HMMT Problems Compilation

During my undergraduate years, I was actively involved in HMMT as a Problem Staff (2022-23), Problem Czar (2023-24), Problem Czar Advisor (2024-25), and Historian (2025-26). Over four years, 113 of my problems were accepted into contest. All problems can be found at the HMMT Archive.

Here is a compilation of all of my problems appeared in HMMT. Problems marked with ★ are my favorites.

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HMIC 2026

HMIC 2026 P2

Prove that there exist infinitely many positive integers \(n\) such that \(n+100\) divides \(3^n-2^n-1\).

HMIC 2026 P3

Let \(ABC\) be a scalene triangle with circumcenter \(O\) and symmedian point \(K\). Points \(X\) and \(Y\) lie on the interiors of sides \(\overline{AC}\) and \(\overline{AB}\), respectively, such that \(XY\parallel BC\). Suppose that the circumcircles of triangles \(AXB\) and \(AYC\) meet segment \(\overline{BC}\) at points \(D \neq B\) and \(E \neq C\), respectively. Prove that \(D\), \(E\), \(X\), and \(Y\) lie on a circle whose center lies on line \(OK\).

(The symmedian point of triangle \(ABC\) is the intersection of the reflections of the \(B\)-median and \(C\)-median across the angle bisectors of \(\angle ABC\) and \(\angle ACB\), respectively.)

HMIC 2026 P5

Let \(n\) be a positive integer. Determine, in terms of \(n\), the number of ordered \(n\)-tuples \((a_1,a_2,\dots,a_n)\) of real numbers such that

\(0\leq a_i<1\) for all \(1\leq i\leq n\), and
\(a_1\cdot 1^k + a_2\cdot 2^k + \dots + a_n\cdot n^k\) is an integer for all nonnegative integers \(k\).

February 2026

February 2026 Alg/NT P1

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.

February 2026 Alg/NT P4

Let \(a\), \(b\), and \(c\) be pairwise distinct complex numbers such that \begin{align*} a^2+ab+b^2 &= 3(a+b),\\ a^2+ac+c^2 &= 3(a+c), \\ b^2+bc+c^2 &= 5(b+c)+1. \end{align*} Compute \(a\).

February 2026 Alg/NT P5

Compute the largest positive integer \(n\) such that \[n \text{ divides } (\lfloor\sqrt n\rfloor)!^{n!} + 450.\]

February 2026 Alg/NT P10

(joint with Jacopo Rizzo)

Let \[S = \sum_{k = 0}^{2026} k\binom{2k}{k}2^k.\] Compute the remainder when \(S\) is divided by \(2027\). (Note that \(2027\) is prime.)

February 2026 Geo P1

Let \(ABCD\) and \(WXYZ\) be squares such that \(W\) lies on segment \(\overline{AD}\), \(X\) lies on segment \(\overline{AB}\), and points \(Y\) and \(Z\) lie strictly inside \(ABCD\). Given that \(AW=4\), \(AX=5\), and \(AB=12\), compute the area of triangle \(\triangle BCY\).

February 2026 Geo P3

Let \(ABCD\) be a rectangle with \(AB=12\) and \(BC=16\). Points \(W\), \(X\), \(Y\), and \(Z\) lie on sides \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\), respectively, such that \(WXYZ\) is a rhombus with area \(120\). Compute \(XY\).

February 2026 Geo P7

(joint with Andrew Brahms, Jackson Dryg, and Jason Mao)

Let \(ABC\) be an isosceles triangle with \(AB = AC\). Points \(P\) and \(Q\) are located inside triangle \(ABC\) such that \(BP = PQ = QC\). Suppose that \(\angle BAP = 20^{\circ}\), \(\angle PAQ = 46^{\circ}\), and \(\angle QAC = 26^{\circ}\). Compute the measure of \(\angle APC\).

February 2026 Geo P8

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 \(AM\).

February 2026 Team P10

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 \(ab-1\).

February 2026 Guts P21

Compute the largest possible value of \[\gcd\left(\binom n3-1, \binom n4 -1\right)\] as \(n\) ranges through all positive integers greater than \(3\).

February 2026 Guts P25

Let \(p(x)\) be the unique polynomial of degree at most \(8\) and with rational coefficients such that \(p(\sqrt[3] 2 + \sqrt[3] 3) = \sqrt[3] 6\). Compute \(p(1)\).

