Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$. Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved.

Let $n \geq 2$ be an integer and let \[M=\bigg\{\frac{a_1 + a_2 + ... + a_k}{k}: 1 \le k \le n\text{ and }1 \le a_1 < \ldots < a_k \le n\bigg\}\]be the set of the arithmetic means of the elements of all non-empty subsets of $\{1, 2, ..., n\}$. Find \[\min\{|a - b| : a, b \in M\text{ with } a \neq b\}.\]

Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$ and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second time at $Y$, show that $A, Y$, and $M$ are collinear. Proposed by Nikola Velov, North Macedonia

Can every positive rational number $q$ be written as $$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$where $a, b, c, d$ are all positive integers? Proposed by Dominic Yeo, UK

Given is an integer $k\geq 2$. Determine the smallest positive integer $n$, such that, among any $n$ points in the plane, there exist $k$ points among them, such that all distances between them are less than or equal to $2$, or all distances are strictly greater than $1$.

Let $ABC$ be an acute scalene triangle and let $\omega$ be its Euler circle. The tangent $t_A$ of $\omega$ at the foot of the height $A$ of the triangle ABC, intersects the circle of diameter $AB$ at the point $K_A$ for the second time. The line determined by the feet of the heights $A$ and $C$ of the triangle $ABC$ intersects the lines $AK_A$ and $BK_A$ at the points $L_A$ and $M_A$, respectively, and the lines $t_A$ and $CM_A$ intersect at the point $N_A$. Points $K_B, L_B, M_B, N_B$ and $K_C, L_C, M_C, N_C$ are defined similarly for $(B, C, A)$ and $(C, A, B)$ respectively. Show that the lines $L_AN_A, L_BN_B,$ and $L_CN_C$ are concurrent.

Let $ABC$ be an acute triangle and let $B'$ and $C'$ be the feet of the heights $B$ and $C$ of triangle $ABC$ respectively. Let $B_A'$ and $B_C'$ be reflections of $B'$ with respect to the lines $BC$ and $AB$, respectively. The circle $BB_A'B_C'$, centered in $O_B$, intersects the line $AB$ in $X_B$ for the second time. The points $C_A', C_B', O_C, X_C$ are defined analogously, by replacing the pair $(B, B')$ with the pair $(C, C')$. Show that $O_BX_B$ and $O_CX_C$ are parallel.

Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions: For all $i=1,\ldots,n$ we have $a_{ii}=1$ and, For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$ Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$

Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?

Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

Let $ABC$ be a triangle with $AB < AC$ and let $D{}$ be the other intersection point of the angle bisector of $\angle A$ with the circumcircle of the triangle $ABC$. Let $E{}$ and $F{}$ be points on the sides $AB$ and $AC$ respectively, such that $AE = AF$ and let $P{}$ be the point of intersection of $AD$ and $EF$. Let $M{}$ be the midpoint of $BC{}$. Prove that $AM$ and the circumcircles of the triangles $AEF$ and $PMD$ pass through a common point.

Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers suitable if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property. Călin Popescu and Marian Andronache

A finite set $\mathcal{L}$ of coplanar lines, no three of which are concurrent, is called odd if, for every line $\ell$ in $\mathcal{L}$ the total number of lines in $\mathcal{L}$ crossed by $\ell$ is odd. Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. Given a positive integer $n$ determine the smallest nonnegative integer $k$ satisfying the following condition: Every set of $n$ coplanar lines, no three of which are concurrent, extends to an odd set of $n+k$ coplanar lines.

Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$

Let $ABC$ be a triangle and let its incircle $\gamma$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $P$ be a point strictly in the interior of $\gamma.$ The segments $PA,PB,PC$ cross $\gamma$ at $A_0,B_0,C_0$ respectively. Let $S_A,S_B,S_C$ be the centres of the circles $PEF,PFD,PDE$ respectively and let $T_A,T_B,T_C$ be the centres of the circles $PB_0C_0,PC_0A_0,PA_0B_0$ respectively. Prove that $S_AT_A, S_BT_B$ and $S_CT_C$ are concurrent.