Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Ukraine National Mathematical Olympiad
Let $M$ be the midpoint of the side $BC$ of $\triangle ABC$. On the side $AB$ and $AC$ the points $E$ and $F$ are chosen. Let $K$ be the point of the intersection of $BF$ and $CE$ and $L$ be chosen in a way that $CL\parallel AB$ and $BL\parallel CE$. Let $N$ be the point of intersection of $AM$ and $CL$. Show that $KN$ is parallel to $FL$.

Edit:Fixed typographical error.
2014 Ukraine National Mathematical Olympiad
It is known that for natural numbers $a,b,c,d$ and $n$ the following inequalities hold: $a+c<n$ and $a/b+c/d<1$. Prove that $a/b+c/d<1-1/n^3$.
2014 Ukraine National Mathematical Olympiad
There are $100$ cards with numbers from $1$ to $100$ on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number $n$ then Nick has a card with a number $2n+2$.What is the maximal number of cards that could be taken by the two guys?
2014 Ukraine National Mathematical Olympiad
Find the values of $x$ such that the following inequality holds:

$\min\{\sin x,\cos x\}<\min\{1-\sin x,1-\cos x\}$
2014 Ukraine National Mathematical Olympiad
Find all pairs of prime numbers $p$ and $q$ that satisfy the equation $3p^{q}-2q^{p-1}=19$.
2014 Ukraine National Mathematical Olympiad
Is it possible to choose $24$ points in the space,such that no three of them lie on the same line and choose $2013$ planes in such a way that each plane passes through at least $3$ of the chosen points and each triple of points belongs to at least one of the chosen planes?
2014 Ukraine National Mathematical Olympiad
Let $M$ be the midpoint of the internal bisector $AD$ of $\triangle ABC$.Circle $\omega_1$ with diameter $AC$ intersects $BM$ at $E$ and circle $\omega_2$ with diameter $AB$ intersects $CM$ at $F$.Show that $B,E,F,C$ are concyclic.
2018 Ukraine National Mathematical Olympiad
Find all triples of pairwise distinct positive integers $(a,b,c)$, which satisfy the following conditions: $2a-1$ is divisible by $b$, $2b-1$ is divisible $c$ and $2c-1$ is divisible by $a$.



 Proposed by Bohdan Rublyov
2018 Ukraine National Mathematical Olympiad
In acute-angled triangle $ABC$, $AH$ is an altitude and $AM$ is a median. Points $X$ and $Y$ on lines $AB$ and $AC$ respectively are such that $AX=XC$ and $AY=YB$. Prove that the midpoint of $XY$ is equidistant from $H$ and $M$.



Proposed by Danylo Khilko
2018 Ukraine National Mathematical Olympiad
Find all functions $f:[0,+\infty) \mapsto [0,+\infty)$, which for all nonnegative $x,y$ satisfy $$f(f(x)+f(y))=xyf(x+y)$$ Proposed by Igor Voronovich
2018 Ukraine National Mathematical Olympiad
Two players - Andriy and Olesya play the following game. On a table there is a round cake, which is cut by one of the players into $2n$ ($n>1$) sectors (pieces), pairwise distinct in weights. Weight of each piece is known to both players. Then they take pieces according to the following rules. At first, Olesya takes $1$ piece, then Andriy takes $2$ pieces such that the remaining pieces form a sector. Then they in turns take $2$ pieces such that after each turn the remaining pieces form a sector. In their last move, one of the players takes one last piece. Each of the players aims to get more of a cake than the opponent. For which $n$ can Olesya cut the cake such that she wins if in her first move she takes the smallest piece?



 Proposed by Bohdan Rublyov
2018 Ukraine National Mathematical Olympiad
The squadron of $10$ powerful destroyers and $20$ small boats is about to attack the island. All ships are positioned on the straight line, and are equally spaced.



Two torpedo boats with $10$ torpedoes each want to protect the island. However, the first torpedo boat can shoot only $10$ successive boats, whereas the second one can shoot $10$ targets which are next by one. Note that they have to shoot at the same moment, so that some targets may be hit by both torpedoes.