February 2026 Guts P27

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 \(a\).

February 2026 Guts P30

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 \(BX\).

February 2026 Guts P31

Let \(\zeta=\cos\frac{2\pi}{19} + i\sin\frac{2\pi}{19}\). It is given that the polynomial \(x^3 + 16x^2 + 3x - 229\) has three distinct real roots, and its largest root can be uniquely written in the form \[a_1\zeta + a_2\zeta^2 + \dots + a_{18}\zeta^{18}\] for some rational numbers \(a_1\), \(\dots\), \(a_{18}\). Compute \(a_1^2 + a_2^2 + \dots + a_{18}^2\).

February 2026 Guts P35

Estimate \[\log_{10}\left(\sum_{k=0}^{15000} \frac{30000!}{(k!)^2 (30000-2k)!}\right).\] Submit a real number \(E\). If the correct answer is \(A\), you will receive \(\lfloor 20.05e^{-0.69(E-A)^2}\rfloor\) points.

November 2025

November 2025 General P9

(joint 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 \(abc\).

November 2025 Theme P8

Let \(MARS\) be a trapezoid with \(\overline{MA}\) parallel to \(\overline{RS}\) and side lengths \(MA = 11\), \(AR = 17\), \(RS = 22\), and \(SM = 16\). Point \(X\) lies on side \(\overline{MA}\) such that the common chord of the circumcircles of triangles \(MXS\) and \(AXR\) bisects segment \(\overline{RS}\). Compute \(MX\).

November 2025 Theme P10

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.

November 2025 Team P6

Let \(P\) be a point inside triangle \(ABC\) such that \(BP=PC\) and \(\angle ABP + \angle ACP=90^\circ\). Given that \(AB=12\), \(AC=16\), and \(AP=11\), compute the area of the concave quadrilateral \(ABPC\).

November 2025 Team P9

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 \(a+b+c\).

November 2025 Team P10

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 \(\cos\angle PAB\).

November 2025 Guts P12

Let \(ABCD\) be a right trapezoid such that \(\angle ABC=\angle BCD=90^\circ\) and the circle with diameter \(\overline{AD}\) is tangent to side \(\overline{BC}\). Given that \(AB=7\) and \(BC=8\), compute \(CD\).

November 2025 Guts P16

Let \(a_1\), \(a_2\), \(a_3\), \(a_4\), and \(a_5\) be the five distinct complex solutions of \(x^5-20x+25=0\). Compute \(a_1^4+a_2^4+a_3^4+a_4^4+a_5^4\).

November 2025 Guts P17

Let \(P\) be a point inside equilateral triangle \(ABC\) such that \(\angle BPC=150^\circ\). Given that circumradii of triangle \(ABP\) and triangle \(ACP\) are \(3\) and \(5\), respectively, compute \(AP\).

November 2025 Guts P19

(joint with Derek Liu)

Compute the number of ordered triples of positive integers \((a, b, c)\) such that \(b\) is a divisor of \(2025\) and \(\frac ab+\frac bc=\frac ac\).

November 2025 Guts P20

Suppose that \(ABCD\) and \(AXYZ\) are squares with side lengths \(10\) and \(7\), respectively. Given that \(X\) lies inside triangle \(ABY\) and \(Y\) lies on segment \(\overline{BD}\), compute the area of triangle \(BXC\).

November 2025 Guts P22

Suppose that \(a\), \(b\), and \(c\) are pairwise distinct nonzero complex numbers such that \[a^3-4a^2+5bc = b^3-4b^2+5ac = c^3-4c^2+5ab = 67.\] Compute \(abc\).

November 2025 Guts P24

Let \(ABCDE\) be a convex pentagon such that \(ABCD\) is a rectangle and \(\angle AEB = \angle CED = 30^\circ\). Given that \(AB=14\) and \(BC=20\sqrt 3\), compute the area of triangle \(ADE\).

November 2025 Guts P27

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 \(7^3\).

November 2025 Guts P34

(joint with Derek Liu)

Estimate the number of integers in \(\{1,2,3,\dots,10^8\}\) that can be written in the form \(x^2-2025y^2\) for some integers \(x\) and \(y\).

