.
Let $F$ be a field of cardinality $2^\ell$,
hence $F \cong \FF_2^{\oplus \ell}$.
Then, we can think of red/blue as elements of $F$,
so we have some $B \subseteq F$, and an $R \subs... |
sols-TSTST-2012_1 | Determine all infinite strings of letters with the following properties:
\begin{enumerate}
\item[(a)] Each letter is either $T$ or $S$,
\item[(b)] If position $i$ and $j$ both have the letter $T$,
then position $i+j$ has the letter $S$,
\item[(c)] There are infinitely many integers $k$ such that position $2k-... | We wish to find all infinite sequences $a_1, a_2, \dots$
of positive integers satisfying the following properties:
\begin{enumerate}[(a)]
\item $a_1 < a_2 < a_3 < \dotsb$,
\item there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$,
\item there are infinitely many $k$ su... |
sols-TSTST-2012_2 | Let $ABCD$ be a quadrilateral with $AC = BD$.
Diagonals $AC$ and $BD$ meet at $P$.
Let $\omega_1$ and $O_1$ denote the circumcircle and circumcenter of triangle $ABP$.
Let $\omega_2$ and $O_2$ denote the circumcircle and circumcenter of triangle $CDP$.
Segment $BC$ meets $\omega_1$ and $\omega_2$ again at $S$ and $T$
(... | Let $Q$ be the second intersection point of $\omega_1$, $\omega_2$.
Suffice to show $\ol{QP} \perp \ol{MN}$.
Now $Q$ is the center of a spiral \emph{congruence}
which sends $\ol{AC} \mapsto \ol{BD}$.
So $\triangle QAB$ and $\triangle QCD$ are similar isosceles triangles.
Now, \[ \dang QPA = \dang QBA = \dang DCQ = \dan... |
sols-TSTST-2012_3 | Let $\NN$ be the set of positive integers.
Let $f \colon \NN \to \NN$ be a function satisfying the following two conditions:
\begin{enumerate}[(a)]
\ii $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.
\ii $n \le f(n) \le n+2012$ for all $n$.
\end{enumerate}
Prove that for any natural ... | \paragraph{First short solution, by Jeffrey Kwan.}
Let $p_0$, $p_1$, $p_2$, \dots\ denote the
sequence of all prime numbers, in any order.
Pick \emph{any} primes $q_i$ such that
\[ q_0 \mid f(p_0), \quad q_1 \mid f(p_1), \quad
q_2 \mid f(p_2), \; \text{ etc}. \]
This is possible since each $f$ value above exceeds $1$... |
sols-TSTST-2012_4 | In scalene triangle $ABC$, let the feet of the perpendiculars
from $A$ to $\ol{BC}$, $B$ to $\ol{CA}$, $C$ to $\ol{AB}$
be $A_1, B_1, C_1$, respectively.
Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$.
Define $B_2$ and $C_2$ analogously.
Let $D$, $E$, $F$ be the respective midpoints
of sides $\ol{BC}$, $\o... | We claim that they pass through the orthocenter $H$.
Indeed, consider the circle with diameter $\ol{BC}$,
which circumscribes quadrilateral $BCB_1C_1$ and has center $D$.
Then by Brocard theorem, $\ol{AA_2}$ is the polar of line $H$.
Thus $\ol{DH} \perp \ol{AA_2}$. |
sols-TSTST-2012_5 | A rational number $x$ is given.
Prove that there exists a sequence
$x_0, x_1, x_2, \dots$ of rational numbers
with the following properties:
\begin{enumerate}[(a)]
\item $x_0=x$;
\item for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \frac{1}{n}$;
\item $x_n$ is an integer for some $n$.
\end{enumer... | Think of the sequence as a process over time.
We'll show that:
\begin{claim*}
At any given time $t$,
if the denominator of $x_t$ has some odd prime power $q = p^e$,
then we can delete a factor of $p$ from the denominator,
while only adding powers of two to the denominator.
\end{claim*}
(Thus we can just delete ... |
sols-TSTST-2012_6 | Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$.
