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.
Consider the sequence $(a_n)$ satisfying
$$a_1=\frac12,\qquad a_{n+1}=\sqrt[3]{3a_{n+1}-a_n},\qquad0\le a_n\le1\quad(n\ge1).$$
a) Prove that the sequence is uniquely determined and has a finite limit.
b) Define $b_n=(1+2a_1)(1+2^2a_2)\cdots(1+2^na_n)$ for $n\ge1$. Prove that $(b_n)$ has a finite limit.
Let $a,b,c,\alpha,\beta$ be integers and define $u_1=\alpha$, $u_2=\beta$, $u_{n+2}=au_{n+1}+bu_n+c$ for $n\ge1$.
a) Prove that for $a=3,b=-2,c=-1$, there are infinitely many integer pairs $(\alpha,\beta)$ such that $u_{2023}=2^{2022}$.
b) Prove that some positive integer $n_0$ has exactly one of the following properties:
i) There are infinitely many positive integers $m$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+m}$ is divisible by $7^{2023}$ or $17^{2023}$.
ii) There are infinitely many positive integers $k$ such that $u_{n_0}u_{n_0+1}\cdots u_{n_0+k}-1$ is divisible by 2023.
Find the largest positive real number $k$ such that
$$\frac1{kab+c^2}+\frac1{kbc+a^2}+\frac1{kca+b^2}\ge\frac{k+3}{a^2+b^2+c^2}$$
for every positive triple $(a,b,c)$ satisfying $a^2+b^2+c^2=2(ab+bc+ca)$.
Let $ABCD$ be a cyclic quadrilateral with $DB=DC$. Let $M,N$ be the midpoints of $AB,AC$, and let $J,E,F$ be the contact points of the incircle $(I)$ of triangle $ABC$ with $BC,CA,AB$, respectively. Line $MN$ meets $JE,JF$ at $K,H$, respectively. Line $IJ$ meets circle $(IBC)$ again at $G$, and $DG$ meets $(IBC)$ again at $T$.
a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$.
b) Let $R,S$ be the perpendicular projections of $D$ on $AB,AC$. Choose $P\in IF$, $Q\in IE$ so that $KP,HQ$ are perpendicular to $MN$. Prove that $MP,NQ,RS$ are concurrent.
Let $f,g:\mathbb R\to\mathbb R$ satisfy $f(0)=2022$ and
$$f(x+g(y))=xf(y)+(2023-y)f(x)+g(x)\quad(x,y\in\mathbb R).$$
a) Prove that $f$ is surjective and $g$ is injective.
b) Find all pairs of functions $f,g$ satisfying the conditions.
There are $n\ge2$ classes organizing $m\ge1$ extracurricular clubs. Every class has students participating in at least one club. Each club has participants from exactly $a$ classes. For any two clubs, at most $b$ classes have participants in both.
a) Find $m$ when $n=8,a=4,b=1$.
b) Prove $n\ge20$ when $m=6,a=10,b=4$.
c) Find the smallest $n$ when $m=20,a=4,b=1$.
Let $ABC$ be an acute scalene triangle, with orthocenter $H$ and circumcenter $O$. Its incircle $(I)$ touches $BC,CA,AB$ at $M,N,P$, respectively. Let $\Omega_A$ be a circle through $A$, externally tangent to $(I)$ at $A'$, and meeting $AB,AC$ again at $A_b,A_c$, respectively. Define $\Omega_B,\Omega_C$ and $B',B_a,B_c,C',C_a,C_b$ analogously.
a) Prove
$$B_cC_b+C_aA_c+A_bB_a\ge NP+PM+MN.$$
b) Suppose $A',B',C'$ lie on $AM,BN,CP$, respectively. Let $K$ be the circumcenter of the triangle whose side lines are $A_bA_c,B_cB_a,C_aC_b$. Prove that $OH\parallel IK$.