Question point values reproduce the supplied source labels; no automatic proof grading is implied.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days.
For a positive integer $k$, a set $S$ of positive integers is called $k$-Olympic if both of the following hold:
(i) $S\ne\varnothing$;
(ii) for every $n\in S$, every positive divisor of $(25^n-3^n)k^n$ also belongs to $S$.
Find all positive integers $k$ for which exactly one $k$-Olympic set exists.
Let $n$ be a positive integer. A country has $8n+3$ airports. Between any two airports there is either a direct flight route or no direct route. Whenever two airports have no direct route between them, their numbers of direct routes differ by exactly $2$. Determine the minimum total number of direct routes.
Let $ABC$ be an acute scalene triangle with altitudes $AD,BE,CF$. From $A$, drop perpendiculars to the lines $EF,FD,DE$, with feet $X,Y,Z$, respectively. Let the line $BZ$ meet the circle $(BDY)$ again at $P$, and let $CY$ meet $(CDZ)$ again at $Q$. Prove that $X$ has equal powers with respect to the circles $(YFP)$ and $(ZEQ)$.
Let $ABC$ be a triangle, and let $O$ be the midpoint of $BC$. Draw the tangents $AE,AF$ to the circle $(O)$ with diameter $BC$, where $E,F\in(O)$. The rays $AE,AF$ meet $BC$ at $K,L$, respectively. Let $KF,LE$ meet $(O)$ again at $M,N$, respectively. The circumcircle of triangle $MON$ meets the circles with diameters $AB,AC$ again at $X,Y$, respectively. Prove that $\angle XAB=\angle YAC$.
Let $k,n$ be positive integers with $k\lt n$. Find all real-coefficient polynomials $P$ of degree $kn$ with leading coefficient $1$ such that
$$Q(x)=P(x^n+1)-(P(x))^n$$
has degree at most $kn(n-1)$.
Let $\mathcal H$ be a family of nonempty subsets of $\{1,2,3,\ldots,2027\}$ with the following property: for every $A\in\mathcal H$ and every subset $B\subset A$, one has $B\in\mathcal H$. Let $\ell_{\mathcal H},c_{\mathcal H}$ denote, respectively, the numbers of sets in $\mathcal H$ having an even number of elements and an odd number of elements. Prove that
$$\ell_{\mathcal H}-c_{\mathcal H}\le\binom{2026}{1013}.$$
For each positive integer $n$, consider the polynomial $P_n(x)=x^{3n}-3\cdot4^{n-1}x^n-2^{3n-3}$.
(a) Prove that $P_n(x)$ has exactly one positive real root, denoted by $a_n$.
(b) Put $b_n=(2-a_n)/n$ and $c_n=b_1+b_2+\cdots+b_n$. Prove that the sequence $(c_n)_{n=1}^{\infty}$ has a finite limit.
To explore space, scientists often observe distant objects such as comets, asteroids and other astronomical phenomena. For this purpose they design and launch observation satellites into orbit around Earth. Most satellites do not move in perfect circles but have elliptical orbits, with Earth at one of the two foci. As a satellite moves in an elliptical orbit, its distance from the object being observed continually changes. Usually, the satellite's sensors receive the best signal when this distance is smallest.
Consider a satellite, regarded as a point, orbiting Earth along an ellipse. In three-dimensional Cartesian coordinates $Oxyz$, with each axis unit equal to $6400$ km, suppose the satellite moves in the coordinate plane $(Oxy)$ along $x^2+3y^2=17$. It must observe an object, also regarded as a point, moving in space. Research indicates that the best observation occurs when the object reaches $A(2;16/\sqrt3;8)$. Determine the coordinates of the point $C$ on the satellite's elliptical orbit for which the distance from $C$ to $A$ is smallest.
Let $n,a,b$ be positive integers satisfying $1\lt n^2\lt a\lt b\lt n^2+n+3$. Find all positive divisors of $ab$ that belong to the interval $(n^2,n^2+n+3)$.
