[b]p1.[/b] Consider a normal $8 \times 8$ chessboard, where each square is labelled with either $1$ or $-1$. Let $a_k$ be the product of the numbers in the $k$th row, and let $b_k$ be the product of the numbers in the $k$th column. Find, with proof, all possible values of $\sum^8_{k=1}(a_kb_k)$. [b]p2.[/b] Let $\overline{AB}$ be a line segment with $AB = 1$, and $P$ be a point on $\overline{AB}$ with $AP = x$, for some $0 < x < 1$. Draw circles $C_1$ and $C_2$ with $\overline{AP}$, $\overline{PB}$ as diameters, respectively. Let $\overline{AB_1}$, $\overline{AB_2}$ be tangent to $C_2$ at $B_1$ and $B_2$, and let $\overline{BA_1}$;$\overline{BA_2}$ be tangent to $C_1$ at $A_1$ and $A_2$. Now $C_3$ is a circle tangent to $C_2$, $\overline{AB_1}$, and $\overline{AB_2}$; $C_4$ is a circle tangent to $C_1$, $\overline{BA_1}$, and $\overline{BA_2}$. (a) Express the radius of $C_3$ as a function of $x$. (b) Prove that $C_3$ and $C_4$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/a/fd28ad91ed0a4893608b92f5ccbd01088ae424.png[/img] [b]p3.[/b] Suppose that the graphs of $y = (x + a)^2$ and $x = (y + a)^2$ are tangent to one another at a point on the line $y = x$. Find all possible values of $a$. [b]p4.[/b] You may assume without proof or justification that the infinite radical expressions $\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a-...}}}}$ and $\sqrt{a-\sqrt{a+\sqrt{a-\sqrt{a+...}}}}$ represent unique values for $a > 2$. (a) Find a real number $a$ such that $$\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a+...}}}}= 2017$$ (b) Show that $$\sqrt{2018-\sqrt{2018+\sqrt{2018-\sqrt{2018+...}}}}=\sqrt{2017-\sqrt{2017-\sqrt{2017-\sqrt{2017-...}}}}$$ [b]p5.[/b] (a) Suppose that $m, n$ are positive integers such that $7n^2 - m^2 > 0$. Prove that, in fact, $7n^2 - m^2 \ge 3$. (b) Suppose that $m, n$ are positive integers such that $\frac{m}{n} <\sqrt7$. Prove that, in fact, $\frac{m}{n}+\frac{1}{mn} <\sqrt7$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We say that a rational number is [i]spheric[/i] if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that: [b]a)[/b] $ 7 $ is not spheric. [b]b)[/b] a rational spheric number raised to the power of any natural number greater than $ 1 $ is spheric.

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.

A convex hexagon $A_1B_1A_2B_2A_3B_3$ it is inscribed in a circumference $\Omega$ with radius $R$. The diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concurrent in $X$. For each $i=1,2,3$ let $\omega_i$ tangent to the segments $XA_i$ and $XB_i$ and tangent to the arc $A_iB_i$ of $\Omega$ that does not contain the other vertices of the hexagon; let $r_i$ the radius of $\omega_i$. $(a)$ Prove that $R\geq r_1+r_2+r_3$ $(b)$ If $R= r_1+r_2+r_3$, prove that the six points of tangency of the circumferences $\omega_i$ with the diagonals $A_1B_2$, $A_2B_3$, $A_3B_1$ are concyclic

A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden? $ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $

Georg has a set of sticks. From these sticks he must create a closed figure with the property that each stick makes right angles with its neighbouring sticks. All the sticks must be used. If the sticks have the lengths $1, 1, 2, 2, 2, 3, 3$ and $4$, the figure might for example look like this: [img]https://cdn.artofproblemsolving.com/attachments/9/7/c16a3143a52ec6f442208c63b41f2df1ae735c.png[/img] (a) Prove that he can create such a figure if the sticks have the lengths $1, 1, 1, 2, 2, 3, 4$ and $4$. (b) Prove that it cannot be done if the sticks have the lengths $1, 2, 2, 3, 3, 3, 4, 4$ and $4$. (c) Determine whether it is doable if the sticks have the lengths $1, 2, 2, 2, 3, 3, 3, 4, 4$ and $5$.