Prove that
\[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}}
+ \sqrt{\frac{1+y^2}{1+y}}
+ \sqrt{\frac{1+z^2}{1+z}} \right)
\le \left( \frac{x+y+z}{3} \right)^{5/8} . \] | The key is the identity
\begin{align*}
\frac{x^2+1}{x+1} &= \frac{(x^2+1)(y+1)(z+1)}{(x+1)(y+1)(z+1)} \\
&= \frac{x(xyz+xy+xz)+x^2+yz+y+z+1}{2(1+x+y+z)} \\
&= \frac{x(x+y+z+1-yz) + x^2+yz+y+z+1}{2(1+x+y+z)} \\
&= \frac{(x+y)(x+z) + x^2 + \left( x-xyz+y+z+1 \right)}{2(1+x+y+z)} \\
&= \frac{2(x+y)(x+z)}{2(1+x+y... |
sols-TSTST-2012_7 | Triangle $ABC$ is inscribed in circle $\Omega$.
The interior angle bisector of angle $A$ intersects side $BC$
and $\Omega$ at $D$ and $L$ (other than $A$), respectively.
Let $M$ be the midpoint of side $BC$.
The circumcircle of triangle $ADM$ intersects sides
$AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respect... | By angle chasing, equivalent to show $\ol{MN} \parallel \ol{AD}$,
so discard the point $H$.
We now present a three solutions.
\paragraph{First solution using vectors.}
We first contend that:
\begin{claim*}
We have $QB = PC$.
\end{claim*}
\begin{proof}
Power of a Point gives $BM \cdot BD = AB \cdot QB$.
Then use ... |
sols-TSTST-2012_8 | Let $n$ be a positive integer.
Consider a triangular array of nonnegative integers as follows:
\begin{center}
\begin{asy}
size(8cm);
defaultpen(fontsize(10pt));
label(scale(0.8)*"Row $1$:", (-6, 2.9));
label(scale(0.8)*"Row $2$:", (-6, 2.2));
label(scale(0.8)*"$\vdots$", (-6, 1.5));
label(scale(0.8)*"Row $n... | Firstly, here are illustrative examples showing the arrays
for $(s_1, s_2, s_3, s_4) = (2, 5, 9, x)$ where $9 \le x \le 14$.
(The array has been left justified.)
\[
\begin{bmatrix}
2 & \swarrow \\
4 & 1 & \swarrow \\
5 & 3 & 1 & \swarrow \\
5 & 3 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
2 & \swar... |
sols-TSTST-2012_9 | Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called
\textit{simple} if it satisfies $(x \times y) \times z = x \times (y \times z)$
for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$.
Given a simple operation $\times$ on $S$, any string of elements in $S$
can be reduced to... | The answer is $(n!)^2$.
In fact it is possible to essentially find all $\times$:
one assigns a real number to each variable in $S$.
Then $x \times y$ takes the larger of $\{x,y\}$,
and in the event of a tie picks either ``left'' or ``right'',
where the choice of side is fixed among elements of each size.
\paragraph{Fi... |
sols-TSTST-2013_1 | Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle.
Line $\ell_a$ passes through the feet of the perpendiculars
from $A$ to $\ol{DB}$ and $\ol{DC}$.
Line $m_a$ passes through the feet of the perpendiculars from $D$ to $\ol{AB}$ and $\ol{AC}$.
Let $A_1$ denote the int... | In fact, it is true for any points $D$, $E$, $F$ on the circumcircle.
More strongly we contend:
\begin{claim*}
Point $A_1$ is the midpoint of $\ol{HD}$.
\end{claim*}
\begin{proof}
Lines $m_a$ and $\ell_a$ are Simson lines,
so they both pass through the point $(a+b+c+d)/2$
in complex coordinates.
\end{proof}
\b... |
sols-TSTST-2013_2 | A finite sequence of integers $a_1, a_2, \dots, a_n$ is called
\emph{regular} if there exists a real number $x$ satisfying
\[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \]
Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that
the term $a_k$ is \emph{forced} if the fo... | The answer is $985$. WLOG, by shifting $a_1 = 0$ (clearly $a_1$ isn't forced).
Now, we construct regular sequences inductively using the following procedure.
Start with the inequality \[ \tfrac01 \le x < \tfrac11. \]
Then for each $k = 2, 3, \dots, 1000$ we perform the following procedure.
If there is no fraction of th... |
sols-TSTST-2013_3 | Divide the plane into an infinite square grid by drawing
all the lines $x=m$ and $y=n$ for $m,n \in \ZZ$.
Next, if a square's upper-right corner has both coordinates even, color it black;
otherwise, color it white (in this way, exactly $1/4$ of the squares are black
and no two black squares are adjacent).
Let $r$ and $... | Here is Sammy Luo's solution.
Fix the speed of light at $\sqrt{r^2+s^2}$ units per second.
We prove periodicity every six seconds.
