Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 USA TSTST
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$  has total area at most $c$.



Linus Hamilton
2017 USA TSTST
Let $ABC$ be a triangle with circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. Assume that $AB\neq AC$ and that $\angle A \neq 90^{\circ}$. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively, and let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ in $\triangle ABC$, respectively. Let $P$ be the intersection of line $MN$ with the tangent line to $\Gamma$ at $A$. Let $Q$ be the intersection point, other than $A$, of $\Gamma$ with the circumcircle of $\triangle AEF$. Let $R$ be the intersection of lines $AQ$ and $EF$. Prove that $PR\perp OH$.



Proposed by Ray Li
2017 USA TSTST
Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses.



For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word.



Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses?



(The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.)



Proposed by Kevin Sun
2017 USA TSTST
Consider solutions to the equation $$x^2-cx+1 = \dfrac{f(x)}{g(x)},$$where $f$ and $g$ are polynomials with nonnegative real coefficients. For each $c>0$, determine the minimum possible degree of $f$, or show that no such $f,g$ exist.



Proposed by Linus Hamilton and Calvin Deng
2017 USA TSTST
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$.



Proposed by Mark Sellke
2017 USA TSTST
Let $ABC$ be a triangle with incenter $I$. Let $D$ be a point on side $BC$ and let $\omega_B$ and $\omega_C$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. Suppose that $\omega_B$ and $\omega_C$ are tangent to segment $BC$ at points $E$ and $F$, respectively. Let $P$ be the intersection of segment $AD$ with the line joining the centers of $\omega_B$ and $\omega_C$. Let $X$ be the intersection point of lines $BI$ and $CP$ and let $Y$ be the intersection point of lines $CI$ and $BP$. Prove that lines $EX$ and $FY$ meet on the incircle of $\triangle ABC$.



Proposed by Ray Li
2017 USA TSTST
A sequence of positive integers $(a_n)_{n \ge 1}$ is of Fibonacci type if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences?



Proposed by Ivan Borsenco
2016 USA TSTST
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 coefficients such that $A = B \cdot C$.



Proposed by Victor Wang
2016 USA TSTST
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$.  Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$.  Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$.  The tangent to $\gamma$ at $G$ meets line $OM$ at $P$.  Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.



Proposed by Evan Chen
2016 USA TSTST
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 $n$.



Proposed by Yang Liu
2016 USA TSTST
Suppose that $n$ and $k$ are positive integers such that $$ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). $$Prove that $n \le 3^k$.



Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$.



Proposed by Linus Hamilton
2016 USA TSTST
In the coordinate plane are finitely many 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 the bulldozer always moves parallel to the axes.)



Prove that it is impossible for the bulldozer to hit both sides of every wall.



Proposed by Linus Hamilton and David Stoner
2016 USA TSTST
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively.  Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$.  Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$.  Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.



Proposed by Danielle Wang
2015 USA TSTST
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 good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$.



Proposed by Mark Sellke
2015 USA TSTST
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 point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.

(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)



Proposed by Ivan Borsenco
2015 USA TSTST
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 ${p_1,p_2,...,p_k}$ of $M$, all prime factors of $p_1p_2...p_k+1$ are also in $M$. Prove that $M=P$.



Proposed by Alex Zhai
2015 USA TSTST
Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$.

Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$.



Proposed by Alyazeed Basyoni
2015 USA TSTST
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 $n$.



Proposed by Iurie Boreico
2015 USA TSTST
A 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 $(n, 0, 0, ..., 0)$ is written on the blackboard.

A legal move consists of erasing the tuple $(a_1,a_2,...,a_k)$ which is written on the blackboard and replacing it with $(a_1+b_1, a_2+b_2, ..., a_k+b_k)$, where $(b_1, b_2, ..., b_k)$ is an element of the set $S$. Two players take turns making legal moves, and the first to write a negative integer loses. In the event that neither player is ever forced to write a negative integer, the game is a draw.

Prove that there is a choice of $k$ and $S$ with the following property: the first player has a winning strategy if $n$ is a power of 2, and otherwise the second player has a winning strategy.



Proposed by Linus Hamilton
2014 USA TSTST
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 reachable from a string $A$ if it is possible to insert some amount of $\leftarrow$'s in $A$, such that typing the resulting characters produces $B$. So, our example shows that "faecdb" is reachable from "abcdef".



Prove that for any two strings $A$ and $B$, $A$ is reachable from $B$ if and only if $B$ is reachable from $A$.
2014 USA TSTST
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 gergonnians.

(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.

(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.
2014 USA TSTST
Find all polynomials $P(x)$ with real coefficients that satisfy $$P(x\sqrt{2})=P(x+\sqrt{1-x^2})$$for all real $x$ with $|x|\le 1$.
2014 USA TSTST
Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients, and let $d$ be the degree of $P(x)$. Assume that $P(x)$ is not the zero polynomial. Prove that there exist polynomials $A(x)$ and $B(x)$ such that:

(i)    both $A$ and $B$ have degree at most $d/2$

(ii)   at most one of $A$ and $B$ is the zero polynomial.