Submit a positive integer \(E\). If the correct answer is \(A\), you will receive \(\left\lfloor 20.99\max\left(0,1-\frac{|E-A|}A\right)^{2.5}\right\rfloor\) points.

February 2025

February 2025 Alg/NT P2

Mark writes the expression \(\sqrt{\underline{abcd}}\) on the board, where \(\underline{abcd}\) is a four-digit number. Derek, a toddler, decides to move the \(a\), changing Mark's expression to \(a\sqrt{\underline{bcd}}\). Surprisingly, these two expressions are equal. Compute the only possible four-digit number \(\underline{abcd}\).

February 2025 Alg/NT P9

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.)

February 2025 Geo P5

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 \(XP\).

February 2025 Geo P6

(joint with Karthik Vedula)

Trapezoid \(ABCD\), with \(AB\parallel CD\), has side lengths \(AB=11\), \(BC=8\), \(CD=19\), and \(DA=4\). Compute the area of the convex quadrilateral whose vertices are the circumcenters of \(\bigtriangleup ABC\), \(\bigtriangleup BCD\), \(\bigtriangleup CDA\), and \(\bigtriangleup DAB\).

February 2025 Geo P7

Point \(P\) is inside triangle \(\bigtriangleup ABC\) such that \(\angle ABP = \angle ACP\). Given that \(AB=6\), \(AC=8\), \(BC=7\), and \(\frac{BP}{PC}=\frac 12\), compute \(\tfrac{[BPC]}{[ABC]}\).

(Here, \([XYZ]\) denotes the area of \(\bigtriangleup XYZ\)).

February 2025 Geo P9

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 \(AB\).

February 2025 Team P3

Let \(\omega_1\) and \(\omega_2\) be two circles intersecting at distinct points \(A\) and \(B\). Point \(X\) varies along \(\omega_1\), and point \(Y\) on \(\omega_2\) is chosen such that \(AB\) bisects the angle \(\angle XAY\). Prove that as \(X\) varies along \(\omega_1\), the circumcenter of \(\triangle AXY\) (if it exists) varies along a fixed line.

February 2025 Team P5

Let \(\bigtriangleup ABC\) be an acute triangle with orthocenter \(H\). Points \(E\) and \(F\) are on segments \(\overline{AC}\) and \(\overline{AB}\), respectively, such that \(\angle EHF=90^\circ\). Let \(X\) be the foot of the altitude from \(H\) to \(\overline{EF}\). Prove that \(\angle BXC = 90^\circ\).

February 2025 Team P8

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 \(\angle IPM = 90^\circ\).

February 2025 Guts P6

Let \(\triangle ABC\) be an equilateral triangle. Point \(D\) lies on segment \(\overline{BC}\) such that \(BD=1\) and \(DC=4\). Points \(E\) and \(F\) lie on rays \(\overrightarrow{AC}\) and \(\overrightarrow{AB}\), respectively, such that \(D\) is the midpoint of \(\overline{EF}\). Compute \(EF\).

February 2025 Guts P11

Let \(f(n) = n^2+100\). Compute the remainder when \(\underbrace{f(f(\cdots f(f(}_{2025\ f\text{'s}}1))\cdots ))\) is divided by \(10^4\).

February 2025 Guts P15

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 \(DE\).

February 2025 Guts P17

Let \(f\) be a quadratic polynomial with real coefficients, and let \(g_1\), \(g_2\), \(g_3\), \(\dots\) be a geometric progression of real numbers. Define \(a_n=f(n)+g_n\). Given that \(a_1\), \(a_2\), \(a_3\), \(a_4\), and \(a_5\) are equal to \(1\), \(2\), \(3\), \(14\), and \(16\), respectively, compute \(\frac {g_2} {g_1}\).

February 2025 Guts P22

Let \(a\), \(b\), and \(c\) be real numbers such that \(a^2(b+c)=1\), \(b^2(c+a)=2\), and \(c^2(a+b)=5\). Given that there are three possible values for \(abc\), compute the minimum possible value of \(abc\).

February 2025 Guts P24

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 \(101^2\).

February 2025 Guts P25

Let \(ABCD\) be a trapezoid such that \(AB\parallel CD\), \(AD=13\), \(BC=15\), \(AB=20\), and \(CD=34\). Point \(X\) lies inside the trapezoid such that \(\angle XAB = 2\angle XBA\) and \(\angle XDC = 2\angle XCD\). Compute \(XD-XA\).