We re-color the white squares as red, blue, or green
according as to whether they have a black square directly
to the left/right, above/below, or neither, as shown below.
Finally, we fix t... |
sols-TSTST-2013_4 | Circle $\omega$, centered at $X$, is internally tangent to circle $\Omega$,
centered at $Y$, at $T$.
Let $P$ and $S$ be variable points on $\Omega$ and $\omega$,
respectively, such that line $PS$ is tangent to $\omega$ (at $S$).
Determine the locus of $O$ --- the circumcenter of triangle $PST$. | The answer is a circle centered at $Y$
with radius $\sqrt{YX \cdot YT}$,
minus the two points on line $XY$ itself.
We let $PS$ meet $\Omega$ again at $P'$,
and let $O'$ be the circumcenter of $\triangle TPS'$.
Note that $O'$, $X$, $O$ are collinear on the perpendicular
bisector of line $\ol{TS}$
Finally, we let $M$ de... |
sols-TSTST-2013_5 | Let $p$ be a prime.
Prove that in a complete graph with $1000p$ vertices
whose edges are labelled with integers,
one can find a cycle whose sum of labels is divisible by $p$. | Select $p-1$ disjoint triangles arbitrarily. If any of these triangles have $0$ sum modulo $p$ we are done. Otherwise, we may label the vertices $u_i$, $x_i$, and $v_i$ (where $1 \le i \le p-1$) in such a way that $u_ix_i + x_iv_i \neq u_iv_i$.
Let $A_i = \left\{ u_ix_i+x_iv_i, u_iv_i \right\}$. We can show that $\... |
sols-TSTST-2013_6 | Let $\NN$ be the set of positive integers.
Find all functions $f \colon \NN \to \NN$ that satisfy the equation
\[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \]
for all $a,b,c \ge 2$. (Here $f^k$ means $f$ applied $k$ times.) | The answer is $f(n) = n-1$ for $n \ge 3$ with $f(1)$ and $f(2)$ arbitrary;
check these work.
\begin{lemma*}
We have $f^{t^2-t}(t^2) = t$ for all $t$.
\end{lemma*}
\begin{proof}
We say $1 \le k \le 8$ is good if $f^{t^9-t^k}(t^9) = t^k$ for all $t$.
First, we observe that
\[ f^{t^9-t^3}(t^9) = t^3 \quad\text{and}\q... |
sols-TSTST-2013_7 | A country has $n$ cities, labelled $1,2,3,\dots,n$.
It wants to build exactly $n-1$ roads between certain pairs of cities
so that every city is reachable from every other city via some sequence of roads.
However, it is not permitted to put roads between pairs of cities
that have labels differing by exactly $1$,
and it ... | You can just spin the tree!
Fixing $n$, the group $G = \ZZ/n\ZZ$ acts on the set of trees
by rotation (where we imagine placing $1,2,\dots,n$ along a circle).
\begin{claim*}
For odd $n$, all trees have trivial stabilizer.
\end{claim*}
\begin{proof}
One way to see this is to look at the degree sequence.
Suppose ... |
sols-TSTST-2013_8 | Define a function $f \colon \NN \to \NN$ by $f(1) = 1$,
$f(n+1) = f(n) + 2^{f(n)}$ for every positive integer $n$.
Prove that $f(1)$, $f(2)$, \dots, $f(3^{2013})$
leave distinct remainders when divided by $3^{2013}$. | I'll prove by induction on $k \ge 1$ that any $3^k$
consecutive values of $f$ produce distinct residues modulo $3^k$.
The base case $k=1$ is easily checked
($f$ is always odd, hence $f$ cycles $1$, $0$, $2$ mod $3$).
For the inductive step, assume it's true for $k$.
Note that $2^\bullet \pmod{3^{k+1}}$ cycles every $2... |
sols-TSTST-2013_9 | Let $r$ be a rational number in the interval $[-1,1]$
and let $\theta = \cos^{-1} r$.
Call a subset $S$ of the plane good if $S$ is unchanged
upon rotation by $\theta$ around any point of $S$
(in both clockwise and counterclockwise directions).
Determine all values of $r$ satisfying the following property:
The midpoint... | The answer is that $r$ has this property
if and only if $r = \frac{4n-1}{4n}$ for some integer $n$.
