and primes 
such that 
(there is at least 1 zero) can't be a perfect square.
such that![\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]](http://latex.artofproblemsolving.com/d/3/5/d355f091797570bdc6c023eedb30e8ee9deac168.png)
Note: 
denotes the bitwise XOR operation. For example, 
.
is divisible by 2024.
is solvable over 
for every 
.
,
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center 
,
.![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
.
be a positive integer and ![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
,
.![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
be an odd number not divisible by 
be a field with 
elements, and 
be the set of elements of 
,
.
has at least 
.
and 
be a continuous function for which there exists an antiderivative 
,
,
,
.
for all 
be a positive integer, ![$g \in \mathbb{R}[X]$](http://latex.artofproblemsolving.com/0/1/e/01e7ebcebcb9f16935716ee859f7acfaa66c29d4.png)
,
be a polynomial with all of its roots being real, and 
for any 
for all 
be two matrices such that 
.
;
,
matrices which has only 
and 
,
:![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
![$f\colon[0,1]\rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/f/d/b/fdb1fb0216ab9d17007b1b7b03d20d1060b88531.png)
be a continuous function. Suppose that for each 
,![\[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]](http://latex.artofproblemsolving.com/4/f/c/4fc6a66d0600a999c000136a26ea23dfcc269cc3.png)
has an unique global minimum point, which we will denote by 
.
,
is constant zero.
Y by Adventure10
Let 
be a triangle with circumcircle 
. Let 
be the midpoints of the arcs 
of containing exactly 2 points of the triangle. Let the reflections of 
over respectively be 
. Let 
meet 
in 
. Let 
. Define 
similarly. Prove that 
concur at a point.
![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15.46cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.38, xmax = 8.08, ymin = -4.74, ymax = 4.46; /* image dimensions */
pen ttffqq = rgb(0.2,1.,0.); pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); pen wwzzff = rgb(0.4,0.6,1.);
filldraw((-0.42,1.42)--(-0.72,-0.1)--(0.84,-0.08)--cycle, ttffqq + opacity(0.1), ttffqq);
filldraw((0.5495813513850182,-2.3473454080314182)--(2.3568579513549004,2.4621606791381163)--(-2.700193434858612,1.5421434410905153)--cycle, wwzzff + opacity(0.1), wwzzff);
/* draw figures */
draw((-0.42,1.42)--(-0.72,-0.1), ttffqq);
draw((-0.72,-0.1)--(0.84,-0.08), ttffqq);
draw((0.84,-0.08)--(-0.42,1.42), ttffqq);
draw(circle((0.05195839675291731,0.5372450532724504), 1.0010000121066647));
draw((0.5495813513850182,-2.3473454080314182)--(2.3568579513549004,2.4621606791381163), wwzzff);
draw((2.3568579513549004,2.4621606791381163)--(-2.700193434858612,1.5421434410905153), wwzzff);
draw((-2.700193434858612,1.5421434410905153)--(0.5495813513850182,-2.3473454080314182), wwzzff);
draw((-2.700193434858612,1.5421434410905153)--(1.4043043264301158,-0.07276532914833184));
draw((-1.3216922153131268,-0.1077140027604247)--(2.3568579513549004,2.4621606791381163));
draw((-1.3216922153131268,-0.1077140027604247)--(1.4043043264301158,-0.07276532914833184));
draw((-0.8112649979811452,1.8857916642632677)--(0.5495813513850182,-2.3473454080314182));
draw((1.2280103450579969,-0.5419170774499965)--(-2.700193434858612,1.5421434410905153));
draw((-0.8112649979811452,1.8857916642632677)--(1.2280103450579969,-0.5419170774499965));
