Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$. [i](Trung Phan, 10 points)[/i]

okay this one is from Prof. Mircea Lascu from Zalau, Romaniaand Prof. V. Cartoaje from Ploiesti, Romania. It goes like this: given being a triangle ABC for every point M inside we construct the points A[size=67]M[/size], B[size=67]M[/size], C[size=67]M[/size] on the circumcircle of the triangle ABC such that A, A[size=67]M[/size], M are collinear and so on. Find the locus of these points M for which the area of the triangle A[size=67]M[/size] B[size=67]M[/size] C[size=67]M[/size] is bigger than the area of the triangle ABC.

We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$

Let $ABC$ be an acute triangle, and let $H$ be its orthocentre. Denote by $H_A$, $H_B$, and $H_C$ the second intersection of the circumcircle with the altitudes from $A$, $B$, and $C$ respectively. Prove that the area of triangle $H_A H_B H_C$ does not exceed the area of triangle $ABC$.

The lateral sidelines $AB$ and $CD$ of trapezoid $ABCD$ meet at point $S$. The bisector of angle $ASC$ meets the bases of the trapezoid at points $K$ and $L$ ($K$ lies inside segment $SL$). Point $X$ is chosen on segment $SK$, and point $Y$ is selected on the extension of $SL$ beyond $L$ such a way that $\angle AXC - \angle AYC = \angle ASC$. Prove that $\angle BXD - \angle BYD = \angle BSD$.

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$ [asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); }[/asy]

You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled? [asy] import graph; size(15cm); pen dps = linewidth(1) + fontsize(10); defaultpen(dps); draw((-3,3)--(-3,1)); draw((-3,3)--(2,3)); draw((2,3)--(2,1)); draw((-3,1)--(2,1)); draw((-3,2)--(2,2)); draw((-2,3)--(-2,1)); draw((-1,3)--(-1,1)); draw((0,3)--(0,1)); draw((1,3)--(1,1)); draw((4,3)--(4,2)); draw((4,3)--(5,3)); draw((5,3)--(5,2)); draw((4,2)--(5,2)); draw((5.5,3)--(5.5,1)); draw((5.5,3)--(6.5,3)); draw((6.5,3)--(6.5,1)); draw((5.5,1)--(6.5,1)); draw((7,3)--(7,1)); draw((7,1)--(9,1)); draw((7,3)--(8,3)); draw((8,3)--(8,2)); draw((8,2)--(9,2)); draw((9,2)--(9,1)); draw((11,3)--(11,1)); draw((11,3)--(16,3)); draw((16,3)--(16,1)); draw((11,1)--(16,1)); draw((12,3)--(12,2)); draw((11,2)--(12,2)); draw((12,2)--(13,2)); draw((13,2)--(13,1)); draw((14,3)--(14,1)); draw((14,2)--(15,2)); draw((15,3)--(15,1));[/asy]

Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

Let $ABC$ be a triangle with $AB = AC$ with centroid $G$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$ respectively and $O$ be the circumcenter of triangle $BCN$ . Prove that $MBOG$ is a cyclic quadrilateral .

A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron.

Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.

Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above. (I. Voronovich)

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

Given a circle centered at $ O $ and two points $ A $ and $ B $ lying on it. $ A $ and $ B $ do not form a diameter. The point $ C $ is chosen on the circle so that the line $ AC $ divides the segment $ OB $ in half. Let lines $ AB $ and $ OC $ intersect at $ D $, and let lines $ BC $ and $ AO $ intersect at $ F $. Prove that $ AF = CD $.

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

Through the midpoint $M$ of the side $BC$ of the triangle $ABC$ passes a line which intersects the rays $AB$ and $AC$ at $D$ and $E$, respectively, such that $AD=AE$. Let $F$ be the foot of the perpendicular from $A$ onto $BC$ and $P{}$ the circumcenter of triangle $ADE$. Prove that $PF=PM$.

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

Let $O$ be the center and $[AB]$ be the diameter of a semicircle. $E$ is a point between $O$ and $B$. The perpendicular to $[AB]$ at $E$ meets the semicircle at $D$. A circle which is internally tangent to the arc $\overarc{BD}$ is also tangent to $[DE]$ and $[EB]$ at $K$ and $C$, respectively. Prove that $\widehat{EDC}=\widehat{BDC}$.

Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP, R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $

A square $ABCD$ of side length $2a$ is given on a plane $\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$ $(a)$ Let $M \in BC$ and $N \in CD$ be two variable points. $i$. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum. $ii$. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values. $(b)$ Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$. $i$. Find the locus of $Q'$. $ii$. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.

Given a convex pentagon $ABCDE$, with $\angle BAC = \angle ABE = \angle DEA - 90^o$, $\angle BCA = \angle ADE$ and also $BC = ED$. Prove that $BCDE$ is parallelogram.

Consider a rectangle $A_1A_2A_3A_4$ and a circle $\mathcal{C}_i$ centered at $A_i$ with radius $r_i$ for $i=1,2,3,4$. Suppose that $r_1+r_3=r_2+r_4<d$, where $d$ is the diagonal of the rectangle. The two pairs of common outer tangents of $\mathcal{C}_1$ and $\mathcal{C}_3$, and of $\mathcal{C}_2$ and $\mathcal{C}_4$ form a quadrangle. Prove that this quadrangle has an inscribed circle.