Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to $\sqrt2$? (There are no tools, the sheet can only be bent.)

Let $D$ be the foot of the altitude from $A$ in a triangle $ABC$. The angle bisector at $C$ intersects $AB$ at a point $E$. Given that $\angle CEA=\frac\pi4$, compute $\angle EDB$.

Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.

3) $f$ cont diff, $R\rightarrow ]0,+\infty[$, prove $|\int_{0}^{1}f^{3}-{f(0)}^{2}\int_{0}^{1}f| \leq \max_{[0,1]} |f'|(\int_{0}^{1}f)^{2}$

Let $ H$ be an arbitrary subgroup of the diffeomorphism group $ \mathsf{Diff}^\infty(M)$ of a differentiable manifold $ M$. We say that an $ \mathcal C^\infty$-vector field $ X$ is [i]weakly tangent[/i] to the group $ H$, if there exists a positive integer $ k$ and a $ \mathcal C^\infty$-differentiable map $ \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that (i) for fixed $ t_1,\dots,t_k$ the map \[ \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)\] is a diffeomorphism of $ M$, and $ \varphi_{t_1,\dots,t_k}\in H$; (ii) $ \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id}$ whenever $ t_j \equal{} 0$ for some $ 1\leq j\leq k$; (iii) for any $ \mathcal C^\infty$-function $ f: M\to \mathbb R$ \[ X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}.\] Prove, that the commutators of $ \mathcal C^\infty$-vector fields that are weakly tangent to $ H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $ H$.

A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? $\textbf{(A)}R\qquad \textbf{(B)}\frac{R}{2}\qquad \textbf{(C)}\frac{R}{4}\qquad \textbf{(D)}\frac{3R}{8}\qquad \textbf{(E)}\frac{R}{8}$

Suppose $r$ is a real number for which \[ \left\lfloor r + \frac{19}{100} \right\rfloor + \left\lfloor r + \frac{20}{100} \right\rfloor + \left\lfloor r + \frac{21}{100} \right\rfloor + \cdots + \left\lfloor r + \frac{91}{100} \right\rfloor = 546. \] Find $\lfloor 100r \rfloor$. (For real $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

We are given a rectangular piece of white paper with length $25$ and width $20$. On the paper we color blue the interiors of $120$ disjoint squares of side length $1$ (the sides of the squares do not necessarily have to be parallel to the sides of the paper). Prove that we can draw a circle of diameter $1$ on the remaining paper such that the entire interior of the circle is white.

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is: $\text{(A)} \ 3 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \frac{27}{19} \qquad \text{(D)} \ \frac{13}9 \qquad \text{(E)} \ \frac{23}{38}$

Two convex quadrilaterals are called [i]partners[/i] if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent? [center][img]https://cdn.artofproblemsolving.com/attachments/6/e/cc9da12a49043410c50733cb6843e5ec1005d3.jpeg[/img][/center]

Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$. Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers. Dang Hung Thang, University of Natural Sciences, Hanoi National University.

Let $\triangle ABC$ a scalene triangle with incenter $I$, circumcenter $O$ and circumcircle $\Gamma$. The incircle of $\triangle ABC$ is tangent to $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. The line $AI$ meet $EF$ and $\Gamma$ at $N$ and $M\neq A$, respectively. $MD$ meet $\Gamma$ at $L\neq M$ and $IL$ meet $EF$ at $K$. The circumference of diameter $MN$ meet $\Gamma$ at $P\neq M$. Prove that $AK, PN$ and $OI$ are concurrent.

Let $C$ is at a circle. $[AB]$ is a diameter this circle. $D$ is a point at $[AB]$. Perpendicular from $C$ to $[AB]$'s foot on the $[AB]$ is $E$, perpendicular from $A$ to $[CD]$'s foot on the $[CD]$ is $F$. Prove that, $$[DC][FC]=[BD][EA]$$

A triangle $ABC$ with an obtuse angle at the vertex $C$ is inscribed in a circle with a center at point $O$. Circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$, line $BC$ at points $B$ and $B_1$, and the perpendicular bisector of the segment $PC$ at points $D$ and $E$. Prove that points $D$ and $E$ together with the centers of the circumscribed circles of triangles $A_1OC$ and $B_1OC$ lie on one circle.

Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$.

Given are real $y>1$ and positive integer $n \leq y^{50}$ such that all prime divisors of $n$ do not exceed $y$. Prove that $n$ is a product of $99$ positive integer factors (not necessarily primes) not exceeding $y$.

Given are natural numbers $n>m>1$. We draw $m$ numbers from the set $\{1,2,...,n\}$ one by one without putting the drawn numbers back. Find the expected value of the difference between the largest and the smallest of the drawn numbers.

Let $ABC$ be a triangle and $D$ be a point inside triangle $ABC$. $\Gamma$ is the circumcircle of triangle $ABC$, and $DB$, $DC$ meet $\Gamma$ again at $E$, $F$ , respectively. $\Gamma_1$, $\Gamma_2$ are the circumcircles of triangle $ADE$ and $ADF$ respectively. Assume $X$ is on $\Gamma_2$ such that $BX$ is tangent to $\Gamma_2$. Let $BX$ meets $\Gamma$ again at $Z$. Prove that the line $CZ$ is tangent to $\Gamma_1$ . [i]Proposed by HakureiReimu[/i].

Let \( \varphi \) be the Euler's totient function. 1. For any given integer \( a > 1 \), does there exist \( l \in \mathbb{N}_+ \) such that for any \( k \in \mathbb{N}_+ \), \( l \mid k \) and \( a^2 \nmid l \), \( \frac{\varphi(k)}{\varphi(l)} \) is a non-negative power of \( a \)? 2. For integer \( x > a \), are there integers \( k_1 \) and \( k_2 \) satisfying: \[ \varphi(k_i) \in \left ( \frac{x}{a} ,x \right ], i = 1,2; \quad \varphi(k_1) \neq \varphi(k_2). \] And these two different \( k_i \) correspond to the same \( l_1 \) and \( l_2 \) as described in (1), yet \( \varphi(l_1) = \varphi(l_2) \). 3. Define \( \#E \) as the number of elements in set \( E \). For integer \( x > a \), let \( V(x) = \#\{v \in \mathbb{N}_+ \mid v = \varphi(k) \leq x\} \) and \( W(x) = \#\{w \in \mathbb{N}_+ \mid w = \varphi(l) \leq x, a^2 \mid l\} \). Compare \( V\left( \frac{x}{a} \right) \) with \( W(x) \).

The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.

A sequence of reals $a_1, a_2, \cdots$ satisfies for all $m>1$, $$a_{m+1}a_{m-1}=a_m^2-a_1^2$$ Prove that for all $m>n>1$, the sequence satisfies the equation $$a_{m+n}a_{m-n}=a_m^2-a_n^2$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Let $a, b, c, d$ be positive reals with $a - c = b - d > 0$. Show that $$\frac{ab}{cd} \ge \left(\frac{\sqrt{a} +\sqrt{b}}{\sqrt{c}+\sqrt{d}}\right)^4$$

Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.