2022 China Team Selection Test

Day 1 - Test 1

1

In a cyclic convex hexagon ABCDEF, AB and DC intersect at G, AF and DE intersect at H. Let M,N be the circumcenters of BCG and EFH, respectively. Prove that the BE, CF and MN are concurrent.

2

Let p be a prime, A is an infinite set of integers. Prove that there is a subset B of A with 2p2 elements, such that the arithmetic mean of any pairwise distinct p elements in B does not belong to A.

3

Let a,b,c,p,q,r be positive integers with p,q,r2. Denote Q={(x,y,z)Z3:0xa,0yb,0zc}.Initially, some pieces are put on the each point in Q, with a total of M pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove p pieces on (x,y,z) and place a piece on (x1,y,z) ; (2) Remove q pieces on (x,y,z) and place a piece on (x,y1,z) ; (3) Remove r pieces on (x,y,z) and place a piece on (x,y,z1). Find the smallest positive integer M such that one can always perform a sequence of operations, making a piece placed on (0,0,0), no matter how the pieces are distributed initially.

Day 2 - Test 1

4

Let ABC be an acute triangle with ACB>2ABC. Let I be the incenter of ABC, K is the reflection of I in line BC. Let line BA and KC intersect at D. The line through B parallel to CI intersects the minor arc BC on the circumcircle of ABC at E(EB). The line through A parallel to BC intersects the line BE at F. Prove that if BF=CE, then FK=AD.

5

Let C={zC:|z|=1} be the unit circle on the complex plane. Let z1,z2,,z240C (not necessarily different) be 240 complex numbers, satisfying the following two conditions: (1) For any open arc Γ of length π on C, there are at most 200 of j (1j240) such that zjΓ. (2) For any open arc γ of length π/3 on C, there are at most 120 of j (1j240) such that zjγ. Find the maximum of |z1+z2++z240|.

6

Let m be a positive integer, and A1,A2,,Am (not necessarily different) be m subsets of a finite set A. It is known that for any nonempty subset I of {1,2,m}, |iIAi||I|+1.Show that the elements of A can be colored black and white, so that each of A1,A2,,Am contains both black and white elements.

Day 1 - Test 2

1

Find all pairs of positive integers (m,n), such that in a m×n table (with m+1 horizontal lines and n+1 vertical lines), a diagonal can be drawn in some unit squares (some unit squares may have no diagonals drawn, but two diagonals cannot be both drawn in a unit square), so that the obtained graph has an Eulerian cycle.

2

Given a non-right triangle ABC with BC>AC>AB. Two points P1P2 on the plane satisfy that, for i=1,2, if APi,BPi and CPi intersect the circumcircle of the triangle ABC at Di,Ei, and Fi, respectively, then DiEiDiFi and DiEi=DiFi0. Let the line P1P2 intersects the circumcircle of ABC at Q1 and Q2. The Simson lines of Q1, Q2 with respect to ABC intersect at W. Prove that W lies on the nine-point circle of ABC.

3

Let a1,a2,,an be n positive integers that are not divisible by each other, i.e. for any ij, ai is not divisible by aj. Show that a1+a2++an1.1n22n. Note: A proof of the inequality when n is sufficient large will be awarded points depending on your results.

Day 2 - Test 2

4

Given a positive integer n, find all n-tuples of real number (x1,x2,,xn) such that f(x1,x2,,xn)=2k1=02k2=02kn=0|k1x1+k2x2++knxn1|attains its minimum.

5

Given a positive integer n, let D is the set of positive divisors of n, and let f:DZ be a function. Prove that the following are equivalent: (a) For any positive divisor m of n, n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. (b) For any positive divisor k of n, k ~\Big|~ \sum_{d|k} f(d).

6

Let m,n be two positive integers with m \ge n \ge 2022. Let a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n be 2n real numbers. Prove that the numbers of ordered pairs (i,j) ~(1 \le i,j \le n) such that |a_i+b_j-ij| \le m does not exceed 3n\sqrt{m \log n}.