Throughout the solution, we will let $r = \frac ab$ with $b > 0$ and
$\gcd(a,b) = 1$. We also let
\[ \omega = e^{i\theta} = \frac ab \pm \frac{\sqrt{b^2-a^2}}{b} i. \]
This means we may work with complex multiplication ... |
sols-TSTST-2014_1 | Let $\leftarrow$ denote the left arrow key on a standard keyboard. If
one opens a text editor and types the keys
``ab$\leftarrow$ cd $\leftarrow \leftarrow$ e $\leftarrow \leftarrow$ f'',
the result is ``faecdb''.
We say that a string $B$ is \emph{reachable} from a string $A$ if
it is possible to insert some amount of ... | Obviously $A$ and $B$ should have the
same multiset of characters, and we focus only on that situation.
\begin{claim*}
If $A = 123 \dots n$
and $B = \sigma(1) \sigma(2) \dots \sigma(n)$
is a permutation of $A$,
then $B$ is reachable if and only if
it is \textbf{213-avoiding},
i.e.\ there are no indices $i ... |
sols-TSTST-2014_2 | Consider a convex pentagon circumscribed about a circle.
We name the lines that connect vertices of the pentagon with
the opposite points of tangency with the circle \emph{gergonnians}.
\begin{enumerate}
\ii[(a)] Prove that if four gergonnians are concurrent,
then all five of them are concurrent.
\ii[(b)] Prove... | This problem is insta-killed by taking a homography
sending the concurrency point (in either part)
to the center of the circle while fixing the incircle.
Alternatively, one may send any four of the tangency points to a rectangle.
Here are the details.
Let $ABCDE$ be a pentagon with gergonnians
$\ol{AV}$, $\ol{BW}$, $\... |
sols-TSTST-2014_3 | Find all polynomials $P(x)$ with real coefficients that satisfy
\[ P(x \sqrt 2) = P(x + \sqrt{1-x^2}) \]
for all real numbers $x$ with $\lvert x \rvert \le 1$. | The answer is any polynomial of the form
$P(x) = f(U(x/\sqrt2))$, where $f \in \RR[x]$
and $U$ is the unique polynomial satisfying $U(\cos\theta) = \cos(8\theta)$.
Let $Q(x) = P(x \sqrt 2)$; then the condition reads
\[
Q(\cos\theta) = Q\left( \frac{1}{\sqrt2}(\cos\theta+\sin\theta) \right)
= Q(\cos(\theta-45\dg)) ... |
sols-TSTST-2014_4 | Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients,
with $P \neq 0$, and let $d = \deg P$.
Prove that there exist polynomials $A(x)$ and $B(x)$,
not both zero, such that $\max \{ \deg A, \deg B \} \le d/2$
and $P(x) \mid A(x) + Q(x) \cdot B(x)$. | Let $V$ be the vector space of real polynomials with degree at most $d/2$.
Consider maps of linear spaces
\begin{align*}
V^{\oplus 2} &\to \RR[x] / (P(x)) \\
\text{by} \qquad (A,B) &\mapsto A+QB \pmod P.
\end{align*}
The domain has dimension
\[ 2 \left( \left\lfloor d/2 \right\rfloor + 1 \right) \]
while the codoma... |
sols-TSTST-2014_5 | Find the maximum number $E$ such that the following holds:
there is an edge-colored graph with $60$ vertices and $E$ edges,
with each edge colored either red or blue, such that in that coloring,
there is no monochromatic cycles of length $3$
and no monochromatic cycles of length $5$. | The answer is $E = 30^2 + 2 \cdot 15^2 = 6 \cdot 15^2 = 1350$.
First, we prove $E \le 1350$. Observe that:
\begin{claim*}
$G$ contains no $K_5$.
\end{claim*}
\begin{proof}
It's a standard fact that the only triangle-free two-coloring
of the edges of $K_5$ is the union of two monochromatic $C_5$'s.
\end{proof}
H... |
sols-TSTST-2014_6 | Suppose we have distinct positive integers $a$, $b$, $c$, $d$
and an odd prime $p$ not dividing any of them,
and an integer $M$ such that if one considers the infinite sequence
\begin{align*}
ca &- db \\
ca^2 &- db^2 \\
ca^3 &- db^3 \\
ca^4 &- db^4 \\
&\vdots
\end{align*}
and looks at the highest power of $p$... | By orders, the indices of terms divisible by $p$
is an arithmetic subsequence of $\NN$:
say they are $\kappa$, $\kappa + \lambda$, $\kappa + 2\lambda$, \dots,
where $\lambda$ is the order of $a/b$.