What is the biggest number of destroyers that can avoid the torpedoes no matter which targets the torpedo boats choose?



 Proposed by Bohdan Rublyov
2018 Ukraine National Mathematical Olympiad
The sequence $(x_n)$ is given by $x_1=a$, $x_{n+1}=\dfrac{1}{2}\left(x_n-\dfrac{1}{x_n}\right)$. Prove that there is $a$ such that the sequence $(x_n)$ has exactly $2018$ pairwise distinct elements. (If some element of the sequence is equal to $0$, it stops on that element)



 Proposed by Andriy Hoholev
2018 Ukraine National Mathematical Olympiad
Given $N$ positive integers such that the greatest common divisors of all nonempty subsets of them are pairwise distinct. What is the smallest number of prime factors of the product of all $N$ numbers?



 Proposed by Aleksandr Golovanov
2018 Ukraine National Mathematical Olympiad
Given an acute-angled triangle $ABC$, $AA_1$ and $CC_1$ are its angle bisectors, $I$ is its incenter, $M$ and $N$ are the midpoints of $AI$ and $CI$. Points $K$ and $L$ in the interior of triangles $AC_1I$ and $CA_1I$ respectively are such that $\angle AKI = \angle CLI = \angle AIC$, $\angle AKM = \angle ICA$, $\angle CLN = \angle IAC$. Prove that the circumradii of triangles $KIL$ and $ABC$ are equal.



 Proposed by Anton Trygub
2021 Ukraine National Mathematical Olympiad
Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains $2021$ stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins?



(Oleksii Masalitin)
2021 Ukraine National Mathematical Olympiad
Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$  for which there exists a $n$ such that for any polynomial  $p \in P^{(n)}$,  number $P(k)$ is divisible by the number $d$.



(Oleksii Masalitin)
2021 Ukraine National Mathematical Olympiad
For arbitrary positive reals $a\ge b \ge c$ prove the inequality:

$$\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}$$(Anton Trygub)
2021 Ukraine National Mathematical Olympiad
Let $O, I, H$  be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$.



(Fedir Yudin)
2021 Ukraine National Mathematical Olympiad
Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$  where  $a\in Z$, in which one of the numbers is equal to the sum of all the others.



(Bogdan Rublev)
2021 Ukraine National Mathematical Olympiad
Circles $w_1$ and $w_2$ intersect at points $P$ and $Q$ and touch a circle $w$ with center at point $O$ internally at points $A$ and $B$, respectively. It is known that the points $A,B$ and $Q$ lie on one line. Prove that the point $O$ lies on the external bisector $\angle APB$.



(Nazar Serdyuk)
2021 Ukraine National Mathematical Olympiad
The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs  exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition  $a_{b_i} \ne a_{c_i}$

.

(Anton Trygub)
2021 Ukraine National Mathematical Olympiad
Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$.



Here $ \phi (n)$  denotes the number of natural numbers less than $n$ that are coprime with $n$.



(Fedir Yudin)
2021 Ukraine National Mathematical Olympiad
It is known that for some integers $a_{2021},a_{2020},...,a_1,a_0$ the expression

$$a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0$$is divisible by $2021$ for any arbitrary integer $n$. Is it required that each of the numbers $a_{2021},a_{2020},...,a_1,a_0$ also divisible by $2021$?
2021 Ukraine National Mathematical Olympiad
Find all natural numbers $n \ge 3$ for which in an arbitrary $n$-gon one can choose $3$ vertices dividing its boundary into three parts, the lengths of which can be the lengths of the sides of some triangle.



(Fedir Yudin)
2021 Ukraine National Mathematical Olympiad
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$(Vadym Koval)
2021 Ukraine National Mathematical Olympiad
Are there natural numbers $(m,n,k)$ that satisfy the equation  $m^m+ n^n=k^k$ ?
2021 Ukraine National Mathematical Olympiad
The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half.



(Anton Trygub)
2021 Ukraine National Mathematical Olympiad
There are $101$ not necessarily different weights, each of which weighs an integer number of grams from $1$ g to $2020$ g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than $2020$. What is the largest number of grams can weigh all $101$ weights?



