Problem Writing
I'm actively involved in writing problems for various math contests. Here is a selection of my creations.
Olympiad
USA TST 2026 Problem 6
A positive integer is called chaotic if it can be expressed as \(a^3+b^3+abc\) for positive integers \(a\geq b\geq c\). Show that there is no infinite increasing arithmetic progression consisting of only chaotic positive integers.
(AoPS) (Official Sol)
IMO Shortlist 2024 A4
Let \(\mathbb{Z}_{>0}\) be the set of all positive integers. Determine all subsets \(\mathcal{S}\) of \(\{2^{0},2^{1},2^{2},\ldots\}\) for which there exists a function \(f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}\) such that
\[\mathcal{S}=\{f(a+b)-f(a)-f(b)\mid a,b\in\mathbb{Z}_{>0}\}.\]
(AoPS) (Official Sol)
USA TST 2025 Problem 6
Prove that there exists a real number \(\varepsilon>0\)
such that there are infinitely many sequences of integers \(0 < a_1 < a_2 <\dots <a_{2025}\) satisfying
\[\gcd(a_1^2+1, a_2^2+1,\dots, a_{2025}^2+1) > a_{2025}^{1+\varepsilon}.\]
(AoPS) (Official Sol)
USA TSTST 2025 Problem 8
Find all polynomials \(f\) with integer coefficients such that for all positive integers \(n\),
\[ n\text{ divides } \underbrace{f(f(\dots(f(0))\dots )}_{n+1\ f\text{'s}} - 1. \]
(AoPS) (Official Sol)
Thailand TST 2025 Problem 12
Prove that there exists a constant \(C>0\) such that
for any odd prime numbers \(p\) and \(q\), there exists a prime number \(r < C\max(p,q)^{3/2}\) such that
- \(r\) does not divide \(2pq\);
- there exists an integer \(a\) such that \(r\) divides \(a^2-p\); and
- there exists an integer \(b\) such that \(r\) divides \(b^2-q\).
(AoPS)
Thailand TST 2025 Problem 16
Let \(ABC\) be a triangle.
Let \(D\) be a point on the interior of segment \(BC\).
The perpendicular bisector of \(AD\) intersects sides \(AC\) and \(AB\) at points \(E\) and \(F\), respectively.
Let \(\omega_B\) and \(\omega_C\) be the circumcircles of triangles \(BDF\) and \(CDE\), respectively.
The common external tangents of \(\omega_B\) and \(\omega_C\) meet at \(T\).
- Prove that \(T\) lies on line \(EF\).
- Prove that line \(AT\) is tangent to the circumcircle of triangle \(ABC\).
(AoPS)
Thailand TSTST 2025 Problem 3
Find all functions \(f\colon\mathbb{Z}\to\mathbb{Z}\) such that
\[f\big(f(f(a))+b\big)+f(a-b)=2f(f(a))\]
for every integers \(a\) and \(b\).
(AoPS)
Thailand TSTST 2025 Problem 8
Let \(ABC\) be an acute scalene triangle.
Let \(D\) be the foot of altitude from \(A\) to \(\overline{BC}\).
Let \(E\) be a point on the segment \(AC\) such that \(CD=CE\) and
\(F\) be a point on the segment \(AB\) such that \(BD=BF\).
Construct point \(P\) on the line \(DE\) such that \(PA=PE\).
Similarly, construct point \(Q\) on the line \(DF\) such that \(QA=QF\).
Prove that the circumcenter of triangle \(DPQ\) is on the internal angle bisector of \(\angle{BAC}\).
(AoPS)
Thailand TST 2024 Problem 13
Determine all polynomials \(P\) with integer coefficients for which
there exists an integer \(a_n\) such that \(P(a_n)=n^n\) for all positive integers \(n\).
(AoPS)
HMMT
I'm involved in HMMT as a Problem Staff (2022-23), Problem Czar (2023-24), Problem Czar Advisor (2024-25), and Historian (2025-26). Below are some of my favorite problems that made into the contest.