February 2025 Guts P27

Compute the number of ordered pairs \((m,n)\) of odd positive integers both less than \(80\) such that \[\gcd(4^m+2^m+1, 4^n+2^n+1) > 1.\]

February 2025 Guts P30

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 \(a\).

November 2024

November 2024 General P9

(joint with Ethan Liu and Isabella Zhu)

Let \(ABCDEF\) be a regular hexagon with center \(O\) and side length \(1\). Point \(X\) is placed in the interior of the hexagon such that \(\angle BXC = \angle AXE =90^\circ\). Compute all possible values of \(OX\).

November 2024 Theme P8

(joint with Srinivas Arun, David Dong, Jackson Dryg, Derek Liu, Henrick Rabinovitz, Krishna Pothapragada, 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 \(\operatorname{Top}(a,z) = \operatorname{Top}(z,a-1)\).

November 2024 Theme P10

Isabella the geologist discovers a diamond deep underground via an X-ray machine. The diamond has the shape of a convex cyclic pentagon \(PABCD\) with \(AD\parallel BC\). Soon after the discovery, her X-ray breaks, and she only recovers partial information about its dimensions. She knows that \(AD=70\), \(BC=55\), \(PA:PD=3:4\), and \(PB:PC=5:6\). Compute \(PB\).

November 2024 Team P8

Compute the unique real number \(x<3\) such that \[\sqrt{(3-x)(4-x)} + \sqrt{(4-x)(6-x)} + \sqrt{(6-x)(3-x)} = x.\]

November 2024 Team P9

(joint 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 \(ABCD\).

November 2024 Team P10

(joint with Jordan Lefkowitz)

For each positive integer \(n\), let \(f(n)\) be either the unique integer \(r\in\{0,1,\dots,n-1\}\) such that \(n\) divides \(15r-1\), or \(0\) if such \(r\) does not exist. Compute \[f(16)+f(17)+f(18)+\dots+f(300).\]

November 2024 Guts P5

Let \(ABCD\) be a trapezoid with \(AB\parallel CD\), \(AB=20\), \(CD=24\), and area \(880\). Compute the area of the triangle formed by the midpoints of \(AB\), \(AC\), and \(BD\).

November 2024 Guts P9

Compute the remainder when \[1\,002\,003\,004\,005\,006\,007\,008\,009\] is divided by \(13\).

November 2024 Guts P14

Let \(ABCD\) be a trapezoid with \(AB\parallel CD\). Point \(X\) is placed on segment \(\overline{BC}\) such that \(\angle BAX = \angle XDC\). Given that \(AB=5\), \(BX=3\), \(CX=4\), and \(CD=12\), compute \(AX\).

November 2024 Guts P20

There exists a unique line tangent to the graph of \(y=x^4-20x^3+24x^2-20x+25\) at two distinct points. Compute the product of the \(x\)-coordinates of the two tangency points.

November 2024 Guts P27

For any positive integer \(n\), let \(f(n)\) be the number of ordered triples \((a,b,c)\) of positive integers such that

\(\max(a,b,c)\) divides \(n\) and
\(\gcd(a,b,c)=1\).

Compute \(f(1)+f(2)+\dots+f(100)\).

November 2024 Guts P29

Let \(ABC\) be a triangle such that \(AB=3\), \(AC=4\), and \(\angle BAC=75^\circ\). Square \(BCDE\) is constructed outside triangle \(ABC\). Compute \(AD^2+AE^2\).

November 2024 Guts P34

The largest known prime number as of October 2024 is \(2^{136\,279\,841} - 1\). It happens to be an example of a prime number of the form \(2x^2-1\). Estimate the number of positive integers \(x \leq 10^6\) such that \(2x^2-1\) is prime.

Submit a positive integer \(E\). If the correct answer is \(A\), you will receive \(\left\lfloor 20.99\max\left(0,1-\frac{|E-A|}A\right)^{2.5}\right\rfloor\) points.

