MathDB

2018 South East Mathematical Olympiad

Part of South East Mathematical Olympiad

Subcontests

(8)
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Exponent of five in a certain expression

Given a positive integer mm. Let Al=(4l+1)(4l+2)...(4(5m+1)l)A_l = (4l+1)(4l+2)...(4(5^m+1)l) for any positive integer ll. Prove that there exist infinite number of positive integer ll which 55ml∣Al and 55ml+1∤Al5^{5^ml}\mid A_l\text{ and } 5^{5^ml+1}\nmid A_l and find the minimum value of ll satisfying the above condition.

Pvove sum less than 3Cy

Given a positive real C≥1C \geq 1 and a sequence a1,a2,a3,⋯a_1, a_2, a_3, \cdots satisfying for any positive integer n,n, an≥0a_n \geq 0 and for any real x≥1x \geq 1, ∣xlg⁡x−∑k=1[x][xk]ak∣≤Cx,\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx, where [x][x] is defined as the largest integer that does not exceed xx. Prove that for any real y≥1y \geq 1, ∑k=1[y]ak<3Cy.\sum_{k=1}^{[y]}a_k < 3Cy.
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Minimum number of K_3 under some constraints.

There are 2424 participants attended a meeting. Each two of them shook hands once or not. A total of 216216 handshakes occured in the meeting. For any two participants who have shaken hands, at most 1010 among the rest 2222 participants have shaken hands with exactly one of these two persons. Define a friend circle to be a group of 33 participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

Does there exist positive integers m,n such that f(m,n)=2018?

For positive integers m,n,m,n, define f(m,n)f(m,n) as the number of ordered triples (x,y,z)(x,y,z) of integers such that {xyz=x+y+z+m,max⁡{∣x∣,∣y∣,∣z∣}≤n \begin{cases} xyz=x+y+z+m, \\ \max\{|x|,|y|,|z|\} \leq n \end{cases} Does there exist positive integers m,n,m,n, such that f(m,n)=2018?f(m,n)=2018? Please prove your conclusion.
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Another Chinese projective cookie

In the isosceles triangle ABCABC with AB=ACAB=AC, the center of ⊙O\odot O is the midpoint of the side BCBC, and AB,ACAB,AC are tangent to the circle at points E,FE,F respectively. Point GG is on ⊙O\odot O with ∠AGE=90∘\angle AGE = 90^{\circ}. A tangent line of ⊙O\odot O passes through GG, and meets ACAC at KK. Prove that line BKBK bisects EFEF.

&quot;n-interesting&quot; meeting

Assume integer m≥2.m \geq 2. There are 3m3m people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "nn-interesting", only if there exists n(n≤3m−1)n(n\leq 3m-1) people of them, the time everyone of whom shakes hands with other 3m−13m-1 people is exactly 1,2,⋯ ,n,1,2,\cdots,n, respectively. If in any "nn-interesting" meeting, there exists 33 people of them who shake hands with each other, find the minimum value of n.n.
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Rational points in regular pentagon.

In a Cartesian plane, if both horizontal coordinate and vertical coordinate of a point are rational numbers, we call the point rational point. Otherwise, we call it irrational point. Consider an arbitrary regular pentagon on the Cartesian plane. Please compare the number of rational point and the number of irrational point among the five vertices of the pentagon. Prove your conclusion.

Find all initial values which make the sequence bounded.

Suppose that aa is real number. Sequence a1,a2,a3,....a_1,a_2,a_3,.... satisfies a1=a,an+1={an−1an,an≠00,an=0(n=1,2,3,..)a_1=a, a_{n+1} = \begin{cases} a_n - \frac{1}{a_n}, & a_n\ne 0 \\ 0, & a_n=0 \end{cases} (n=1,2,3,..) Find all possible values of aa such that ∣an∣<1|a_n|<1 for all positive integer nn.
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Prove line tangent to circle given segment is bisected

Let OO be the circumcenter of acute △ABC\triangle ABC(AB<ACAB<AC), the angle bisector of ∠BAC\angle BAC meets BCBC at TT and MM is the midpoint of ATAT. Point PP lies inside △ABC\triangle ABC such that PB⊥PCPB\perp PC. D,ED,E distinct from PP lies on the perpendicular to APAP through PP such that BD=BP,CE=CPBD=BP, CE=CP. If AOAO bisects segment DEDE, prove that AOAO is tangent to the circumcircle of △AMP\triangle AMP.

Prove angle relation given parallelogram

Let OO be the circumcenter of △ABC\triangle ABC, where ∠ABC>90∘\angle ABC> 90^{\circ} and MM is the midpoint of BCBC. Point PP lies inside △ABC\triangle ABC such that PB⊥PCPB\perp PC. D,ED,E distinct from PP lies on the perpendicular to APAP through PP such that BD=BP,CE=CPBD=BP, CE=CP. If quadrilateral ADOEADOE is a parallelogram, prove that ∠OPE=∠AMB.\angle OPE = \angle AMB.
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