Solve the equation $a^3+b^3+c^3=2001$ in positive integers. Mircea Becheanu, Romania

Click for solution $\forall t\in\mathbb N t^3\equiv0,\pm1(mod9).$ $2001\equiv3(mod9).$ Hence, $a=3x+1,b=3y+1,c=3z+1$ for $\{x,y,z\}\subset\mathbb N_0.$ $a^3\leq2001.$ Hence, $x\leq3$ and $y\leq3,z\leq3.$ Let $x\geq y\geq z.$ Then $3a^3\geq2001\Rightarrow x>2.$ Hence, $x=3.$ Hence, $b^3+c^3=1001.$ Hence, $2(3y+1)^3\geq1001\Rightarrow y>2.$ Hence, $2<y\leq3.$ Hence, $y=3 \Rightarrow z=0 \Rightarrow a=10,b=10,c=1.$ Well $\{(10,10,1),(10,1,10),(1,10,10)\}.$

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$. Bulgaria

Click for solution (1) Let $O_A, O_B$ be the circumcenters and $R_A, R_B$ the circumradii of the triangles $\triangle XAC, \triangle XBC$ with the common side XC. It is sufficient to show that the circumradii $R_A \neq R_B$ are not equal. The perpendicular bisectors of the sides CA, CB meet at the circumcenter O of the right angle triangle $\triangle ABC$, i.e., at the midpoint of the hypotenuse AB. The angle $\angle OO_AO_B = 90^\circ$ is right, because $OO_A, OO_B$ are the perpendicular bisectors of the triangle sides CA, CB and $CA \perp CB$. The angle $\angle OO_AO_B = OO_BO_A = 45^\circ$, because the $O_AO_B$ is the perpendicular bisector of the angle bisector segment CX, which forms the angle $45^\circ$ with both CA, CB and $OO_A \perp CA, OO_B \perp CB$. Thus the triangle $\triangle OO_AO_B$ is isosceles and its symmetry axis, the perpendicular bisector of the segment $O_AO_B$ is parallel to the angle bisector CL, identical with the radical axis of the circles $(O_A), (O_B)$. Therefore, these 2 circles do not intersect on the perpendicular bisector of the segment $O_AO_B$, which implies that their radii are different, $R_A \neq R_B$. (2) Let $O_1, O_2$ be the circumcenters and $R_1, R_2$ the circumradii of the triangles $\triangle YAC, \triangle YBC$ with the common side YC. It is sufficient to show that the circumradii $R_1 \neq R_2$ are not equal. The perpendicular bisectors of the sides CA, CB meet at the circumcenter O of the right angle triangle $\triangle ABC$, i.e., at the midpoint of the hypotenuse AB. The angle $\angle OO_1O_2 = 90^\circ$ is right, because $OO_1, OO_2$ are the perpendicular bisectors of the triangle sides CA, CB and $CA \perp CB$. The angles $\angle OO_1O_2 = \angle CBA$ are equal and the angles $\angle OO_2O_1 = \angle CAB$ are equal, because the the perpendicular bisector $O_1O_2$ of the altitude segment CY is parallel to the hypotenuse AB, both being perpendicular to the altitude CH. Thus the triangles $\triangle O_2O_1O \sim \triangle ABC$ are centrally similar, having their corresponding sides parallel. The lines connecting the corresponding vertices C, O pass through their similarity center. Since O is also the midpoint od AB, the line CO intersects the segment $O_1O_2$ at its midpoint M. Since CO is not parallel to the altitude CH and these 2 lines meet at C, the midpoint M does not lie on the altitude CH. The perpendicular bisector of the segment $O_1O_2$ passing through M is parallel to the altitude CH, identical with the radical axis of the circles $(O_1), (O_2)$, both being perpendicular to $AB \parallel O_1O_2$. Therefore, these 2 circles do not intersect on the perpendicular bisector of the segment $O_1O_2$, which implies that their radii are different, $R_1 \neq R_2$.

Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold? Greece

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.