If $P(x)=x^n+a_1x^{n-1}+...+a_{n-1}$ be a polynomial with real coefficients and $a_1^2<a_2$ then prove that not all roots of $P(x)$ are real.

Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.

Let $n$ be a natural number. Solve in whole numbers the equation \[x^{n}+y^{n}=(x-y)^{n+1}.\]

$\odot A$, centered at point $A$, has radius $14$ and $\odot B$, centered at point $B$, has radius $15$. $AB = 13$. The circles intersect at points $C$ and $D$. Let $E$ be a point on $\odot A$, and $F$ be the point where line $EC$ intersects $\odot B$, again. Let the midpoints of $DE$ and $DF$ be $M$ and $N$, respectively. Lines $AM$ and $BN$ intersect at point $G$. If point $E$ is allowed to move freely on $\odot A$, what is the radius of the locus of $G$?

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[-1,1].$ Find the probability that $$|x|+|y|+1 \le 3\min\{|x+y+1|, |x+y-1|\}.$$

Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.

Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers. Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$ such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \] Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that $a_k \in A$ for all $k \ge 2025$ and, for every positive integer $d$, there exist infinitely many positive integers $n$ satisfying \[ \sum_{k=1}^n a_k k^d = 0. \] [i]Daniel Zhu[/i]

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Let $n$ be a fixed positive integer. The points $A_1$, $A_2$, $\ldots$, $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color. Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.

Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

Consider a $10 \times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10\%$ of the burrito's original size and accidentally throw it into a random square (possibly the one it is already in). Otherwise, the pigeon will move to an adjacent square, decreasing the distance between it and the burrito. What is the expected number of minutes before the pigeon has eaten the entire burrito?

Let $ ABC$ be a triangle, and $ D$ be a point where incircle touches side $ BC$. $ M$ is midpoint of $ BC$, and $ K$ is a point on $ BC$ such that $ AK\perp BC$. Let $ D'$ be a point on $ BC$ such that $ \frac{D'M}{D'K}=\frac{DM}{DK}$. Define $ \omega_{a}$ to be circle with diameter $ DD'$. We define $ \omega_{B},\omega_{C}$ similarly. Prove that every two of these circles are tangent.

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

\begin{quote} Ted quite likes haikus, \\ poems with five-seven-five, \\ but Ted knows few words. He knows $2n$ words \\ that contain $n$ syllables \\ for every int $n$. Ted can only write \\ $N$ distinct haikus. Find $N$. \\ Take mod one hundred. \end{quote} Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$. [i]Proposed by Lewis Chen[/i]

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 $j,k$.