(Bogdan Rublev)
2021 USA TSTST
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $ADX$ and $BCY$ meet line $XY$ again at $P$ and $Q$, respectively. Show that $OP=OQ$.



Holden Mui
2021 USA TSTST
Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence

$$ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.$$

Merlijn Staps
2021 USA TSTST
Find all positive integers $k > 1$ for which there exists a positive integer $n$ such that $\tbinom{n}{k}$ is divisible by $n$, and $\tbinom{n}{m}$ is not divisible by $n$ for $2\leq m < k$.



Merlijn Staps
2021 USA TSTST
Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m,n)$ for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares. Prove that $a$ divides $2b$.



Holden Mui
2021 USA TSTST
Let $T$ be a tree on $n$ vertices with exactly $k$ leaves. Suppose that there exists a subset of at least $\frac{n+k-1}{2}$ vertices of $T$, no two of which are adjacent. Show that the longest path in $T$ contains an even number of edges. *A tree is a connected graph with no cycles. A leaf is a vertex of degree 1



Vincent Huang
2021 USA TSTST
Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly.



(a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel.



(b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear.



Nikolai Beluhov
2021 USA TSTST
Let $M$ be a finite set of lattice points and $n$ be a positive integer. A $\textit{mine-avoiding path}$ is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n,$ that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2^{n-|M|}$ mine-avoiding paths. *A lattice point is a point $(x,y)$ where $x$ and $y$ are integers. A path of lattice points with length $n$ is a sequence of lattice points $P_0,P_1,\ldots, P_n$ in which any two adjacent points in the sequence have distance 1 from each other.
2021 USA TSTST
Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar.
2021 USA TSTST
Let $q=p^r$ for a prime number $p$ and positive integer $r$. Let $\zeta = e^{\frac{2\pi i}{q}}$. Find the least positive integer $n$ such that

$$\sum_{\substack{1\leq k\leq q\\ \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}$$is not an integer. (The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.)
2020 USA TSTST
Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a

circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks

picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors.

A move consists of an operation of one of the following three forms:



If a duck picking rock sits behind a duck picking scissors, they switch places.

If a duck picking paper sits behind a duck picking rock, they switch places.

If a duck picking scissors sits behind a duck picking paper, they switch places.



Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take

place, over all possible initial configurations.
2020 USA TSTST
Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$.



Zack Chroman and Daniel Liu
2020 USA TSTST
We say a nondegenerate triangle whose angles have measures $\theta_1$, $\theta_2$, $\theta_3$ is quirky if there exists integers $r_1,r_2,r_3$, not all zero, such that

$$r_1\theta_1+r_2\theta_2+r_3\theta_3=0.$$Find all integers $n\ge 3$ for which a triangle with side lengths $n-1,n,n+1$ is quirky.



Evan Chen and Danielle Wang
2020 USA TSTST
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:



$a$ divides $b^4+1$,

$b$ divides $a^4+1$,

$\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.





Yang Liu
2020 USA TSTST
Let $\mathbb{N}^2$ denote the set of ordered pairs of positive integers. A finite subset $S$ of $\mathbb{N}^2$ is stable if whenever $(x,y)$ is in $S$, then so are all points $(x',y')$ of $\mathbb{N}^2$ with both $x'\leq x$ and $y'\leq y$.



Prove that if $S$ is a stable set, then among all stable subsets of $S$ (including the empty set and $S$ itself), at least half of them have an even number of elements.



Ashwin Sah and Mehtaab Sawhney
2020 USA TSTST
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.



Andrew Gu
2020 USA TSTST
Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1.



Ankan Bhattacharya
2020 USA TSTST
For every positive integer $N$, let $\sigma(N)$ denote the sum of the positive integer divisors of $N$. Find all integers $m\geq n\geq 2$ satisfying $$\frac{\sigma(m)-1}{m-1}=\frac{\sigma(n)-1}{n-1}=\frac{\sigma(mn)-1}{mn-1}.$$

Ankan Bhattacharya
2020 USA TSTST
Ten million fireflies are glowing in $\mathbb{R}^3$ at midnight. Some of the fireflies are friends, and friendship is always mutual. Every second, one firefly moves to a new position so that its distance from each one of its friends is the same as it was before moving. This is the only way that the fireflies ever change their positions. No two fireflies may every occupy the same point.



