2026 Western China Mathematics Invitational

This is an edited English translation of the 2026 中国西部数学邀请赛 paper. The original paper consists of four problems on each of two days.

Day 1

Problem 1

Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be positive real numbers such that, for every $1\le i\le n$ ,

\frac{1}{a_i}-\frac{1}{b_i}\le 1.

Prove that

\frac{1}{a_1+a_2+\cdots+a_n} -\frac{1}{b_1+b_2+\cdots+b_n} \le 1.

Problem 2

Let $l$ be the tangent at a point $A$ on a circle $\omega$ . Choose points $A'$ and $A''$ on $l$ so that they are symmetric about $A$ .

A secant through $A'$ meets $\omega$ at $P$ and $Q$ , and a secant through $A''$ meets $\omega$ at $S$ and $T$ . The lines through $A$ parallel to $PQ$ and $ST$ meet $\omega$ again at $X$ and $Y$ , respectively. Let $M$ be the midpoint of $QS$ .

Prove that the three lines $AM$ , $QY$ , and $SX$ are concurrent.

Problem 3

Let $S=\{1,2,\ldots,n\}$ , and let $f:S\to S$ be a bijection with the following property: for all $a,b\in S$ ,

a\mid b \quad\Longrightarrow\quad f(a)\mid f(b).

Prove that if every prime divisor of $k\in S$ is less than $\sqrt n$ , then

f(k)=k.

Problem 4

Let $n$ be a positive even integer. Suppose that real numbers $x_1,x_2,\ldots,x_n$ satisfy

\sum_{i=1}^{n}x_i=0, \qquad \sum_{i=1}^{n}|x_i|=1.

For $1\le i,j\le n$ , define

D_{ij} =\max\{x_i,x_{i+1},\ldots,x_{i+j-1}\} -\min\{x_i,x_{i+1},\ldots,x_{i+j-1}\},

where subscripts are interpreted modulo $n$ . Find the minimum possible value of

\sum_{i=1}^{n}\sum_{j=1}^{n}D_{ij}.

Day 2

Problem 5

Let $H$ be the orthocenter of an acute triangle $ABC$ , and let $BD$ and $CE$ be altitudes, with $D\in AC$ and $E\in AB$ . A point $P$ inside $ABC$ lies on the circumcircle of triangle $BHC$ . The line $BP$ meets $AC$ at $Q$ , and the line $CP$ meets $AB$ at $R$ .

Prove that the line $DE$ passes through the midpoint of $QR$ .

Problem 6

Complex numbers $z_1,z_2,z_3,z_4$ satisfy

z_1+z_2+z_3+z_4=0

and

|z_i-\overline{z_j}|\le 1 \qquad (1\le i,j\le 4).

Find the maximum possible value of

\left|\sum_{1\le i<j\le 4}z_i z_j\right|.

Problem 7

Let $n$ be a positive integer. On an infinite square grid, $n^2$ cells are selected so that no two selected cells share a vertex.

Among the unselected cells, find the minimum possible number of cells that share at least one vertex with a selected cell.

Problem 8

Find the smallest positive integer $m$ for which there exists a set $F$ of $m$ ordered triples $(i,j,k)$ satisfying both of the following conditions:

For every $(i,j,k)\in F$ , the indices $i,j,k$ are pairwise distinct elements of $\{1,2,\ldots,100\}$ .
For any positive integers $a_1,a_2,\ldots,a_{100}$ , if
$a_i\mid a_j+a_k$
for every $(i,j,k)\in F$ , then at least $99$ of the numbers $a_1,a_2,\ldots,a_{100}$ are equal.