Let $T=\text{TNFTPP}$. Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y|\leq T-500$ and $|y|\leq T-500$. Find the area of region $R$.

Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$. The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$. Let $D$ be the point of tangency of the $A$-excircle of the triangle $ABC$ with the side $BC$. Let $E$ be the reflection of $C$ with respect to the point $D$. Prove that $PE\perp CQ$.

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$ [asy] size(6cm); draw((0,0)--(10,0)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((0,0)--(0,10)); dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0)); label("$D$",(0,8.5),W); label("$A$",(3.5,10),NE); label("$B$",(10,3.5),E); label("$C$",(3.5,0),S); draw((0,8.5)--(3.5,10)); draw((3.5,10)--(10,3.5)); draw((10,3.5)--(3.5,0)); draw((3.5,0)--(0,8.5)); [/asy]

Square corners, $5$ units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is [asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy] $\text{(A)}\ 300 \qquad \text{(B)}\ 500 \qquad \text{(C)}\ 550 \qquad \text{(D)}\ 600 \qquad \text{(E)}\ 1000$

Let $ABC$ be a triangle and let $N$ and $M$ be the midpoints of $AB$ and $CA$, respectively. Let $H$ be the foot of altitude from $A$. The circumcircle of $ABH$ intersects $MN$ at $P$, with $P$ and $M$ on the same side relative to $N$, and the circumcircle of $ACH$ intersects $MN$ at $Q$, with $Q$ and $N$ on the same side relative to $M$. $BP$ and $CQ$ intersect at $X$. Prove that $AX$ is the angle bisector of $\angle CAB$.

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

Triangle $ABC$ has $AB=10$, $BC=17$, and $CA=21$. Point $P$ lies on the circle with diameter $AB$. What is the greatest possible area of $APC$?

Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.

A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

In $\mathbb{R}^3$ some $n$ points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point $P \in \mathbb{R}^3$ in a finite number of steps (possibly depending on $P$). Find the minimal value of $n$ for which this could happen.

A triangle $\vartriangle ABC$ has $AC = 16$ and $BC = 12$. $E$ and $F$ are points on $AC$ and $BC$, respectively, so that $CE = 3CF$. Let $M$ be the midpoint of $AB$, and let lines $EF$ and $CM$ intersect at $G$. Compute the ratio $EG : GF$.

Four lines intersecting at six points form four triangles. Prove that the circles circumscribed around out these triangles have a common point.

We have a convex polygon $P$ in the plane and two points $S,T$ in the boundary of $P$, dividing the perimeter in a proportion $1:2$. Three distinct points in the boundary, denoted by $A,B,C$ start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments $AB, BC, CA$ will have a length smaller or equal than $ST$.

Two triangles $ACE, BDF$ are given which intersect at six points: $G, H, I, J, K, L$ as in the picture. It is known that in each of the quadrilaterals \[ABIK ,BCJL ,CDKG ,DELH ,EFGI\] it is possible to inscribe a circle. Is it possible for the quadrilateral $FAHJ$ is also circumscribed around a circle?

We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10\times 11$ rectangle, we can find a domino that has an inner point above it. How many stories is the lowest [i]stable[/i] tower?

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Show that for every convex polygon whose area is less than or equal to $1$, there exists a parallelogram with area $2$ containing the polygon.

In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.

Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves \[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \] partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)

Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.

A circle $(K)$ is through the vertices $B, C$ of the triangle $ABC$ and intersects its sides $CA, AB$ respectively at $E, F$ distinct from $C, B$. Line segment $BE$ meets $CF$ at $G$. Let $M, N$ be the symmetric points of $A$ about $F, E$ respectively. Let $P, Q$ be the reflections of $C, B$ about $AG$. Prove that the circumcircles of triangles $BPM , CQN$ have radii of the same length. Trần Quang Hùng