Day 1 - Test 3

1

Given two circles \omega_1 and \omega_2 where \omega_2 is inside \omega_1. Show that there exists a point P such that for any line \ell not passing through P, if \ell intersects circle \omega_1 at A,B and \ell intersects circle \omega_2 at C,D, where A,C,D,B lie on \ell in this order, then \angle APC=\angle BPD.

2

Two positive real numbers \alpha, \beta satisfies that for any positive integers k_1,k_2, it holds that \lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor, where \lfloor x \rfloor denotes the largest integer less than or equal to x. Prove that there exist positive integers m_1,m_2 such that \frac{m_1}{\alpha}+\frac{m_2}{\beta}=1.

3

Given a positive integer n \ge 2. Find all n-tuples of positive integers (a_1,a_2,\ldots,a_n), such that 1<a_1 \le a_2 \le a_3 \le \cdots \le a_n, a_1 is odd, and (1) M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n is a positive integer; (2) One can pick n-tuples of integers (k_{i,1},k_{i,2},\ldots,k_{i,n}) for i=1,2,\ldots,M such that for any 1 \le i_1 <i_2 \le M, there exists j \in \{1,2,\ldots,n\} such that k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}.

Day 2 - Test 3

4

Find all positive integer k such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length k (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.

5

Show that there exist constants c and \alpha > \frac{1}{2}, such that for any positive integer n, there is a subset A of \{1,2,\ldots,n\} with cardinality |A| \ge c \cdot n^\alpha, and for any x,y \in A with x \neq y, the difference x-y is not a perfect square.

6

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of z^{20}+63z+22=0 is greater than \pi. (2) Let a_1,a_2,\ldots,a_n be complex numbers with sum 1, and k_1<k_2<\cdots<k_n be odd positive integers. Let \omega be a complex number with norm at least 1. Prove that the equation a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w has at least one complex root with norm at most 3n|\omega|.

Day 1 - Test 4

1

Initially, each unit square of an n \times n grid is colored red, yellow or blue. In each round, perform the following operation for every unit square simultaneously: For a red square, if there is a yellow square that has a common edge with it, then color it yellow. For a yellow square, if there is a blue square that has a common edge with it, then color it blue. For a blue square, if there is a red square that has a common edge with it, then color it red. It is known that after several rounds, all unit squares of this n \times n grid have the same color. Prove that the grid has became monochromatic no later than the end of the (2n-2)-th round.

2

Let ABCD be a convex quadrilateral, the incenters of \triangle ABC and \triangle ADC are I,J, respectively. It is known that AC,BD,IJ concurrent at a point P. The line perpendicular to BD through P intersects with the outer angle bisector of \angle BAD and the outer angle bisector \angle BCD at E,F, respectively. Show that PE=PF.

3

Find all functions f: \mathbb R \to \mathbb R such that for any x,y \in \mathbb R, the multiset \{(f(xf(y)+1),f(yf(x)-1)\} is identical to the multiset \{xf(f(y))+1,yf(f(x))-1\}. Note: The multiset \{a,b\} is identical to the multiset \{c,d\} if and only if a=c,b=d or a=d,b=c.

Day 2 - Test 4

4

Find all positive integers a,b,c and prime p satisfying that 2^a p^b=(p+2)^c+1.

5

Let n be a positive integer, x_1,x_2,\ldots,x_{2n} be non-negative real numbers with sum 4. Prove that there exist integer p and q, with 0 \le q \le n-1, such that \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, where the indices are take modulo 2n. Note: If q=0, then \sum_{i=1}^q x_{p+2i-1}=0; if q=n-1, then \sum_{i=q+1}^{n-1} x_{p+2i}=0.

6

Given a positive integer n, let D be the set of all positive divisors of n. The subsets A,B of D satisfies that for any a \in A and b \in B, it holds that a \nmid b and b \nmid a. Show that \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}.