HMIC 2024

HMIC 2024 P5

Let \(ABC\) be an acute, scalene triangle with circumcenter \(O\) and symmedian point \(K\). Let \(X\) be the point on the circumcircle of triangle \(BOC\) such that \(\angle AXO = 90^\circ\). Assume that \(X\neq K\). The hyperbola passing through \(B\), \(C\), \(O\), \(K\), and \(X\) intersects the circumcircle of triangle \(ABC\) at points \(U\) and \(V\), distinct from \(B\) and \(C\). Prove that \(UV\) is the perpendicular bisector of \(AX\).

The symmedian point of triangle \(ABC\) is the intersection of the reflections of \(B\)-median and \(C\)-median across the angle bisectors of \(\angle ABC\) and \(\angle ACB\), respectively.

February 2024

February 2024 Alg/NT P4

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 \(f(0).\)

February 2024 Alg/NT P5

Compute the unique ordered pair \((x,y)\) of real numbers satisfying the system of equations \[\frac{x}{\sqrt{x^2+y^2}} - \frac 1x = 7 \quad \text{and} \quad \frac{y}{\sqrt{x^2+y^2}} + \frac 1y = 4.\]

February 2024 Alg/NT P6

Compute the sum of all positive integers \(n\) such that \(50\leq n\leq 100\) and \(2n+3\) does not divide \(2^{n!}-1\).

February 2024 Alg/NT P10

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.

February 2024 Geo P3

(joint with Benjamin Kang and Holden Mui)

Let \(\Omega\) and \(\omega\) be circles with radii \(123\) and \(61\), respectively, such that the center of \(\Omega\) lies on \(\omega\). A chord of \(\Omega\) is cut by \(\omega\) into three segments, whose lengths are in the ratio \(1:2:3\) in that order. Given that this chord is not a diameter of \(\Omega\), compute the length of this chord.

February 2024 Geo P5

Let \(ABCD\) be a convex trapezoid such that \(\angle DAB=\angle ABC=90^\circ\), \(DA=2\), \(AB=3\), and \(BC=8\). Let \(\omega\) be a circle passing through \(A\) and tangent to segment \(\overline{CD}\) at point \(T\). Suppose that the center of \(\omega\) lies on line \(BC\). Compute \(CT\).

February 2024 Geo P8

Let \(ABTCD\) be a convex pentagon with area \(22\) such that \(AB=CD\) and the circumcircles of triangles \(TAB\) and \(TCD\) are internally tangent. Given that \(\angle ATD=90^\circ\), \(\angle BTC=120^\circ\), \(BT=4\), and \(CT=5\), compute the area of triangle \(TAD\).

February 2024 Geo P9

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 \(CP\).

February 2024 Team P3

Let \(ABC\) be a scalene triangle and \(M\) be the midpoint of \(BC\). Let \(X\) be the point such that \(CX\parallel AB\) and \(\angle AMX=90^\circ\). Prove that \(AM\) bisects \(\angle BAX\).

February 2024 Team P8

Let \(P\) be a point in the interior of quadrilateral \(ABCD\) such that the circumcircles of triangles \(PDA\), \(PAB\), and \(PBC\) are pairwise distinct but congruent. Let the lines \(AD\) and \(BC\) meet at \(X\). If \(O\) is the circumcenter of triangle \(XCD\), prove that \(OP\perp AB\).

February 2024 Team P10

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 \(i^2 = -1\).

February 2024 Guts P5

Let \(a\), \(b\), and \(c\) be real numbers such that \begin{align*} a+b+c &= 100, \\ ab+bc+ca &= 20,\text{ and } \\ (a+b)(a+c) &= 24. \end{align*} Compute all possible values of \(bc\).

February 2024 Guts P9

(joint with Rishabh Das and Holden Mui)

Compute the sum of all positive integers \(n\) such that \(n^2-3000\) is a perfect square.

February 2024 Guts P11

Let \(ABCD\) be a rectangle such that \(AB=20\) and \(AD=24\). Point \(P\) lies inside \(ABCD\) such that triangles \(PAC\) and \(PBD\) have areas \(20\) and \(24\), respectively. Compute all possible areas of triangle \(PAB\).

February 2024 Guts P18

An ordered pair \((a,b)\) of positive integers is called spicy if \(\gcd(a+b, ab+1)=1\). Compute the probability that both \((99,n)\) and \((101,n)\) are spicy when \(n\) is chosen from \(\{1,2,\dots,2024!\}\) uniformly at random.