draw((-0.8362044832515221,-0.6887693818077115)--(2.3568579513549004,2.4621606791381163));
draw((-0.3100711868956388,1.9769726530620972)--(0.5495813513850182,-2.3473454080314182));
draw((-0.3075640492964691,0.6007668334291895)--(0.5495813513850182,-2.3473454080314182));
draw((-0.28966430714849944,0.263266665517739)--(2.3568579513549004,2.4621606791381163));
draw((0.04666976516840058,0.18245536445196603)--(-2.700193434858612,1.5421434410905153));
draw((-0.3100711868956388,1.9769726530620972)--(-0.8362044832515221,-0.6887693818077115));
/* dots and labels */
dot((-0.42,1.42),blue);
label("$A$", (-0.56,1.66), NE * labelscalefactor,blue);
dot((-0.72,-0.1),blue);
label("$B$", (-0.62,-0.04), NE * labelscalefactor,blue);
dot((0.84,-0.08),blue);
label("$C$", (1.02,-0.32), NE * labelscalefactor,blue);
dot((0.0647906756925091,-0.46367270401570915),linewidth(3.pt) + uuuuuu);
label("$M_A$", (0.14,-0.34), NE * labelscalefactor,uuuuuu);
dot((0.8184289756774501,1.1810803395690581),linewidth(3.pt) + uuuuuu);
label("$M_B$", (0.98,0.98), NE * labelscalefactor,uuuuuu);
dot((-0.9300967174293059,0.7310717205452576),linewidth(3.pt) + uuuuuu);
label("$M_C$", (-1.14,0.92), NE * labelscalefactor,uuuuuu);
dot((0.5495813513850182,-2.3473454080314182),blue);
label("$D$", (0.7,-2.38), NE * labelscalefactor,blue);
dot((2.3568579513549004,2.4621606791381163),blue);
label("$E$", (2.44,2.66), NE * labelscalefactor,blue);
dot((-2.700193434858612,1.5421434410905153),blue);
label("$F$", (-2.62,1.74), NE * labelscalefactor,blue);
dot((-1.3216922153131268,-0.1077140027604247),linewidth(3.pt) + uuuuuu);
dot((1.4043043264301158,-0.07276532914833184),linewidth(3.pt) + uuuuuu);
dot((-0.3075640492964691,0.6007668334291895),linewidth(3.pt) + uuuuuu);
label("$X$", (-0.34,0.74), NE * labelscalefactor,uuuuuu);
dot((-0.8112649979811452,1.8857916642632677),linewidth(3.pt) + uuuuuu);
dot((1.2280103450579969,-0.5419170774499965),linewidth(3.pt) + uuuuuu);
dot((-0.28966430714849944,0.263266665517739),linewidth(3.pt) + uuuuuu);
label("$Y$", (-0.46,0.02), NE * labelscalefactor,uuuuuu);
dot((-0.3100711868956388,1.9769726530620972),linewidth(3.pt) + uuuuuu);
dot((-0.8362044832515221,-0.6887693818077115),linewidth(3.pt) + uuuuuu);
dot((0.04666976516840058,0.18245536445196603),linewidth(3.pt) + uuuuuu);
label("$Z$", (0.18,0.08), NE * labelscalefactor,uuuuuu);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](//latex.artofproblemsolving.com/f/0/c/f0cdc578926784cce3d30fb6d819adb1018da49b.png)
be a triangle with circumcircle 
. Let 
be the midpoints of the arcs 
of containing exactly 2 points of the triangle. Let the reflections of 
over respectively be 
. Let 
meet 
in 
. Let 
. Define 
similarly. Prove that 
concur at a point.![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(15.46cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.38, xmax = 8.08, ymin = -4.74, ymax = 4.46; /* image dimensions */
pen ttffqq = rgb(0.2,1.,0.); pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); pen wwzzff = rgb(0.4,0.6,1.);
filldraw((-0.42,1.42)--(-0.72,-0.1)--(0.84,-0.08)--cycle, ttffqq + opacity(0.1), ttffqq);
filldraw((0.5495813513850182,-2.3473454080314182)--(2.3568579513549004,2.4621606791381163)--(-2.700193434858612,1.5421434410905153)--cycle, wwzzff + opacity(0.1), wwzzff);
/* draw figures */
draw((-0.42,1.42)--(-0.72,-0.1), ttffqq);
draw((-0.72,-0.1)--(0.84,-0.08), ttffqq);
draw((0.84,-0.08)--(-0.42,1.42), ttffqq);
draw(circle((0.05195839675291731,0.5372450532724504), 1.0010000121066647));