(iii)  $\frac{A(x)+Q(x)B(x)}{P(x)}$ is a polynomial with real coefficients. That is, there is some polynomial $C(x)$ with real coefficients such that $A(x)+Q(x)B(x)=P(x)C(x)$.
2014 USA TSTST
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.
2014 USA TSTST
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 $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
2013 USA TSTST
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 $DB$ and $DC$.  Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$.  Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$.  Define points $B_1$ and $C_1$ similarly.  Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.
2013 USA TSTST
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called 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 forced if the following condition is satisfied: the sequence $$ a_1, a_2, \dots, a_{k-1}, b $$ is regular if and only if $b = a_k$.  Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.
2013 USA TSTST
Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$.  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 $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational.  Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.
2013 USA TSTST
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 $PST$.
2013 USA TSTST
Let $p$ be a prime.  Prove that any complete graph with $1000p$ vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by $p$.
2013 USA TSTST
Let $\mathbb N$ be the set of positive integers.  Find all functions $f: \mathbb N \to \mathbb N$ 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 $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)
2013 USA TSTST
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 is also not permitted to put a road between cities $1$ and $n$.  Let $T_n$ be the total number of possible ways to build these roads.

(a) For all odd $n$, prove that $T_n$ is divisible by $n$.

(b) For all even $n$, prove that $T_n$ is divisible by $n/2$.
2013 USA TSTST
Define a function $f: \mathbb N \to \mathbb N$ 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 $3^{2013}$.
2013 USA TSTST
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 of any two points in a good set also lies in the set.
2012 USA TSTST
Find all infinite sequences $a_1, a_2, \ldots$ of positive integers satisfying the following properties:

(a) $a_1 < a_2 < a_3 < \cdots$,

(b) there are no positive integers $i$, $j$, $k$, not necessarily distinct, such that $a_i+a_j=a_k$,

(c) there are infinitely many $k$ such that $a_k = 2k-1$.
2012 USA TSTST
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 the 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 $T$ (other than $B$ and $C$), respectively.  Let $M$ and $N$ be the midpoints of minor arcs $\widehat {SP}$ (not including $B$) and $\widehat {TP}$ (not including $C$).  Prove that $MN \parallel O_1O_2$.
2012 USA TSTST
Let $\mathbb N$ be the set of positive integers.  Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions:

(a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime.

(b) $n \le f(n) \le n+2012$ for all $n$.



Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.
2012 USA TSTST
In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ 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 $BC, CA, AB$.  Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.
2012 USA TSTST
A rational number $x$ is given.  Prove that there exists a sequence $x_0, x_1, x_2, \ldots$ of rational numbers with the following properties:

(a) $x_0=x$;

(b) for every $n\ge1$, either $x_n = 2x_{n-1}$ or $x_n = 2x_{n-1} + \textstyle\frac{1}{n}$;

(c) $x_n$ is an integer for some $n$.
2012 USA TSTST
Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$.  Prove that $$ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . $$
2012 USA TSTST
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 $A$), respectively.  Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$.  Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
2012 USA TSTST
Let $n$ be a positive integer.  Consider a triangular array of nonnegative integers as follows: $$

	\begin{array}{rccccccccc}

		\text{Row } 1:  &&&&& a_{0,1} &&&& \smallskip\\
		\text{Row } 2:  &&&& a_{0,2} && a_{1,2} &&& \smallskip\\
		&&& \vdots && \vdots && \vdots && \smallskip\\
		\text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\
		\text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n}

	\end{array}

$$ Call such a triangular array stable if for every $0 \le i < j < k \le n$ we have $$ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. $$ For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.
2012 USA TSTST
Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called 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 a single element, such as $xyz \to x \times (y \times z)$.  A string of variables in $S$ is called full if it contains each variable in $S$ at least once, and two strings are equivalent if they evaluate to the same variable regardless of which simple $\times$ is chosen.  For example $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n=1$.  Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$.  Determine the number of elements of $T$.
2011 USA TSTST
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 median of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
2011 USA TSTST
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 $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.
2011 USA TSTST
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).

$$
2011 USA TSTST
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 $OA\perp RA$.
2011 USA TSTST
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, but no pair of enemies does, and no three parents are in a love triangle (where each pair of them has a child).
2011 USA TSTST
Let $a, b, c$ be positive 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\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}.

$$
2011 USA TSTST
Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.
2011 USA TSTST
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 $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define

$$

x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor.

$$

Show that the sequence $\{x_n\}$ is eventually constant.
2011 USA TSTST
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 symmetric difference of two sets as the set of objects belonging to exactly one of the two sets.  Prove that there are at least $2^n$ different sets which can be obtained as the symmetric difference of a red set and a blue set.
2021 USAMO
Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that $$\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.$$Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.
2021 USAMO
The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)







A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?