2026 Southeast China Mathematical Olympiad

Grade 10 - Day 1

Problem 1

The positive sequence $\{a_{n}\}$ satisfies $a_{1}=1$, and for every positive integer $n$,
$$a_{n}^{2}+a_{n+1}^{2}+a_{n+2}^{2}+a_{n}a_{n+1}a_{n+2}=4.$$

(1) Find the general formula for $\{a_{n}\}$ (expressed in terms of $a_{2}$).
(2) Prove that there exists a positive integer $N$ such that for every integer $n \ge N$,
$$\sum_{i=1}^{n}a_{i}\le n.$$

Problem 2

Find all integers $n > 1$ such that there exists an integer $m$ satisfying
$$n \mid (23p + m^{p^{2026}}),$$
where $p$ is the largest prime factor of $n$.

Answer:

Problem 3

In $\triangle ABC$, $AB < AC$. Let $M$ be the midpoint of side $BC$, and let $D$ be the intersection of the angle bisector of $\angle BAC$ with $BC$. Let $P$ be the reflection of $D$ over $A$, and let $Q$ be a point on the circumcircle of $\triangle PBC$ such that $MQ \parallel AD$, with $P$ and $Q$ on opposite sides of $BC$. Let $R$ be the intersection of $PQ$ and $AM$. Prove that
$$\angle RBQ = \angle RCQ.$$

Problem 4

Given an integer $n \ge 4$. Let
$$S_{n} = \{(a,b,c,d) \mid a,b,c,d \in \mathbb{N}_{+}, 1 \le a < b < c < d \le n\}.$$
Find the smallest positive integer $m$ such that for any $m$ distinct elements of $S_{n}$,
$$(a_{1},b_{1},c_{1},d_{1}), (a_{2},b_{2},c_{2},d_{2}), \dots, (a_{m},b_{m},c_{m},d_{m});$$
and for any $n$ distinct points $P_{1}, P_{2}, \dots, P_{n}$ in the plane, if for every $1 \le i \le m$ the four points $P_{a_{i}}, P_{b_{i}}, P_{c_{i}}, P_{d_{i}}$ are concyclic, then $P_{1}, P_{2}, \dots, P_{n}$ are all concyclic.

Grade 10 - Day 2

Problem 5

Let $ABCD$ be a convex quadrilateral, and let $E, F, G, H$ be points moving in the interiors (not including endpoints) of segments $AB, BC, CD, DA$ respectively. Let $l$ denote the perimeter of quadrilateral $EFGH$.

(1) If $l$ attains a minimum value, prove that $A, B, C, D$ are concyclic.
(2) Suppose $ABCD$ is inscribed in a circle of radius $R$. Prove that
$$l \cdot R \ge 2\sqrt{AB \cdot BC \cdot CD \cdot DA}.$$

Problem 6

The real sequence $\{a_{n}\}$ satisfies: for all positive integers $i, j$, $a_{i+j} \ge a_{i} + a_{j}$. Prove that for every positive integer $n$,
$$\frac{2}{n+1}\sum_{i=1}^{n}a_{i} \ge \sum_{1\le i\le j\le n}\frac{a_{i}}{i\cdot j}.$$

Problem 7

Let
$$M = \{(a_{1},a_{2},\dots,a_{12}) \mid a_{i}\in\mathbb{N}_{+}, 1 \le a_{1} < a_{2} < \dots < a_{12} \le 28\}.$$
For a positive integer $m$ and an array $A = (a_{1}, a_{2}, \dots, a_{12}) \in M$, let $N_{A}(m)$ denote the number of elements in the set
$$\{(i,j) \mid 1 \le i < j \le 12, a_{j} - a_{i} = m\}.$$
Find
$$\min_{A\in M} \max_{1\le m\le 27} N_{A}(m).$$

Problem 8

Prove that there exist infinitely many quintuples of positive integers $(a, b, c, d, e)$ with $\gcd(a, b) = 1$, satisfying

Grade 11 - Day 1

Problem 1

If real numbers $x_{1}, x_{2}, x_{3}$ satisfy $0 \le x_{1} \le x_{2} \le x_{3}$ and $x_{1} + x_{2} + x_{3} = 1$, call $(x_{1}, x_{2}, x_{3})$ a “fusion triple.” Find the smallest real number $c$ such that for every fusion triple $(x_{1}, x_{2}, x_{3})$,
$$\sum_{i\le s-c}x_{i}\le\frac{1}{2},$$
where $s = x_{1} + 2x_{2} + 3x_{3}$.

