Found problems: 46
2023 USA TSTST, 1
Let $ABC$ be a triangle with centroid $G$. Points $R$ and $S$ are chosen on rays $GB$ and $GC$, respectively, such that
\[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\]
Prove that $\angle RAS+\angle BAC=\angle BGC$.
[i]Merlijn Staps[/i]
2022 USA TSTST, 2
Let $ABC$ be a triangle. Let $\theta$ be a fixed angle for which \[\theta<\frac12\min(\angle A,\angle B,\angle C).\]
Points $S_A$ and $T_A$ lie on segment $BC$ such that $\angle BAS_A=\angle T_AAC=\theta$. Let $P_A$ and $Q_A$ be the feet from $B$ and $C$ to $\overline{AS_A}$ and $\overline{AT_A}$ respectively. Then $\ell_A$ is defined as the perpendicular bisector of $\overline{P_AQ_A}$.
Define $\ell_B$ and $\ell_C$ analogously by repeating this construction two more times (using the same value of $\theta$). Prove that $\ell_A$, $\ell_B$, and $\ell_C$ are concurrent or all parallel.
2022 USA TSTST, 3
Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_0<s_1<s_2<\cdots$ satisfying the following properties:
[list=disc]
[*]the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, $\ldots$ is periodic; and
[*]$s_{s_n}-s_{s_{n-1}}\leq N<s_{1+s_n}-s_{s_{n-1}}$ for all positive integers $n$
[/list]
2020 USA TSTST, 2
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 $D$.
[i]Zack Chroman and Daniel Liu[/i]
2021 USA TSTST, 9
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 $k$.)
[i]Victor Wang[/i]
2022 USA TSTST, 1
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that for any set $S$ of $n$ points in the interior of the unit square, there exists a set of $k$ rectangles such that the following hold:
[list=disc]
[*]The sides of each rectangle are parallel to the sides of the unit square.
[*]Each point in $S$ is [i]not[/i] in the interior of any rectangle.
[*]Each point in the interior of the unit square but [i]not[/i] in $S$ is in the interior of at least one of the $k$ rectangles
[/list]
(The interior of a polygon does not contain its boundary.)
[i]Holden Mui[/i]
2023 USA TSTST, 7
The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera has a row of 2023 such coins alternately tails-up and heads-up, with the leftmost coin tails-up.
In a [i]move[/i], Vera may flip over one of the coins in the row, subject to the following rules:
[list=disc]
[*] On the first move, Vera may flip over any of the $2023$ coins.
[*] On all subsequent moves, Vera may only flip over a coin adjacent to the coin she flipped on the previous move. (We do not consider a coin to be adjacent to itself.)
[/list]
Determine the smallest possible number of moves Vera can make to reach a state in which every coin is heads-up.
[i]Luke Robitaille[/i]
2015 USA TSTST, 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 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.)
[i]Proposed by Ivan Borsenco[/i]
2021 USA TSTST, 7
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. [hide=*]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.[/hide]
[i]Ankit Bisain and Holden Mui[/i]
2021 USA TSTST, 3
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 $2\leq m < k$.
[i]Merlijn Staps[/i]
2017 USA TSTST, 3
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.
[i]Proposed by Linus Hamilton and Calvin Deng[/i]
2021 USA TSTST, 8
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.
[i]Fedir Yudin [/i]
2016 USA TSTST, 2
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 $\overline{PN}$.
[i]Proposed by Evan Chen[/i]
2024 USA TSTST, 9
Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$.
[i]Anonymous[/i]
2024 USA TSTST, 8
Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel.
[i]Michael Ren[/i]
2022 USA TSTST, 9
Let $k>1$ be a fixed positive integer. Prove that if $n$ is a sufficiently large positive integer, there exists a sequence of integers with the following properties:
[list=disc]
[*]Each element of the sequence is between $1$ and $n$, inclusive.
[*]For any two different contiguous subsequence of the sequence with length between $2$ and $k$ inclusive, the multisets of values in those two subsequences is not the same.
[*]The sequence has length at least $0.499n^2$
[/list]
2020 USA TSTST, 1
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:
[list]
[*] 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.
[/list]
Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take
place, over all possible initial configurations.
2023 USA TSTST, 3
Find all positive integers $n$ for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells.
[i]Proposed by Merlijn Staps[/i]
2022 USA TSTST, 6
Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$.
Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic.
[i]Hongzhou Lin[/i]
2017 USA TSTST, 1
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 $PR\perp OH$.
[i]Proposed by Ray Li[/i]
2024 USA TSTST, 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]
[i]Jaedon Whyte[/i]