There are two classes: class A has $m$ pupils and class B has $n$ pupils, where $m,n\gt 1$. All pupils sit around a circular table. Each pupil X receives as many candies as the number of consecutive pupils immediately to X's left belonging to the same class as X (if there are none, X receives no candy). Pupils receiving the same number of candies are placed in one group.
(a) What is the largest possible size of a group?
(b) If the group receiving no candy is excluded, what is the largest possible size of a group?
The source gives one shared 270-minute header for the two day sections. This edition applies it to each session, totaling 540 minutes; it does not claim the source prints a 540-minute annual limit or two separate timing headers. The source gives no point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
Consider the following functions on the nonzero real numbers:
$$P(x)=(x^2-1)^{2023},\qquad Q(x)=(2x+1)^{14},\qquad R(x)=\left(2x+1+\frac2x\right)^{34}.$$
Initially a list contains exactly two of these three functions. At each operation, one may add, subtract or multiply functions in the list, or raise a listed function to a positive integer power. One may also add, subtract or multiply a listed function by any real constant, obtaining a new function and adding it to the list. These operations may be repeated. Prove that, whichever two functions are chosen initially, the remaining function cannot be obtained.
Let $ABC$ be an acute scalene triangle inscribed in circle $(O)$. The altitudes $BE,CF$ meet at the orthocenter $H$, and $M$ is the midpoint of $AH$. Let $K$ be the orthogonal projection of $H$ onto $EF$. A line not passing through $A$ and parallel to $BC$ meets the minor arcs $AB,AC$ at $P,Q$, respectively. Prove that the tangent at $E$ to the circumcircle of triangle $CQE$ and the tangent at $F$ to the circumcircle of triangle $BPF$ meet at a point on line $MK$.
Let $a,b$ be relatively prime positive integers with $b$ odd and $a\gt 2$. Define the sequence by $x_0=2$, $x_1=a$, and $x_{n+2}=ax_{n+1}+bx_n$ for every positive integer $n$.
(a) If $a$ is even, prove that there are no positive integers $m,n,p$ for which $x_m/(x_nx_p)$ is an integer.
(b) If $a$ is odd, prove that there are no positive integers $m,n,p$ such that $mnp$ is even and $x_m/(x_nx_p)$ is a perfect square.
Let $ABCD$ be a convex quadrilateral with $\angle B\lt \angle A\lt 90^\circ$. Let $I$ be the midpoint of $AB$ and $S=AD\cap BC$. A point $R$ moves inside triangle $SAB$ with $\angle ASR=\angle BSR$. On the lines $AR,BR$, choose $E,F$, respectively, so that $BE$ and $AF$ are parallel to $RS$. Suppose $EF$ meets the circumcircle of triangle $SAB$ at $H,K$. On segment $AB$, choose $M,N$ such that $\angle AHM=\angle BHI$ and $\angle BKN=\angle AKI$.
(a) Prove that the circumcenter $J$ of triangle $SMN$ lies on a fixed line.
(b) On $BE,AF$, choose $P,Q$, respectively, so that $CP\parallel SE$ and $DQ\parallel SF$. The lines $SE,SF$ meet circle $(O)$ again at $U,V$, respectively. Let $G=AU\cap BV$. Prove that the median from $G$ in triangle $GPQ$ always passes through a fixed point.
Let $n\ge3$ be an integer and $A=\{1,2,\ldots,n\}$. Determine the largest $k$ such that, for every collection of $k$ three-element subsets of $A$, one can color every element of $A$ either blue or red (one color per element) so that none of the selected subsets has all three elements of the same color.