HMMT February 2026 Alg/NT 1
A line intersects the graph of \(y=x^2+\frac 2x\) at three distinct points. Given that the \(x\)-coordinates of two of the points are \(6\) and \(7\), respectively, compute the \(x\)-coordinate of the third point.
HMMT February 2026 Geo 8
Let \(ABC\) be a triangle with orthocenter \(H\). The internal angle bisector of \(\angle BAC\) meets the circumcircles of triangles \(ABH\), \(ACH\), and \(ABC\) again at points \(P\), \(Q\), and \(M\), respectively. Suppose that points \(A\), \(P\), \(Q\), and \(M\) are distinct and lie on the internal angle bisector of \(\angle BAC\) in that order. Given that \(AP=4\), \(AQ=5\), and \(BC=7\), compute \(AM\).
HMMT February 2026 Team 10
Prove that there exists a real constant \(M\) such that for every prime \(p\geq M\) and any positive integer \(2\leq m\leq p-1\), there exist positive integers \(a\) and \(b\) such that \(m \leq a \leq 1.01m\), \(p^{0.99} \leq b \leq p\), and \(p\) divides \(ab-1\).
HMMT February 2026 Guts 27
Let \(a\), \(b\), and \(c\) be positive real numbers such that
\begin{align*}
\sqrt{ab+1}+\sqrt{ca+1} &= 2a, \\
\sqrt{bc+1}+\sqrt{ab+1} &= 3b, \\
\sqrt{ca+1}+\sqrt{bc+1} &= 5c.
\end{align*}
Compute \(a\).
HMMT February 2026 Guts 30
Let \(ABC\) be a triangle with \(AB=60\), \(AC=67\), and \(BC=69\). The incircle \(\omega\) of triangle \(ABC\) touches sides \(\overline{BC}\), \(\overline{CA}\), and \(\overline{AB}\) at \(D\), \(E\), and \(F\), respectively. Let \(D'\) be the point diametrically opposite to \(D\) in \(\omega\). Let the common chord of the circumcircles of triangles \(BD'F\) and \(CD'E\) meet line \(BC\) at \(X\). Compute \(BX\).
HMMT November 2025 General 9
(with Qiao Zhang)
Let \(a\), \(b\), and \(c\) be pairwise distinct nonzero complex numbers such that
\begin{align*}
(10a+b)(10a+c) &= a+\tfrac 1a, \\
(10b+a)(10b+c) &= b+\tfrac 1b, \\
(10c+a)(10c+b) &= c+\tfrac 1c.
\end{align*}
Compute \(abc\).
HMMT November 2025 Theme 10
The orbits of Pluto and Charon are given by the ellipses
\[x^2+xy+y^2=20 \quad \text{and}\quad 2x^2-xy+y^2=25,\]
respectively. These orbits intersect at four points that form a parallelogram. Compute the largest of the slopes of the four sides of this parallelogram.
HMMT November 2025 Team 9
Let \(a\), \(b\), and \(c\) be positive real numbers such that
\begin{align*}
\sqrt a + \sqrt b + \sqrt c &= 7, \\
\sqrt{a+1} + \sqrt{b+1} + \sqrt{c+1} &= 8, \\
(\sqrt{a+1}+\sqrt a)(\sqrt{b+1}+\sqrt b)(\sqrt{c+1}+\sqrt c) &= 60.
\end{align*}
Compute \(a+b+c\).
HMMT November 2025 Team 10
Let \(ABCD\) be an isosceles trapezoid with \(\overline{AB}\) parallel to \(\overline{CD}\), and let \(P\) be a point in the interior of \(ABCD\) such that
\[\angle PBA = 3\angle PAB\quad\text{and}\quad \angle PCD = 3\angle PDC.\]
Given that \(BP=8\), \(CP=9\), and \(\cos\angle APD = \tfrac 23\), compute \(\cos\angle PAB\).
HMMT November 2025 Guts 27
Let \(a_1\), \(a_2\), \(a_3\), \(\dots\) be a sequence of integers such that \(a_1=2\) and \(a_{n+1}=a_n^{7}-a_n+1\) for all \(n\ge 1\). Compute the remainder when \(a_{500}\) is divided by \(7^3\).
