Let $P(x)$ be a nonconstant monic polynomial with real coefficients. Find every continuous function $f:\mathbb R\to\mathbb R$ such that
$$f\bigl(f(P(x))+y+2023f(y)\bigr)=P(x)+2024f(y)$$
for all $x,y\in\mathbb R$.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days. The source gives no point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
A garden is a $2024\times2024$ grid of unit squares. A gardener places flowerpots subject to all three conditions:
1. Every pot contains exactly one of three species: chrysanthemum, rose, or orchid.
2. Each $1\times1$ square contains at most one pot.
3. For every pot, the number of pots of a different species in its row plus the number of pots of a different species in its column is $3$.
What is the largest possible number of pots if all three species must be present?
Let $ABC$ be an acute scalene triangle. Its incircle touches $BC,CA,AB$ at $D,E,F$, respectively. Let $X,Y,Z$ be the feet of the altitudes from $A,B,C$, respectively. Let $A'$ be the reflection of $X$ across $EF$, let $B'$ be the reflection of $Y$ across $FD$, and let $C'$ be the reflection of $Z$ across $DE$. Prove that triangles $ABC$ and $A'B'C'$ are similar.
Let $\alpha>1$ be real, and let $P(x)$ be a real polynomial of degree $24$ whose leading and constant coefficients are both $1$. Suppose $P(x)$ has $24$ positive real roots, all at most $\alpha$. Prove that
$$|P(1)|\le\left(\frac{19}{5}\right)^5(\alpha-1)^{24}.$$
Let $ABC$ be an acute scalene triangle inscribed in circle $(O)$. Its incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$, respectively. Ray $EF$ meets $(O)$ at $M$. The tangents to $(O)$ at $A,M$ meet at $S$, and the tangents at $B,C$ meet at $T$. Suppose $IT$ meets $OA$ at $J$. Prove that
$$\angle ASJ=\angle TSI.$$
Let $P(x)$ be a nonconstant polynomial with integer coefficients. Find all polynomials $Q(x)$ with integer coefficients such that, for every positive integer $n$, there is a polynomial $R_n(x)$ with integer coefficients satisfying
$$Q(x)^{2n}-1=R_n(x)\bigl(P(x)^{2n}-1\bigr).$$