An plays a game in which an ordered triple is written on a board from left to right. Initially the triple is $(1,1,1)$. On each turn, An performs one of these operations on the current triple $(x,y,z)$:
(i) erase it and write $(y,z,x+z)$;
(ii) erase it and write $(x+z+1,x+y+z+1,x+y+2z+1)$.
(a) Prove that An needs exactly four turns to write a triple $(a,b,6)$.
(b) Find the smallest nonnegative integer $k$ for which some play sequence lets An write $(c,d,129)$ after $k$ turns.
Let $ABC$ be an acute triangle with $AB\lt AC$, inscribed in a circle $(O)$. Let $E,F$ be the reflections of $A$ in $B,C$, respectively, and let $M$ be the midpoint of $BC$. On the line $AM$, take points $I,P$ such that triangles $AIE$ and $APF$ are isosceles at $I$ and $P$, respectively. Let $T$ be the intersection of the lines $IE$ and $PF$.
(a) Prove that $T$ belongs to $(O)$.
(b) Let $D=BI\cap CP$, $J=TD\cap IP$, and $X=OT\cap BC$. The lines $AT,AX$ meet $EF$ at $L,U$, respectively. Prove that the four circumcircles of triangles $TAJ,TIP,TEF,TLU$ have a second common point distinct from $T$.
(a) Find the number of distinct positive real roots of $Q(x)=x^3-5x^2+4x-1$.
(b) Suppose $a,b,c,d$ are integers such that $P(x)=ax^3+bx^2+cx+d$ has three distinct positive real roots and no rational root. Find the minimum value of $S=|a|+|b|+|c|+|d|$.
In a game, Mario moves forward by jumps of length $1$ metre, $a$ metres or $b$ metres, where $a,b$ are positive integers with $1\lt a\lt b$. Given a route of length $m$ metres, where $m$ is a positive integer, Mario always uses this strategy: first take as many jumps of length $b$ as possible without passing the destination; if the destination has not been reached, take as many jumps of length $a$ as possible without passing it; finally take jumps of length $1$. Mario wins if there is no way of reaching the destination in fewer jumps than this strategy. Otherwise Mario loses.
(a) Show that for $(a,b)=(6,14)$, there exists an $m$ for which Mario loses.
(b) Find the number of pairs $(a,b)$, with $1\lt a\lt b\le100$, for which Mario always wins with this strategy for every positive integer $m$.
Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $D,E,F$ be the reflections of $H$ in $BC,CA,AB$, respectively, and let $A^{\prime},B^{\prime},C^{\prime}$ be the reflections of $A,B,C$ in those same respective lines. Let $S$ be the circumcenter of $A^{\prime} B^{\prime} C^{\prime}$, and $H^{\prime}$ the orthocenter of $DEF$. Let $J$ be the center of the circle through the three perpendicular projections of $H$ onto $B^{\prime} C^{\prime},C^{\prime} A^{\prime},A^{\prime} B^{\prime}$. Prove that $HJ\parallel H^{\prime} S$.
At an applied-mathematics summer camp there are $8m+1$ boys, where $m>5$, and some girls. Each girl knows exactly three boys, and every two boys have exactly one girl whom they both know. Let $n$ be the largest number of girls that can be selected so that each boy knows at most one selected girl. Prove that $n\ge2m+1$.
Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.
An $n\times n$ board has rows and columns numbered $1$ through $n$; the cell in row $i$, column $j$ is denoted $(i,j)$. A subset $A$ of its cells is called good if, for any two cells $(x_1,y),(x_2,y)$ in $A$, none of the cells $(u,v)$ satisfying
$$x_1\lt u\le x_2,\quad v\lt y,$$
or
$$x_1\le u\lt x_2,\quad v>y,$$
belongs to $A$. Find the smallest number of pairwise disjoint good subsets such that every board cell belongs to exactly one subset.
Let $P(x)=x^4-x^3+x$.
a) Prove that for every positive real $a$, the polynomial $P(x)-a$ has exactly one positive root.
b) Define $a_1=1/3$ and, for $n\ge1$, let $a_{n+1}$ be the positive root of $P(x)-a_n$. Prove that $(a_n)$ has a finite limit and find it.