HMMT February 2025 Alg/NT 9
Let \(f\) be the unique polynomial of degree at most \(2026\) such that for all \(n\in\{1,2,3,\dots,2027\}\),
\[f(n) = \begin{cases}
1 & \text{ if } n\text{ is a perfect square,}\\
0 & \text{ otherwise.}
\end{cases}\]
Suppose that \(\tfrac ab\) is the coefficient of \(x^{2025}\) in \(f\), where \(a\) and \(b\) are integers such that \(\gcd(a,b)=1\). Compute the unique integer \(r\) between \(0\) and \(2026\) (inclusive) such that \(a-rb\) is divisible by \(2027\). (Note that \(2027\) is prime.)
HMMT February 2025 Geo 5
Let \(\bigtriangleup ABC\) be an equilateral triangle with side length \(6\). Let \(P\) be a point inside triangle \(\bigtriangleup ABC\) such that \(\angle BPC=120^\circ\). The circle with diameter \(\overline{AP}\) meets the circumcircle of \(\bigtriangleup ABC\) again at \(X\neq A\). Given that \(AX=5\), compute \(XP\).
HMMT February 2025 Geo 9
Let \(ABCD\) be a rectangle with \(BC=24\). Point \(X\) lies inside the rectangle such that \(\angle AXB=90^\circ\). Given that triangles \(\bigtriangleup AXD\) and \(\bigtriangleup BXC\) are both acute and have circumradii \(13\) and \(15\), respectively, compute \(AB\).
HMMT February 2025 Team 8
Let \(\bigtriangleup ABC\) be a triangle with incenter \(I\). The incircle of triangle \(\bigtriangleup ABC\) touches \(\overline{BC}\) at \(D\). Let \(M\) be the midpoint of \(\overline{BC}\), and let line \(AI\) meet the circumcircle of triangle \(\triangle ABC\) again at \(L\neq A\). Let \(\omega\) be the circle centered at \(L\) tangent to \(AB\) and \(AC\). If \(\omega\) intersects segment \(\overline{AD}\) at point \(P\), prove that \(\angle IPM = 90^\circ\).
HMMT February 2025 Guts 15
Right triangle \(\bigtriangleup DEF\) with \(\angle D=90^\circ\) and \(\angle F = 30^\circ\) is inscribed in equilateral triangle \(\bigtriangleup ABC\) such that \(D\), \(E\), and \(F\) lie on segments \(\overline{BC}\), \(\overline{CA}\), and \(\overline{AB}\), respectively. Given that \(BD=7\) and \(DC=4\), compute \(DE\).
HMMT February 2025 Guts 24
For any integer \(x\), let
\[f(x) = 100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^{100}}{100!}\right).\]
A positive integer \(a\) is chosen such that \(f(a)-20\) is divisible by \(101^2\). Compute the remainder when \(f(a+101)\) is divided by \(101^2\).
HMMT February 2025 Guts 30
Let \(a\), \(b\), and \(c\) be real numbers satisfying the system of equations
\begin{align*}
a\sqrt{1+b^2} + b\sqrt{1+a^2} &= \tfrac 34, \\
b\sqrt{1+c^2} + c\sqrt{1+b^2} &= \tfrac 5{12} ,\text{ and}\\
c\sqrt{1+a^2} + a\sqrt{1+c^2} &= \tfrac{21}{20}.
\end{align*}
Compute \(a\).
HMMT November 2024 Theme 8
(with Srinivas Arun, Jackson Dryg, David Dong, Derek Liu, Krishna Pothapragada, Henrick Rabinovitz, and Linus Yifeng Tang)
For all positive integers \(r\) and \(s\), let \(\operatorname{Top}(r,s)\) denote the top number (i.e., numerator) when \(\tfrac rs\) is written in simplified form. For instance, \(\operatorname{Top}(20,24)=5\). Compute the number of ordered pairs of positive integers \((a,z)\) such that \(200 \leq a \leq 300\) and \(\operatorname{Top}(a,z) = \operatorname{Top}(z,a-1)\).
