Here are some math problems (not necessarily "hard") (full document in both latex and image - can't upload pdf)
Latex (Compiled with Overleaf)
\documentclass[10pt,a4paper]{article}
\usepackage[a4paper, total={6in, 8in}]{geometry}
\usepackage{graphicx}
\usepackage{amsmath} % For the equation* environment
\usepackage{amssymb}
\usepackage{gensymb}
\usepackage{pgfplots}
\pgfplotsset{width=10cm,compat=1.9}
\usepackage{tikz}
\usetikzlibrary{calc} % To perform calculations within coordinates
% We will externalize the figures
\usepgfplotslibrary{external}
\tikzexternalize
\title{'25 April Whatsapp Math Problem}
\author{Totient 4breakfast}
\date{April 2025}
\begin{document}
\maketitle
\section{Photon Madness}
A top-opened, rectangular container exists in a 2d universe. A photon $P$ is fired somewhere in the container towards a side with a slight downward tilt. As the surfaces of the container are perfectly reflective, $\theta_{incidence} = \theta_{reflection}$ for all reflections. The rectangle has a height of 2025u and a width of 11u, and the fired photon first hits the surface of the container at $A$, which is at a height of 1012u with an angle of incidence of $45\degree$. How many times does the photon hit the surface before it escapes?
\begin{center}
\begin{tikzpicture}
% Draw a circle with center at (0,0) and radius 2
\draw (0,8) -- (0,0) -- (2,0) -- (2,8);
\draw[->] (1,4.4) -- (2,4) -- (0,3.2);
\draw[dotted] (0,3.2) -- (2,2.4) -- (0,1.6);
\draw[loosely dotted] (0,4) -- (2, 4);
\node at (1,4.4) [left] {$P$};
\node at (2,4) [right] {$A$};
\node at (2,2) [right] {$1012$};
\node at (2,6) [right] {$1013$};
\node at (1.7,4.4) {$45\degree$};
\node at (1,0) [below] {$11$};
\end{tikzpicture}
\end{center}
\clearpage
\section{Algebraic Manipulations}
Consider the equation
\[\frac{a^2}{b^2+c^2} + \frac{b^2}{a^2+c^2} + \frac{c^2}{a^2+b^2} = 2025\]
Given that $ab + bc + ca = 2$ and $a+b+c=7$, find the value of $\frac{1}{b^2+c^2} + \frac{1}{a^2+c^2} + \frac{1}{b^2+a^2}$.
\section{Polynomials}
Consider a 19-degree polynomial $P(x)=a_{19}x^{19}+a_{18}x^{18}+...+a_{1}x^{1}+a_{0}x^{0}$. Given $P(1) = 2025$ and $P(-1) = 1205$, find the value of $a_{19} + a_{17} + a_{15} +...+a_{5}+a_{3}+a_{1}$.
\section{Modular Arithmetic}
$D(x)$ represents the sum of digits of $x$. For example, $D(456)=4+5+6=15$. Find the value of $D(D(2025^{2025}))$
\section{Modular Arithmetic Too}
Find the last two digits of $123^{456^{789}}$.
\section{Induction (Proof)}
$k$ number of bacteria live in a pond. Every day, their number grows according to the formula $2n+2$, where $n$ is the current number of bacteria, if there is an odd number of bacteria and half in number if there is an even number of bacteria. If the number of bacteria becomes 1, it is considered endangered. Will the bacteria always become endangered no matter what $k$ is?
\end{document}
Find the smallest positive integer n such that among n irrational numbers, there exist three numbers such that the sum of any two of them is still an irrational number.