Math
I like writing math problemsHere are some of my favorite problems I've written, sorted by contest date
2026 WAIME P5:
Non-degenerate octagon \(ABCDEFGH\) with area \(30\) satisfies that \(ACEG\) is a square and \(\angle ABC=\angle CDE=\angle EFG=\angle GHA=90^\circ\). Square \(JKLM\) with side lengths parallel to \(ACEG\) is constructed such that points \(B\), \(D\), \(F\), and \(H\) lie on the square. The value of \(\frac{[JKLM]}{[ACEG]}\) is a positive integer. Find the sum of all possible values of \([ACEG]^2\), given that \(ACEG\) is inside \(ABCDEFGH\).
Non-degenerate octagon \(ABCDEFGH\) with area \(30\) satisfies that \(ACEG\) is a square and \(\angle ABC=\angle CDE=\angle EFG=\angle GHA=90^\circ\). Square \(JKLM\) with side lengths parallel to \(ACEG\) is constructed such that points \(B\), \(D\), \(F\), and \(H\) lie on the square. The value of \(\frac{[JKLM]}{[ACEG]}\) is a positive integer. Find the sum of all possible values of \([ACEG]^2\), given that \(ACEG\) is inside \(ABCDEFGH\).
2026 WAIME P12:
Find the minimum value of \(k\) such that $$\sum_{x=1}^{k}\sum_{m=1}^{3^x} 3^m \prod _{n=1}^{3^x}\left(\left \lfloor \frac{m}{3^n} \right \rfloor -3 \left \lfloor \frac{m}{3^{n+1}} \right \rfloor-1\right)^2$$ is divisible by \(1000\).
Find the minimum value of \(k\) such that $$\sum_{x=1}^{k}\sum_{m=1}^{3^x} 3^m \prod _{n=1}^{3^x}\left(\left \lfloor \frac{m}{3^n} \right \rfloor -3 \left \lfloor \frac{m}{3^{n+1}} \right \rfloor-1\right)^2$$ is divisible by \(1000\).
2026 WAIME P13:
Triangle \(ABC\) satisfies \(AB=3\), \(BC=5\), and \(AC=7\). A point \(P\) lies in the interior of the triangle. Let \(D\) be the reflection of \(P\) across point \(B\), and let \(E\) be the reflection of \(P\) across point \(C\). Let \(F\) denote the reflection of point \(B\) across \(P\). If \(F\) is the midpoint of segment \(AE\), determine the square of the area of \(\triangle ADE\).
Triangle \(ABC\) satisfies \(AB=3\), \(BC=5\), and \(AC=7\). A point \(P\) lies in the interior of the triangle. Let \(D\) be the reflection of \(P\) across point \(B\), and let \(E\) be the reflection of \(P\) across point \(C\). Let \(F\) denote the reflection of point \(B\) across \(P\). If \(F\) is the midpoint of segment \(AE\), determine the square of the area of \(\triangle ADE\).
2025 WAMC12 P21:
Triangle \(ABC\) has \(AB=11\), \(AC=2\), and circumcircle \(\omega\). Let \(D\) be a point on side \(AB\), and point \(E\) is a point on arc \(BC\) of \(\omega\) not including \(A\) satisfying \(DE\perp BC\). Another point \(F\) on \(\omega\) satisfies \(\angle CBF=90^\circ\). Given that \(BE=BF\sqrt3=EF\sqrt3\), find \(BD\).
Triangle \(ABC\) has \(AB=11\), \(AC=2\), and circumcircle \(\omega\). Let \(D\) be a point on side \(AB\), and point \(E\) is a point on arc \(BC\) of \(\omega\) not including \(A\) satisfying \(DE\perp BC\). Another point \(F\) on \(\omega\) satisfies \(\angle CBF=90^\circ\). Given that \(BE=BF\sqrt3=EF\sqrt3\), find \(BD\).
MathDash Contest 49 Emerald P1:
How many ways are there to go from the bottom left corner to the top right corner of a \(4 \times 4\) grid of squares if the only possible movements are up, down, and right, and the same point cannot be visited twice?
How many ways are there to go from the bottom left corner to the top right corner of a \(4 \times 4\) grid of squares if the only possible movements are up, down, and right, and the same point cannot be visited twice?
MathDash Contest 49 Emerald P4:
Given that the small circles have radius \(1\) and the large circle has radius \(2\), find \(r^2\). Note that the line with length \(r\) is tangent to the smaller circle and that all lines and circles that look tangent are tangent.
Given that the small circles have radius \(1\) and the large circle has radius \(2\), find \(r^2\). Note that the line with length \(r\) is tangent to the smaller circle and that all lines and circles that look tangent are tangent.
MathDash Contest 49 Emerald P5:
How many integer values of \(x, y\) satisfy \(1 \le x \le y \le 100\) and \(100|xy + x + y\)?
How many integer values of \(x, y\) satisfy \(1 \le x \le y \le 100\) and \(100|xy + x + y\)?
MathDash Contest 49 Emerald P8:
Evaluate $$ \sum_{n=1}^{2025} \lfloor \frac{n\lfloor \frac{2025}{n} \rfloor-25}{1000} \rfloor$$
Evaluate $$ \sum_{n=1}^{2025} \lfloor \frac{n\lfloor \frac{2025}{n} \rfloor-25}{1000} \rfloor$$