HMMT November 2024 Team 9
(with Karthik Vedula)
Let \(P\) be a point inside isosceles trapezoid \(ABCD\) with \(AB\parallel CD\) such that
\[\angle PAD = \angle PDA = 90^\circ - \angle BPC.\]
If \(PA = 14\), \(AB= 18\), and \(CD=28\), compute the area of \(ABCD\).
HMMT February 2024 Alg/NT 4
Let \(f(x)\) be a quotient of two quadratic polynomials. Given that \(f(n)=n^3\) for all \(n\in\{1,2,3,4,5\}\), compute \(f(0).\)
HMMT February 2024 Alg/NT 10
A polynomial \(f\in\mathbb Z[x]\) is called splitty if and only if for every prime \(p\), there exist polynomials \(g_p, h_p\in\mathbb Z[x]\) with \(\deg g_p, \deg h_p < \deg f\) and all coefficients of \(f-g_ph_p\) are divisible by \(p\). Compute the sum of all positive integers \(n \leq 100\) such that the polynomial \(x^4+16x^2+n\) is splitty.
HMMT February 2024 Geo 9
Let \(ABC\) be a triangle. Let \(X\) be the point on side \(\overline{AB}\) such that \(\angle BXC = 60^\circ\). Let \(P\) be the point on segment \(\overline{CX}\) such that \(BP\perp AC\). Given that \(AB=6\), \(AC=7\), and \(BP=4\), compute \(CP\).
HMMT February 2024 Team 10
Across all polynomials \(P\) such that \(P(n)\) is an integer for all integers \(n\), determine, with proof, all possible values of \(P(i)\), where \(i^2 = -1\).
HMMT February 2024 Guts 26
It can be shown that there exists a unique polynomial \(P\) in two variables such that for all positive integers \(m\) and \(n\),
\[P(m,n)=\sum_{i=1}^m\sum_{j=1}^n (i+j)^7.\]
Compute \(P(3,-3)\)
HMMT November 2023 General 10
Let \(ABCD\) be a convex trapezoid such that \(\angle ABC=\angle BCD=90^\circ\), \(AB=3\), \(BC=6\), and \(CD=12\). Among all points \(X\) inside the trapezoid satisfying \(\angle XBC = \angle XDA\), compute the minimum possible value of \(CX\).
HMMT November 2023 Theme 4
Let \(LOVER\) be a convex pentagon such that \(LOVE\) is a rectangle. Given that \(OV=20\) and \(LO=VE=RE=RL=23\), compute the radius of the circle passing through \(R\), \(O\), and \(V\).
HMMT November 2023 Guts 21
An integer \(n\) is chosen uniformly at random from the set \(\{1,2,3,\dots,2023!\}\). Compute the probability that \[\gcd(n^n+50, n+1)=1.\]
HMMT November 2023 Guts 23
The points \(A = (4,\frac{1}{4})\) and \(B=(-5,-\frac{1}{5})\) lie on the hyperbola \(xy=1\). The circle with diameter \(AB\) intersects this hyperbola again at points \(X\) and \(Y\). Compute \(XY\).
HMMT February 2023 Alg/NT 8
Let \(S\) be the set of ordered pairs \((a,b)\) of positive integers such that \(\gcd(a,b)=1\). Compute
\[\sum_{(a,b)\in S} \left\lfloor\frac{300}{2a+3b}\right\rfloor.\]
HMMT February 2023 Geo 5
Let \(ABC\) be a triangle with \(AB=13\), \(BC=14\), and \(CA=15\). Suppose \(PQRS\) is a square such that \(P\) and \(R\) lie on line \(BC\), \(Q\) lies on line \(CA\), and \(S\) lies on line \(AB\). Compute the side length of this square.
HMMT November 2022 Team 9
Call an ordered pair \((a,b)\) of positive integers fantastic if and only if \(a,b\leq 10^4\) and \[\gcd(a\cdot n!-1, a\cdot (n+1)!+b) > 1\] for infinitely many positive integers \(n\). Find the sum of \(a+b\) across all fantastic pairs \((a,b)\).