MathDB

Mathley 2014-15

Part of Mathley Magazine

Subcontests

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organizers want to rearrange 2014 students from same school in circle

There are 20142014 students from high schools nationwide communications sit around a round table in arbitrary manner. Then the organizers want to rearrange students from the same school sit next to each other by performing the following swapping: permutation view of two adjacent groups of students (see illustration). Find the smallest kk number so that a result can be obtained results as desired by the organizers with no more than kk swapping permits. Permission to change places like after ...ABCD⏟1EFG⏟2...→...EFG⏟2ABCD⏟1......\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...
Vu The Khoi, Institute of Mathematics, Vietnam Academy of Science and Technology, Cau Giay, Hanoi.
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x_n/ y_n =\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}, y_n not divisible by 2^n

For every nn positive integers we denote xnyn=∑k=1n1k(nk)\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}} where xn,ynx_n, y_n are coprime positive integers. Prove that yny_n is not divisible by 2n2^n for any positive integers nn.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi

circumcircle of triangle formed by t_Q, t_S, AB is also tangent to (W)

Two circles (U)(U) and (V)(V) intersect at A,BA,B. A line d meets (U),(V)(U), (V) at P,QP, Q and R,SR,S respectively. Let tP,tQ,tR,tSt_P, t_Q, t_R,t_S be the tangents at P,Q,R,SP,Q,R, S of the two circles. Another circle (W)(W) passes through through A,BA, B. Prove that if the circumcircle of triangle that is formed by the intersections of tP,tR,ABt_P,t_R, AB is tangent to (W)(W) then the circumcircle of triangle formed by tQ,tS,ABt_Q, t_S, AB is also tangent to (W)(W).
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
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line through N perpendicular to PC crosses a fixed point as P moves along EF

Let the inscribed circle (I)(I) of the triangle ABCABC, touches CA,ABCA, AB at E,FE, F. PP moves along EFEF, PBPB cuts CACA at M,MIM, MI cuts the line, through CC perpendicular to ACAC, at NN. Prove that the line through NN is perpendicular to PCPC crosses a fixed point as PP moves.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University

there is a circle with center O tangent to all 4 circumcircles , bicentric

A quadrilateral is called bicentric if it has both an incircle and a circumcircle. ABCDABCD is a bicentric quadrilateral with (O)(O) being its circumcircle. Let E,FE, F be the intersections of ABAB and CD,ADCD, AD and BCBC respectively. Prove that there is a circle with center OO tangent to all of the circumcircles of the four triangles EAD,EBC,FAB,FCDEAD, EBC, FAB, FCD.
Nguyen Van Linh, a student of the Vietnamese College, Ha Noi
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u_{n + 2} = u_{n + 1} ++u_ n + [(-1)^n-1]/ 2}

Given the sequence (un)n=1∞(u_n)_{n=1}^{\infty}, where u1=1,u2=2u_1 = 1, u_2 = 2, and un+2=un+1+un+(−1)n−12u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2} for any positive integers nn. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers (un)n=1∞(u_n)_{n=1}^{\infty}
Nguyen Duy Thai Son, The University of Danang, Da Nang.

concurrency wanted, cyclic ABCD, 4 more circles related

A quadrilateral ABCDABCD is inscribed in a circle (O)(O). Another circle (I)(I) is tangent to the diagonals AC,BDAC, BD at M,NM, N respectively. Suppose that MNMN meets AB,CDAB,CD at P,QP, Q respectively. The circumcircle of triangle IMNIMN meets the circumcircles of IAB,ICDIAB, ICD at K,LK, L respectively, which are distinct from II. Prove that the lines PK,QLPK, QL, and OIOI are concurrent.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh

Mathley 2014.3 p5 by Tran Quang Hung

Triangle ABCABC has incircle (I)(I) and P,QP,Q are two points in the plane of the triangle. Let QA,QB,QCQA,QB,QC meet BA,CA,ABBA,CA,AB respectively at D,E,FD,E,F. The tangent at DD, other than BCBC, of the circle (I)(I) meets PAPA at XX. The points YY and ZZ are defined in the same manner. The tangent at XX, other than XDXD, of the circle (I)(I) meets (I) (I) at UU. The points V,WV,W are defined in the same way. Prove that three lines (AU,BV,CW)(AU,BV,CW) are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
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(con)cyclic wanted, 2 circles internally tangent to a third one, tangents

The circles γ\gamma and δ\delta are internally tangent to the circle ω\omega at AA and BB. From AA, draw two tangent lines ℓ1,ℓ2\ell_1, \ell_2 to δ\delta, . From BB draw two tangent lines t1,t2t_1, t_2 to γ\gamma . Let ℓ1\ell_1 intersect t1t_1 at XX and ℓ2\ell_2 intersect t2t_2 at YY . Prove that the quadrilateral AXBYAX BY is cyclic.
Nguyen Van Linh, High School of Natural Sciences, Hanoi National University

(p^{2q}+q^{2p})/(p^3-pq+q^3)= r where p,q,r primes

Find all primes p,q,rp,q, r such that p2q+q2pp3−pq+q3=r\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r.
Titu Andreescu, Mathematics Department, College of Texas, USA
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