February 2024 Guts P25

Point \(P\) is inside a square \(ABCD\) such that \(\angle APB=135^\circ\), \(PC=12\), and \(PD=15\). Compute the area of this square.

February 2024 Guts P26

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 \(P(3,-3)\).

February 2024 Guts P29

For each prime \(p\), a polynomial \(P(x)\) with rational coefficients is called \(p\)-good if and only if there exist three integers \(a\), \(b\), and \(c\) such that \(0\leq a < b < c < \tfrac p3\) and \(p\) divides all the numerators of \(P(a)\), \(P(b)\), and \(P(c)\), when written in simplest form. Compute the number of ordered pairs \((r,s)\) of rational numbers such that the polynomial \(x^3 + 10x^2 + rx + s\) is \(p\)-good for infinitely many primes \(p\).

February 2024 Guts P34

Estimate the number of positive integers \(n\leq 10^6\) such that \(n^2+1\) has a prime factor greater than \(n\). Submit a positive integer \(E\). If the correct answer is \(A\), you will receive \(\max\left(0, \left\lfloor 20\cdot \min\left(\frac{E}{A}, \frac{10^6-E}{10^6-A}\right)^5+ 0.5 \right\rfloor\right)\) points.

February 2024 Guts P36

(joint with Kevin Zhao)

Let \(ABC\) be a triangle. The following diagram contains points \(P_1,P_2,\dots,P_7\), which are the following triangle centers of triangle \(ABC\) in some order:

the incenter \(I\);
the circumcenter \(O\);
the orthocenter \(H\);
the symmedian point \(L\), which is the intersections of the reflections of \(B\)-median and \(C\)-median across angle bisectors of \(\angle ABC\) and \(\angle ACB\), respectively;
the Gergonne point \(G\), which is the intersection of lines from \(B\) and \(C\) to the tangency points of the incircle with \(\overline{AC}\) and \(\overline{AB}\), respectively;
the Nagel point \(N\), which is the intersection of line from \(B\) to the tangency point between \(B\)-excircle and \(\overline{AC}\), and line from \(C\) to the tangency point between \(C\)-excircle and \(\overline{AB}\); and
the Kosnita point \(K\), which is the intersection of lines from \(B\) and \(C\) to the circumcenters of triangles \(AOC\) and \(AOB\), respectively.

Note that the triangle \(ABC\) is not shown. Compute which triangle centers \(\{I,O,H,L,G,N,K\}\) corresponds to \(P_k\) for \(k\in\{1,2,3,4,5,6,7\}\).

Your answer should be a seven-character string containing \(I\), \(O\), \(H\), \(L\), \(G\), \(N\), \(K\), or \(X\) for blank. For instance, if you think \(P_2=H\) and \(P_6=L\), you would answer \(XHXXXLX\). If you attempt to identify \(n>0\) points and get them all correct, then you will receive \(\left\lceil (n-1)^{5/3}\right\rceil\) points. Otherwise, you will receive \(0\) points.

November 2023

November 2023 General P3

(joint with Ankit Bisain, Sean Li, and Eric Shen)

Compute the number of positive four-digit multiples of \(11\) whose sum of digits (in base ten) is divisible by \(11\).

November 2023 General P7

(joint with Ethan Liu)

Compute all ordered triples \((x,y,z)\) of real numbers satisfying the following system of equations: \begin{align*} xy + z &= 40 \\ xz + y &= 51 \\ x + y + z &= 19. \end{align*}

November 2023 General P8

Mark writes the expression \(\sqrt d\) for each positive divisor \(d\) of \(8!\) on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form \(a\sqrt b\), where \(a\) and \(b\) are integers such that \(b\) is not divisible by the square of a prime number. (For example, \(\sqrt{20}\), \(\sqrt{16}\), and \(\sqrt{6}\) simplify to \(2\sqrt{5}\), \(4\sqrt{1}\), and \(1\sqrt{6}\), respectively.) Compute the sum of \(a+b\) across all expressions that Rishabh writes.

November 2023 General P10

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 \(CX\).

November 2023 Theme P4

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 \(V\).