Problem 2

Find all integer triples $(x, y, z)$ with $1 < x \le y \le z$ satisfying
$$x^{2} + y^{2} + (x+y)(z-2) = xyz.$$

Problem 3

In acute triangle $\triangle ABC$, $AB > BC > AC$. Let $O$ be the circumcenter and $H$ the orthocenter. Line $AO$ meets lines $BH$ and $CH$ at points $D$ and $E$ respectively; let $O_{1}$ be the circumcenter of $\triangle DEH$. A point $T$ on segment $AO_{1}$ satisfies that $B, T, H, C$ are concyclic. A point $S$ on the extension of $BC$ satisfies $SA = ST$. Prove that
$$\angle STH = \angle OAO_{1}.$$

Problem 4

Given positive integers $n, k$ with $n > 2k \ge 6$. Let $G$ be a simple connected graph on $n$ vertices satisfying:

(i) $G$ remains connected after deleting any single vertex together with all edges incident to it;
(ii) There exist $k$ vertices $v_{1}, v_{2}, \dots, v_{k}$ of $G$ such that for any $1 \le i < j \le k$, deleting $v_{i}, v_{j}$ together with all edges incident to at least one of $v_{i}, v_{j}$ disconnects $G$.

Find the maximum possible number of edges of $G$.

Grade 11 - Day 2

Problem 5

For a positive integer $m$, let $S(m)$ denote the sum of the decimal digits of $m$. Does there exist a positive integer $n$ such that
$$S(23n) = S(n) \cdot S(n + 2026)?$$

Problem 6

The Ming-dynasty poet Ye Xianggao wrote in Ascending Shizhu Rock: “the crags of bamboo-rock pierce the blue sky.”

Given an integer $n \ge 2$. A “stone-bamboo” with $k \, (k \ge 2)$ internodes is called fillable if the numbers $1, 2, 3, \dots, 2^{n}$ can be placed without repetition into the $k$ internodes so that every internode is nonempty, and the sum of the numbers in each internode divides the sum of the numbers in the next internode. For example, when $n = 3$, the arrangement $4 \mid 3,6,7 \mid 1,2,5,8$ shows that a stone-bamboo with 3 internodes is fillable.

Find the maximum number of internodes of a fillable stone-bamboo, in terms of $n$.

Problem 7

Let $n \ge 3$ be an integer, and let $a_{1}, a_{2}, \dots, a_{n}$ be positive reals with $a_{1}a_{2}\cdots a_{n} \le 1$. Prove that
$$\sum_{i=1}^{n}\frac{a_{i}}{1+a_{i}^{2}} \le \frac{n}{2} - \frac{1}{2}\left(1 - \sqrt[n]{a_{1}a_{2}\cdots a_{n}}\right)^{2}.$$

Problem 8

As shown in the figure, in the crossed quadrilateral $ABCD$, sides $BC$ and $DA$ intersect. Circle $\omega$ is tangent to the extensions of $AB$ and $CD$ and to sides $BC$ and $DA$. A circle $I$ passing through $A$ and $C$ intersects $\omega$, with $B, D$ lying inside $I$; let $l_{1}, l_{2}$ be the external common tangents of $I$ and $\omega$. Circle $\alpha$ is tangent to $I$, to the extension of $CB$, and to side $AB$; circle $\beta$ is tangent to $I$, to the extension of $AD$, and to side $CD$. Prove that there exists a circle $\Gamma'$, distinct from $I$, that is tangent to lines $l_{1}, l_{2}$ and to circles $\alpha, \beta$.