That means we want
\[ \nu_p \left( c a^{\kappa + n\lambda} - d b^{\kappa + n \lambda} \right)
= \nu_p \left(
\left( ... |
sols-TSTST-2015_1 | Let $a_1$, $a_2$, \dots, $a_n$ be a sequence of real numbers,
and let $m$ be a fixed positive integer less than $n$.
We say an index $k$ with $1 \le k \le n$ is \textit{good} if there exists
some $\ell$ with $1 \le \ell \le m$ such that
\[ a_k + a_{k + 1} + \dotsb + a_{k + \ell - 1} \ge 0, \]
where the indices are take... | First we prove the result if the indices are not taken modulo $n$.
Call a number $\ell$-good if $\ell$ is the \emph{smallest} number
such that $a_k + a_{k+1} + \dots + a_{k+\ell-1} \ge 0$, and $\ell \le m$.
Then if $a_k$ is $\ell$-good,
the numbers $a_{k+1}, \dots, a_{k+\ell-1}$ are good as well.
Then by greedy from l... |
sols-TSTST-2015_2 | Let $ABC$ be a scalene triangle. Let $K_a$, $L_a$, and
$M_a$ be the respective intersections with $BC$ of the internal
angle bisector, external angle bisector, and the median from
$A$. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time
at a point $X_a$ different from $A$. Define $X_b$ and $X_c$
analogously. ... | The main content of the problem:
\begin{claim*}
$\angle H X_a G = 90\dg$.
\end{claim*}
This implies the result,
since then the desired circumcenter is the midpoint of $\ol{GH}$.
(This is the main difficulty; the Euler line is a red herring.)
In what follows, we abbreviate $K_a$ $L_a$, $M_a$, $X_a$
to $K$, $L$, $M$, ... |
sols-TSTST-2015_3 | Let $P$ be the set of all primes, and let $M$ be a
non-empty subset of $P$. Suppose that for any non-empty subset
$\{p_1, p_2, \dots, p_k\}$ of $M$, all prime factors of
$p_1 p_2 \dots p_k + 1$ are also in $M$. Prove that $M = P$. | The following solution was found by user \texttt{Aiscrim} on AOPS.
Obviously $|M| = \infty$.
Assume for contradiction $p \notin M$.
We say a prime $q \in M$ is \emph{sparse}
if there are only finitely many elements of $M$
which are $q \pmod p$
(in particular there are finitely many sparse primes).
Now let $C$ be the ... |
sols-TSTST-2015_4 | Let $x,y,z$ be real numbers (not necessarily positive)
such that $x^4 + y^4 + z^4 + xyz = 4$.
Prove that $x \le 2$ and \[ \sqrt{2-x} \ge \frac{y+z}{2}. \] | We prove that the condition $x^4+y^4+z^4+xyz = 4$ implies
\[ \sqrt{2-x} \geq \frac{y+z}{2}. \]
We first prove the easy part.
\begin{claim*}
We have $x \le 2$.
\end{claim*}
\begin{proof}
Indeed, AM-GM gives that
\begin{align*}
5 =x^4+y^4+(z^4+1)+xyz
&= \frac{3x^4}{4}+\left(\frac{x^4}{4}+y^4\right)+(z^4... |
sols-TSTST-2015_5 | Let $\varphi(n)$ denote the number of positive integers
less than $n$ that are relatively prime to $n$.
Prove that there exists a positive integer $m$ for which the equation
$\varphi(n) = m$ has at least $2015$ solutions in $n$. | Here are two explicit solutions.
\paragraph{First solution with ad-hoc subsets, by Evan Chen.}
I consider the following eleven prime numbers:
\[ S = \left\{ 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 71 \right\}. \]
This has the property that for any $p \in S$,
all prime factors of $p-1$ are one digit.
Let $N = (210)^{\... |
sols-TSTST-2015_6 | A \emph{Nim-style game} is defined as follows. Two
positive integers $k$ and $n$ are specified, along with a finite
set $S$ of $k$-tuples of integers (not necessarily positive). At
the start of the game, the $k$-tuple $(n,0,0,\dots,0)$ is written
on the blackboard.
A legal move consists of erasing the tuple $(a_1,a_2,... | Here we present a solution with $14$ registers and $22$ moves.
Initially $X=n$ and all other variables are zero.
\begin{center}
\scriptsize
\setlength{\tabcolsep}{3pt}
\begin{tabular}{l|rr|r|rrr|rrr|r|rr|rr}
&$X$&$Y$&Go&$S_X^0$&$S_X$&$S_X'$&$S_Y^0$&$S_Y$&$S_Y'$&Cl&$A$&$B$&Die&Die'\\\hline
Init&-1&&1&&&&&&&&&1&1&1\\
Be... |
sols-TSTST-2016_1 | Let $A = A(x,y)$ and $B = B(x,y)$ be
two-variable polynomials with real coefficients.