draw((0.5495813513850182,-2.3473454080314182)--(2.3568579513549004,2.4621606791381163), wwzzff);
draw((2.3568579513549004,2.4621606791381163)--(-2.700193434858612,1.5421434410905153), wwzzff);
draw((-2.700193434858612,1.5421434410905153)--(0.5495813513850182,-2.3473454080314182), wwzzff);
draw((-2.700193434858612,1.5421434410905153)--(1.4043043264301158,-0.07276532914833184));
draw((-1.3216922153131268,-0.1077140027604247)--(2.3568579513549004,2.4621606791381163));
draw((-1.3216922153131268,-0.1077140027604247)--(1.4043043264301158,-0.07276532914833184));
draw((-0.8112649979811452,1.8857916642632677)--(0.5495813513850182,-2.3473454080314182));
draw((1.2280103450579969,-0.5419170774499965)--(-2.700193434858612,1.5421434410905153));
draw((-0.8112649979811452,1.8857916642632677)--(1.2280103450579969,-0.5419170774499965));
draw((-0.8362044832515221,-0.6887693818077115)--(2.3568579513549004,2.4621606791381163));
draw((-0.3100711868956388,1.9769726530620972)--(0.5495813513850182,-2.3473454080314182));
draw((-0.3075640492964691,0.6007668334291895)--(0.5495813513850182,-2.3473454080314182));
draw((-0.28966430714849944,0.263266665517739)--(2.3568579513549004,2.4621606791381163));
draw((0.04666976516840058,0.18245536445196603)--(-2.700193434858612,1.5421434410905153));
draw((-0.3100711868956388,1.9769726530620972)--(-0.8362044832515221,-0.6887693818077115));
/* dots and labels */
dot((-0.42,1.42),blue);
label("$A$", (-0.56,1.66), NE * labelscalefactor,blue);
dot((-0.72,-0.1),blue);
label("$B$", (-0.62,-0.04), NE * labelscalefactor,blue);
dot((0.84,-0.08),blue);
label("$C$", (1.02,-0.32), NE * labelscalefactor,blue);
dot((0.0647906756925091,-0.46367270401570915),linewidth(3.pt) + uuuuuu);
label("$M_A$", (0.14,-0.34), NE * labelscalefactor,uuuuuu);
dot((0.8184289756774501,1.1810803395690581),linewidth(3.pt) + uuuuuu);
label("$M_B$", (0.98,0.98), NE * labelscalefactor,uuuuuu);
dot((-0.9300967174293059,0.7310717205452576),linewidth(3.pt) + uuuuuu);
label("$M_C$", (-1.14,0.92), NE * labelscalefactor,uuuuuu);
dot((0.5495813513850182,-2.3473454080314182),blue);
label("$D$", (0.7,-2.38), NE * labelscalefactor,blue);
dot((2.3568579513549004,2.4621606791381163),blue);
label("$E$", (2.44,2.66), NE * labelscalefactor,blue);
dot((-2.700193434858612,1.5421434410905153),blue);
label("$F$", (-2.62,1.74), NE * labelscalefactor,blue);
dot((-1.3216922153131268,-0.1077140027604247),linewidth(3.pt) + uuuuuu);
dot((1.4043043264301158,-0.07276532914833184),linewidth(3.pt) + uuuuuu);
dot((-0.3075640492964691,0.6007668334291895),linewidth(3.pt) + uuuuuu);
label("$X$", (-0.34,0.74), NE * labelscalefactor,uuuuuu);
dot((-0.8112649979811452,1.8857916642632677),linewidth(3.pt) + uuuuuu);
dot((1.2280103450579969,-0.5419170774499965),linewidth(3.pt) + uuuuuu);
dot((-0.28966430714849944,0.263266665517739),linewidth(3.pt) + uuuuuu);
label("$Y$", (-0.46,0.02), NE * labelscalefactor,uuuuuu);
dot((-0.3100711868956388,1.9769726530620972),linewidth(3.pt) + uuuuuu);
dot((-0.8362044832515221,-0.6887693818077115),linewidth(3.pt) + uuuuuu);
dot((0.04666976516840058,0.18245536445196603),linewidth(3.pt) + uuuuuu);
label("$Z$", (0.18,0.08), NE * labelscalefactor,uuuuuu);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/f/0/c/f0cdc578926784cce3d30fb6d819adb1018da49b.png)
be a set contain 
points
with 
lie in this order such that 
lie on a conic
concurrent
,
etc. line on a conic, then it's just Brianchon's theorem on Bradley's theorem.