November 2023 Theme P6

(joint with Isabella Quan and Alex Yi)

A function \(g\) is ever more than a function \(h\) if, for all real numbers \(x\), we have \(g(x) \geq h(x)\). Consider all quadratic functions \(f(x)\) such that \(f(1)=16\) and \(f(x)\) is ever more than both \((x+3)^2\) and \(x^2+9\). Across all such quadratic functions \(f\), compute the minimum value of \(f(0)\).

November 2023 Team P5

A complex quartic polynomial \(Q\) is quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of \(k\) for which the polynomial \(Q(x) = x^4-kx^3-x^2-x-45\) is quirky. Compute the product of these four values of \(k\).

November 2023 Team P7

Let \(ABCD\) be a convex trapezoid such that \(\angle BAD = \angle ADC = 90^\circ\), \(AB=20\), \(AD=21\), and \(CD=28\). Point \(P \neq A\) is chosen on segment \(AC\) such that \(\angle BPD=90^\circ\). Compute \(AP\).

November 2023 Guts P10

A real number \(x\) is chosen uniformly at random from the interval \((0,10)\). Compute the probability that \(\sqrt{x}\), \(\sqrt{x+7}\), and \(\sqrt{10-x}\) are the side lengths of a non-degenerate triangle.

November 2023 Guts P17

Let \(ABC\) be an equilateral triangle of side length \(15\). Let \(A_b\) and \(B_a\) be points on side \(AB\), \(A_c\) and \(C_a\) be points on side \(AC\), and \(B_c\) and \(C_b\) be points on side \(BC\) such that \(\triangle AA_bA_c\), \(\triangle BB_cB_a\), and \(\triangle CC_aC_b\) are equilateral triangles with side lengths \(3\), \(4\), and \(5\), respectively. Compute the radius of the circle tangent to segments \(\overline{A_bA_c}\), \(\overline{B_aB_c}\), and \(\overline{C_aC_b}\).

November 2023 Guts P20

Let \(ABCD\) be a square of side length \(10\). Point \(E\) is on ray \(\overrightarrow{AB}\) such that \(AE=17\), and point \(F\) is on ray \(\overrightarrow{AD}\) such that \(AF=14\). The line through \(B\) parallel to \(CE\) and the line through \(D\) parallel to \(CF\) meet at \(P\). Compute the area of quadrilateral \(AEPF\).

November 2023 Guts P21

An integer \(n\) is chosen uniformly at random from the set \(\{1,2,3,\dots,2023!\}\). Compute the probability that \[\gcd(n^n+50, n+1)=1.\]

November 2023 Guts P23

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 \(XY\).

November 2023 Guts P28

There is a unique quadruple of positive integers \((a,b,c,k)\) such that \(c\) is not a perfect square and \(a+\sqrt{b+\sqrt c}\) is a root of the polynomial \(x^4-20x^3+108x^2-kx+9\). Compute \(c\).

November 2023 Guts P29

Let \(A_1A_2\dots A_6\) be a regular hexagon with side length \(11\sqrt 3\), and let \(B_1B_2\dots B_6\) be another regular hexagon completely inside \(A_1A_2\dots A_6\) such that for all \(i\in\{1,2,\dots,5\}\), \(A_iA_{i+1}\) is parallel to \(B_iB_{i+1}\). Suppose that the distance between lines \(A_1A_2\) and \(B_1B_2\) is \(7\), the distance between lines \(A_2A_3\) and \(B_2B_3\) is \(3\), and the distance between lines \(A_3A_4\) and \(B_3B_4\) is \(8\). Compute the side length of \(B_1B_2\dots B_6\).

November 2023 Guts P35

(joint with Rishabh Das, Amy Feng, Vidur Jasuja, and Isabella Quan)

Dorothea has a \(3 \times 4\) grid of dots. She colors each dot red, blue, or dark gray. Compute the number of ways Dorothea can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color.

Submit a positive integer \(A.\) If the correct answer is \(C\) and your answer is \(A\), you will receive \(\left\lfloor 20\left(\min\left(\frac{A}{C}, \frac{C}{A}\right)\right)^2\right\rfloor\) points.