Suppose that $A(x,y)/B(x,y)$ is a polynomial in $x$
for infinitely many values of $y$,
and a polynomial in $y$ for infinitely many values of $x$.
Prove that $B$ divides $A$, meaning there exists a third polynomial $C$
with real coeffi... | This is essentially an application of the division algorithm,
but the details require significant care.
First, we claim that $A/B$ can be written as a polynomial in $x$
whose coefficients are rational functions in $y$.
To see this, use the division algorithm to get
\[ A = Q \cdot B + R \qquad Q,R \in (\RR(y))[x] \]
wh... |
sols-TSTST-2016_2 | Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$
and denote by $M$, $N$ the midpoints of $\ol{AH}$, $\ol{BC}$.
Suppose the circle $\gamma$ with diameter $\ol{AH}$ meets
the circumcircle of $ABC$ at $G \neq A$,
and meets line $\ol{AN}$ at $Q \neq A$.
The tangent to $\gamma$ at $G$ meets line $OM... | We present two solutions,
one using essentially only power of a point,
and the other more involved.
\paragraph{First solution (found by contestants).}
Denote by $\triangle DEF$ the orthic triangle.
Observe $\ol{PA}$ and $\ol{PG}$ are tangents to $\gamma$,
since $\ol{OM}$ is the perpendicular bisector of $\ol{AG}$.
Als... |
sols-TSTST-2016_3 | Decide whether or not there exists a nonconstant polynomial $Q(x)$
with integer coefficients with the following property:
for every positive integer $n > 2$, the numbers
\[ Q(0), \; Q(1), Q(2), \; \dots, \; Q(n-1) \]
produce at most $0.499n$ distinct residues when taken modulo $n$. | We claim that
\[ Q(x) = 420(x^2-1)^2 \]
works.
Clearly, it suffices to prove the result when $n=4$ and when $n$ is an odd prime $p$.
The case $n=4$ is trivial, so assume now $n=p$ is an odd prime.
First, we prove the following easy claim.
\begin{claim*}
For any odd prime $p$, there are at least $\frac12(p-3)$
valu... |
sols-TSTST-2016_4 | Prove that if $n$ and $k$ are positive integers
satisfying $\varphi^k(n) = 1$, then $n \le 3^k$.
(Here $\varphi^k$ denotes $k$ applications of the Euler phi function.) | The main observation is that the exponent of $2$ decreases
by at most $1$ with each application of $\varphi$.
This will give us the desired estimate.
Define the \emph{weight} function $w$ on positive integers as follows: it satisfies
\begin{align*}
w(ab) &= w(a)+w(b); \\
w(2) &= 1; \quad\text{and} \\
w(p) &= w(p... |
sols-TSTST-2016_5 | In the coordinate plane are finitely many \emph{walls},
which are disjoint line segments, none of which are parallel to either axis.
A bulldozer starts at an arbitrary point and moves in the $+x$ direction.
Every time it hits a wall, it turns at a right angle to its path,
away from the wall, and continues moving.
(Thus... | We say a wall $v$ is \emph{above} another wall $w$ if some point on
$v$ is directly above a point on $w$.
(This relation is anti-symmetric, as walls do not intersect).
The critical claim is as follows:
\begin{claim*}
There exists a lowest wall,
i.e.\ a wall not above any other walls.
\end{claim*}
\begin{proof}
A... |
sols-TSTST-2016_6 | Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent
to $\ol{BC}$, $\ol{CA}$, $\ol{AB}$ at $D$, $E$, $F$, respectively.
Let $K$ be the foot of the altitude from $D$ to $\ol{EF}$.
Suppose that the circumcircle of $\triangle AIB$
meets the incircle at two distinct points $C_1$ and $C_2$,
while the cir... | \paragraph{First solution (Allen Liu).}
Let $X$, $Y$, $Z$ be midpoints of $EF$, $FD$, $DE$, and let $G$ be the Gergonne point.
By radical axis on $(AEIF)$, $(DEF)$, $(AIC)$ we see that $B_1$, $X$, $B_2$ are collinear.
Likewise, $B_1$, $Z$, $B_2$ are collinear, so lines $B_1B_2$ and $XZ$ coincide.
Similarly, lines $C_1C... |
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