Five points $K_i, i = 1, 2, 3, 4$ and $P$ are chosen arbitrarily on the same circle. Denote by $P(i, j)$ the distance from $P$ to the line passing through $K_i$ and $K_j$ . Prove that $$P(1, 2)P(3, 4) = P(1, 4)P(2, 3) = P(1, 3)P(2, 4)$$ Bùi Quang Tuấn

The figure on the right shows the ellipse $\frac{(x-19)^2}{19}+\frac{(x-98)^2}{98}=1998$. Let $R_1,R_2,R_3,$ and $R_4$ denote those areas within the ellipse that are in the first, second, third, and fourth quadrants, respectively. Determine the value of $R_1-R_2+R_3-R_4$. [asy] defaultpen(linewidth(0.7)); pair c=(19,98); real dist = 30; real a = sqrt(1998*19),b=sqrt(1998*98); xaxis("x",c.x-a-dist,c.x+a+3*dist,EndArrow); yaxis("y",c.y-b-dist*2,c.y+b+3*dist,EndArrow); draw(ellipse(c,a,b)); label("$R_1$",(100,200)); label("$R_2$",(-80,200)); label("$R_3$",(-60,-150)); label("$R_4$",(70,-150));[/asy]

Points $E$ and $F$ lie inside a square $ABCD$ such that the two triangles $ABF$ and $BCE$ are equilateral. Show that $DEF$ is an equilateral triangle.

Is there a heptagon and a point $P$ inside it such that any vertex of the heptagon has its distance from $P$ equal to the length of the side opposite the vertex? [i]A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).[/i]

Two families of parallel lines are given in the plane, consisting of $15$ and $11$ lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let $V$ be the set of $165$ intersection points of the lines under consideration. Show that there exist not fewer than $1986$ distinct squares with vertices in the set $V .$

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle $ABC$) $F$ is marked in triangle $ABC$. $A_1$ is on the side $BC$ such that $AA_1$ is the altitude of triangle $ABC$. Prove that the line symmetric to $FA_1$ with respect to $BC$ is perpendicular to $IO$, where $O$ is the center of the circumcircle of the triangle $ABC$ and $I$ is the center of its incircle.

It is known that for any point $P$ on the circumcircle of a triangle $ABC$, the orthogonal projections of $P$ onto $AB,BC,CA$ lie on a line, called a [i]Simson line[/i] of $P$. Show that the Simson lines of two diametrically opposite points $P_1$ and $P_2$ are perpendicular.

In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.

Outside of the plane of the triangle $ABC$ is given point $D$. (a) prove that if the segment $DA$ is perpendicular to the plane $ABC$ then orthogonal projection of the orthocenter of the triangle $ABC$ on the plane $BCD$ coincides with the orthocenter of the triangle $BCD$. (b) for all tetrahedrons $ABCD$ with base, the triangle $ABC$ with smallest of the four heights that from the vertex $D$, find the locus of the foot of that height.

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected. Prove that $O_1O_2O_3$ is an acute-angled one.

Let $BF$ and $CN$ be the altitudes of the acute triangle $ABC$. Bisectors the angles $ACN$ and $ABF$ intersect at the point $T$. Find the radius of the circle circumscribed around the triangle $FTN$, if it is known that $BC = a$. (Grigory Filippovsky)

Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$. Prove, that $\angle DEB=\angle DAC$

There are $ n$ points in $ \mathbb R^3$ such that every three form an acute angled triangle. Find maximum of $ n$.

Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. (Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.) [i]Proposed by Sammy Luo[/i]

Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$, and let $M$ be a midpoint of the side $BC$. Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$. Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$. $\omega$, the circumcircle of triangle $ADE$, meets $EM$ again at $F$. Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$.

Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$?