Which number is greater, $\frac{1996^{1995}+1}{1996^{1996}+1}$ or $ \frac{1996^{1996}+1}{1996^{1997}+1}$ ?

A conical surface $C$ is cut by a plane $T$ as shown in the figure on the back of this sheet. Show that $C \cap T$ is an ellipse. You can use as an aid the fact that if you consider the two spheres tangent to $C$ and $T$ as shown in the figure, they intersect $T$ in the bulbs. [asy] // calculate intersection of line and plane // p = point on line // d = direction of line // q = point in plane // n = normal to plane triple lineintersectplan(triple p, triple d, triple q, triple n) { return (p + dot(n,q - p)/dot(n,d)*d); } // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C) { return lineintersectplan(B, B - C, A, B - C); } // calculate area of space triangle with vertices A, B, and C real trianglearea(triple A, triple B, triple C) { return abs(cross(A - C, B - C)/2); } // calculate incentre of space triangle ABC triple triangleincentre(triple A, triple B, triple C) { return (abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(abs(B - C) + abs(C - A) + abs(A - B)); } // calculate inradius of space triangle ABC real triangleinradius(triple A, triple B, triple C) { return 2*trianglearea(A,B,C)/(abs(B - C) + abs(C - A) + abs(A - B)); } // calculate excentre of space triangle ABC triple triangleexcentre(triple A, triple B, triple C) { return (-abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(-abs(B - C) + abs(C - A) + abs(A - B)); } // calculate exradius of space triangle ABC real triangleexradius(triple A, triple B, triple C) { return 2*trianglearea(A,B,C)/(-abs(B - C) + abs(C - A) + abs(A - B)); } unitsize(2 cm); pair project (triple A, real t) { return((A.x, A.y*Sin(t) + A.z*Cos(t))); } real alpha, beta, theta, t; real coneradius = 1, coneheight = 3; real a, b, c; real[] m, r; triple A, B, V; triple ellipsecenter, ellipsex, ellipsey; triple[] F, O, P, R, W; path[] ellipse, spherering; theta = 15; V = (0,0,-coneheight); m[1] = sqrt(Cos(theta)^2*coneheight^2 - Sin(theta)^2*coneradius^2)/coneradius; m[2] = -m[1]; alpha = -aTan(Sin(theta)/m[1]); beta = -aTan(Sin(theta)/m[2]) + 180; A = (coneradius*Cos(alpha), coneradius*Sin(alpha), 0); B = (coneradius*Cos(beta), coneradius*Sin(beta), 0); W[1] = interp(V,(coneradius,0,0),0.6); W[2] = interp(V,(-coneradius,0,0),0.4); O[1] = triangleexcentre(V,W[1],W[2]); O[2] = triangleincentre(V,W[1],W[2]); r[1] = triangleexradius(V,W[1],W[2]); r[2] = triangleinradius(V,W[1],W[2]); F[1] = projectionofpointontoline(O[1],W[1],W[2]); F[2] = projectionofpointontoline(O[2],W[1],W[2]); P[1] = O[1] - (0,0,r[1]*coneradius/sqrt(coneradius^2 + coneheight^2)); P[2] = O[2] - (0,0,r[2]*coneradius/sqrt(coneradius^2 + coneheight^2)); spherering[11] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta); spherering[12] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360); spherering[21] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta); spherering[22] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360); ellipsecenter = (W[1] + W[2])/2; a = abs(W[1] - ellipsecenter); c = abs(F[1] - ellipsecenter); b = sqrt(a^2 - c^2); ellipsex = (W[1] - W[2])/abs(W[1] - W[2]); ellipsey = (0,1,0); ellipse[1] = project(ellipsecenter + a*ellipsex, theta); for (t = 0; t <= 180; t = t + 5) { ellipse[1] = ellipse[1]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta); } ellipse[2] = project(ellipsecenter - a*ellipsex, theta); for (t = 180; t <= 360; t = t + 5) { ellipse[2] = ellipse[2]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta); } R[1] = ellipsecenter + 1*ellipsex + ellipsey; R[2] = ellipsecenter - 1.2*ellipsex + ellipsey; R[3] = ellipsecenter - 1*ellipsex - ellipsey; R[4] = ellipsecenter + 1.2*ellipsex - ellipsey; fill(ellipse[1]--ellipse[2]--cycle, gray(0.9)); draw(yscale(Sin(theta))*Circle((0,0),coneradius)); draw(project(V,theta)--project(A,theta)); draw(project(V,theta)--project(B,theta)); draw(Circle(project(O[1],theta),r[1])); draw(Circle(project(O[2],theta),r[2])); draw(spherering[11], dashed); draw(spherering[12]); draw(spherering[21], dashed); draw(spherering[22]); draw(ellipse[1], dashed); draw(ellipse[2]); draw(project(R[1],theta)--interp(project(R[1],theta),project(R[2],theta),0.13)); draw(interp(project(R[1],theta),project(R[2],theta),0.13)--interp(project(R[1],theta),project(R[2],theta),0.76), dashed); draw(interp(project(R[1],theta),project(R[2],theta),0.76)--project(R[2],theta)); draw(project(R[2],theta)--project(R[3],theta)--project(R[4],theta)--project(R[1],theta)); label("$C$", (-1,0.3)); label("$T$", (1.2,-0.8)); dot(project(F[1],theta)); dot(project(F[2],theta)); //dot("$F_1$", project(F[1],theta)); //dot("$F_2$", project(F[2],theta)); //dot("$O_1$", project(O[1],theta)); //dot("$O_2$", project(O[2],theta)); //dot("$P_1$", project(P[1],theta)); //dot("$V$", project(V,theta)); //dot("$W_1$", project(W[1],theta)); //dot("$W_2$", project(W[2],theta)); [/asy]

[asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((7,0),1),white); fill(circle((3,0),3),white); draw((0,0)--(8,0),black+linewidth(1)); draw((6,0)--(6,sqrt(12)),black+linewidth(1)); MP("A", (0,0), W); MP("B", (8,0), E); MP("C", (6,0), S); MP("D",(6,sqrt(12)), N); [/asy] In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is: $\textbf{(A)}\ 1:2\qquad \textbf{(B)}\ 1:3\qquad \textbf{(C)}\ \sqrt{3}:7\qquad \textbf{(D)}\ 1:4\qquad \textbf{(E)}\ \sqrt{2}:6$

Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.

Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \] we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. [i]Radu Gologan[/i]

Let circle $A$ have radius $9,$ and let circle $B$ have radius $5$ and be internally tangent to circle $A.$ The largest radius $r$ such that there are two circles with radius $r$ that lie inside circle $A,$ are externally tangent to each other, and externally tangent with circle $B$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively and let $I$ be the center of that circle. Furthermore, let $P$ be the foot of the perpendicular from point $I$ to line $AD$ and let $M$ be the midpoint of $DE$. If $N$ is the intersection point of $PM$ and $AC$, prove that $DN \parallel EF$.

Let $ ABC$ be a triangle, line $ l$ cuts its sides $ BC,CA,AB$ at $ D,E,F$, respectively. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ AEF,BFD,CDE$, respectively. Prove that the orthocenter of triangle $ O_{1}O_{2}O_{3}$ lies on line $ l$.

Four dice are thrown. What is the probability that the product of the number equals $ 36?$

Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$

A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.

Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)

There is a set of $50$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $25$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be? [i]K.A.Sukhov[/i]

Given that $0 < a, b, c, d < 1$ and $abcd = (1 - a)(1 - b)(1 - c)(1 - d)$, prove that $(a + b + c + d) -(a + c)(b + d) \ge 1$

Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy \[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]

Let $ S$ be any point on the circumscribed circle of $ PQR.$ Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by $ l(S, PQR).$ Suppose that the hexagon $ ABCDEF$ is inscribed in a circle. Show that the four lines $ l(A,BDF),$ $ l(B,ACE),$ $ l(D,ABF),$ and $ l(E,ABC)$ intersect at one point if and only if $ CDEF$ is a rectangle.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.

Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

Define a sequence $a_n$ such that $a_n = a_{n-1} - a_{n-2}$. Let $a_1 = 6$ and $a_2 = 5$. Find $\Sigma_{n=1}^{1000}a_n$.

Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

Ana chose some unit squares of a $50 \times 50$ board and placed a chip on each of them. Prove that Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.