November 2023 Guts P36

(joint with Rishabh Das, Vidur Jasuja, and Isabella Quan)

Isabella writes the expression \(\sqrt d\) for each positive integer \(d\) not exceeding \(8!\) on the board. Seeing that these expressions might not be worth points on HMMT, Vidur simplifies each expression to the form \(a\sqrt b\), where \(a\) and \(b\) are integers such that \(b\) is not divisible by the square of a prime number. (For example, \(\sqrt{20}\), \(\sqrt{16}\), and \(\sqrt{6}\) simplify to \(2\sqrt{5}\), \(4\sqrt{1}\), and \(1\sqrt{6}\), respectively.) Compute the sum of \(a+b\) across all expressions that Vidur writes.

Submit a positive real number \(A\). If the correct answer is \(C\) and your answer is \(A\), you get \(\max \left(0, \left\lceil 20\left(1-\lvert \log(A/C)\rvert^{1/5}\right)\right\rceil \right)\) points.

February 2023

February 2023 Alg/NT P8

Let \(S\) be the set of ordered pairs \((a,b)\) of positive integers such that \(\gcd(a,b)=1\). Compute \[\sum_{(a,b)\in S} \left\lfloor\frac{300}{2a+3b}\right\rfloor.\]

February 2023 Geo P5

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.

February 2023 Geo P6

Convex quadrilateral \(ABCD\) satisfies \(\angle CAB = \angle ADB = 30^\circ\), \(\angle ABD = 77^\circ\), \(BC=CD\), and \(\angle BCD = n^\circ\) for some positive integer \(n\). Compute \(n\).

February 2023 Team P3

Let \(ABCD\) be a convex quadrilateral such that \(\angle ABC = \angle BCD = \theta\) for some angle \(\theta < 90^{\circ}\). Point \(X\) lies inside the quadrilateral such that \(\angle XAD = \angle XDA = 90^\circ - \theta\). Prove that \(BX=XC\).

February 2023 Team P7

Let \(ABC\) be a triangle. Point \(D\) lies on segment \(BC\) such that \(\angle BAD = \angle DAC\). Point \(X\) lies on the opposite side of line \(BC\) as \(A\) and satisfies \(XB=XD\) and \(\angle BXD=\angle ACB\). Analogously, point \(Y\) lies on the opposite side of line \(BC\) as \(A\) and satisfies \(YC=YD\) and \(\angle CYD=\angle ABC\). Prove that lines \(XY\) and \(AD\) are perpendicular.

February 2023 Guts P17

An equilateral triangle lies in the Cartesian plane such that the \(x\)-coordinates of its vertices are pairwise distinct and all satisfy the equation \(x^3 - 9x^2 + 10x + 5 = 0\). Compute the side length of the triangle.

February 2023 Guts P32

(joint with Luke Robitaille)

Let \(ABC\) be a triangle with \(\angle BAC > 90^{\circ}\). Let \(D\) be the foot of the perpendicular from \(A\) to side \(BC\). Let \(M\) and \(N\) be the midpoints of segments \(BC\) and \(BD\), respectively. Suppose that \(AC=2\), \(\angle BAN = \angle MAC\), and \(AB \cdot BC = AM\). Compute the distance from \(B\) to line \(AM\).

November 2022

November 2022 Theme P9

Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle \(ABC\) and a point \(D\) on segment \(BC\) satisfying \(BD=3\) and \(CD=5\). Then, Alice chooses a point \(P\) on line \(AD\) and challenges Bob to mark a point \(Q \neq P\) on line \(AD\) such that \(\frac{BQ}{QC} = \frac{BP}{PC}\). Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of \(\frac{BP}{PC}\) for the \(P\) she chose?

November 2022 Team P9

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 \((a,b)\).

November 2022 Guts P9

Let \(ABCD\) be a trapezoid such that \(AB\parallel CD\), \(\angle BAC=25^\circ\), \(\angle ABC = 125^\circ\), and \(AB+AD=CD\). Compute \(\angle ADC\).

November 2022 Guts P24

A string consisting of letters \(\texttt A\), \(\texttt C\), \(\texttt G\), and \(\texttt U\) is untranslatable if and only if it has no \(\texttt{AUG}\) as a consecutive substring. For example, \(\texttt{ACUGG}\) is untranslatable.

Let \(a_n\) denote the number of untranslatable strings of length \(n\). It is given that there exists a unique triple of real numbers \((x,y,z)\) such that \(a_n = xa_{n-1} + ya_{n-2} + za_{n-3}\) for all integers \(n\geq 100\). Compute \((x, y, z)\).