Initially, no two fireflies, friends or not, are more than a meter away. Following some finite number of seconds, all fireflies find themselves at least ten million meters away from their original positions. Given this information, find the greatest possible number of friendships between the fireflies.



Nikolai Beluhov
2019 USA TSTST
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,



the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and

if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.



Evan Chen
2019 USA TSTST
Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$.



Show that lines $PE$ and $QD$ meet on $\omega$.

Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well.



Merlijn Staps
2019 USA TSTST
On an infinite square grid we place finitely many cars, which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a move, one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times.



Nikolai Beluhov
2019 USA TSTST
Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is at most $C$.



Merlijn Staps
2019 USA TSTST
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$.



Gunmay Handa
2019 USA TSTST
Suppose $P$ is a polynomial with integer coefficients such that for every positive integer $n$, the sum of the decimal digits of $|P(n)|$ is not a Fibonacci number. Must $P$ be constant?

(A Fibonacci number is an element of the sequence $F_0, F_1, \dots$ defined recursively by $F_0=0, F_1=1,$ and $F_{k+2} = F_{k+1}+F_k$ for $k\ge 0$.)



Nikolai Beluhov
2019 USA TSTST
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying

$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.



Ankan Bhattacharya
2019 USA TSTST
Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$, such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$.



Ankan Bhattacharya
2019 USA TSTST
Let $ABC$ be a triangle with incenter $I$. Points $K$ and $L$ are chosen on segment $BC$ such that the incircles of $\triangle ABK$ and $\triangle ABL$ are tangent at $P$, and the incircles of $\triangle ACK$ and $\triangle ACL$ are tangent at $Q$. Prove that $IP=IQ$.



Ankan Bhattacharya
2018 USA TSTST
As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$,



$\theta(p+1) = \theta(p)+1$, and

if $\theta(p) \neq 0$ 	then $\theta(p)$ divides $\theta(p \cdot q)$.





Evan Chen and Yang Liu
2018 USA TSTST
In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it.



We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of $2$ greater than $1$ (i.e.\ of the form $2^n$ for some integer $n \ge 1$).



Victor Wang
2018 USA TSTST
Let $ABC$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{AB}$. Ray $AI$ meets $\overline{BC}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$, respectively. Line $MO$ meets $\omega$ at $X$ and $Y$, while line $CO$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$.



Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle BAC \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$.



Evan Chen and Yannick Yao
2018 USA TSTST
For an integer $n > 0$, denote by $\mathcal F(n)$ the set of integers $m > 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root.



Let $S$ denote the set of integers $n > 0$ 	for which $\mathcal F(n)$ contains two consecutive integers. 	Show that $S$ is infinite but 	$$ \sum_{n \in S} \frac 1n \le 1. $$

Prove that there are infinitely many positive integers $n$ 	such that $\mathcal F(n)$ contains three consecutive integers.





Ivan Borsenco
2018 USA TSTST
Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$.



Ankan Bhattacharya and Evan Chen
2018 USA TSTST
Let $S = \left\{ 1, \dots, 100 \right\}$, and for every positive integer $n$ define $$ 	T_n = \left\{ (a_1, \dots, a_n) \in S^n 		\mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}. $$Determine which $n$ have the following property: if we color any $75$ elements of $S$ red, then at least half of the $n$-tuples in $T_n$ have an even number of coordinates with red elements.



Ray Li
2018 USA TSTST
Let $n$ be a positive integer. A frog starts on the number line at $0$. Suppose it makes a finite sequence of hops, subject to two conditions:



The frog visits only points in $\{1, 2, \dots, 2^n-1\}$, 	each at most once.

The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$. 	(The hops may be either direction, left or right.)



Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$?



Ashwin Sah
2018 USA TSTST
For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$?



Evan Chen and Ankan Bhattacharya