[/quote]
,
and 
.
?
be a sequence of positive real numbers that satisfies 
and 
for every natural number 
.
for every natural number 
be positive real numbers such that : 
. Prove that : 
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
(ans- 
)
,
,
with center 
the midpoint of side 
intersects the circle 
and 
.
.
satisfying 
Prove that 
When does equality hold?
where 
intersects 
.
and 
are defined similarly. Line 
intersects 
at points 
,
.
,
.
,
,
,
are defined similarly. Prove that points 
.![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
Y by dantx5, Adventure10, Mango247
Consider three parallelograms 
. Parallelogram 
is inside parallelogram 
, and the vertices of are on the edges of . Parallelogram is inside parallelogram 
, and the vertices of are on the edges of . The sides of are parallel to the sides of . Prove that one side of has length at least half the length of the parallel side of .
. Parallelogram 
is inside parallelogram 
, and the vertices of are on the edges of . Parallelogram is inside parallelogram 
, and the vertices of are on the edges of . The sides of are parallel to the sides of . Prove that one side of has length at least half the length of the parallel side of .![[asy]import graph; size(17cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.5600000000000007,xmax=6.254814814814821,ymin=0.02074074074074210,ymax=3.505185185185189;
pair A=(-0.1200000000000001,1.260000000000001), B=(4.420000000000005,1.160000000000001), C=(0.5600000000000007,2.420000000000003), D=(0.3032114060963623,1.981948869223208), F=(1.840574748298308,2.188810859724446), G=(3.564336998641675,2.150842528219086), H=(3.139425251701695,1.391189140275557), I=(0.4425269887945396,2.219604863237746), J=(0.1367885939036383,1.698051130776796), K=(-0.002526988794538982,1.460395136762259), L=(4.982526988794544,2.119604863237746), M=(4.843211406096366,1.881948869223208), O=(4.071246973990832,1.898952491075752);
draw(A--B); draw(A--C); draw(B--(5.100000000000005,2.320000000000002)); draw(C--(5.100000000000005,2.320000000000002)); draw(D--(3.136902823217370,2.363240025920325)); draw((3.136902823217370,2.363240025920325)--(4.676788593903643,1.598051130776796)); draw(D--(1.843097176782635,1.216759974079679)); draw((1.843097176782635,1.216759974079679)--(4.676788593903643,1.598051130776796)); draw((1.415663001358330,1.429157471780918)--F); draw(F--G); draw(G--H); draw(H--(1.415663001358330,1.429157471780918)); draw((xmin,-0.02202643171806172*xmin+1.988627534555727)--(xmax,-0.02202643171806172*xmax+1.988627534555727)); draw((xmin,-0.02202643171806172*xmin+1.701064095400224)--(xmax,-0.02202643171806172*xmax+1.701064095400224)); draw((xmin,-0.02202643171806172*xmin+2.229352153739828)--(xmax,-0.02202643171806172*xmax+2.229352153739828)); draw((xmin,-0.02202643171806172*xmin+1.460339476216124)--(xmax,-0.02202643171806172*xmax+1.460339476216124));
dot(A,ds); label("$A$",(-0.09777777777777803,1.294814814814817),NE*lsf); dot(B,ds); label("$B$",(4.441481481481486,1.194074074074076),NE*lsf); dot(C,ds); label("$C$",(0.5837037037037041,2.456296296296299),NE*lsf); dot((5.100000000000005,2.320000000000002),ds); label("$C'$",(5.122962962962967,2.355555555555558),NE*lsf); dot(D,ds); label("$D$",(0.3288888888888891,2.017777777777781),NE*lsf); dot((3.136902823217370,2.363240025920325),ds); label("$E$",(3.114074074074077,2.397037037037040),NE*lsf); dot((1.843097176782635,1.216759974079679),ds); label("$A'$",(1.869629629629631,1.253333333333335),NE*lsf); dot((4.676788593903643,1.598051130776796),ds); label("$C''$",(4.702222222222226,1.632592592592595),NE*lsf); dot(F,ds); label("$F$",(1.863703703703705,2.225185185185188),NE*lsf); dot(G,ds); label("$G$",(3.588148148148151,2.183703703703706),NE*lsf); dot((1.415663001358330,1.429157471780918),ds); label("$D'$",(1.437037037037038,1.466666666666669),NE*lsf); dot(H,ds); label("$H$",(3.161481481481484,1.425185185185187),NE*lsf); dot(I,ds); label("$I$",(0.4651851851851854,2.254814814814818),NE*lsf); dot(J,ds); label("$J$",(0.1629629629629630,1.733333333333336),NE*lsf); dot(K,ds); label("$K$",(0.02074074074074059,1.496296296296298),NE*lsf); dot(L,ds); label("$L$",(5.004444444444449,2.154074074074077),NE*lsf); dot(M,ds); label("$M$",(4.868148148148152,1.917037037037040),NE*lsf); dot((4.537473011205465,1.360395136762259),ds); label("$N$",(4.560000000000004,1.395555555555558),NE*lsf); dot(O,ds); label("$O$",(4.097777777777782,1.934814814814817),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]](http://latex.artofproblemsolving.com/3/6/f/36f897541459c2c0bbb2104f29f0b1abd0a651bd.png)
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to both sides of Result 1 gives 
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