Consider two segments of length $a, b \ (a > b)$ and a segment of length $c = \sqrt{ab}$. [b](a)[/b] For what values of $a/b$ can these segments be sides of a triangle ? [b](b)[/b] For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled ?

Let $\triangle ABC$ be an acute triangle with angles $\alpha, \beta,$ and $\gamma$. Prove that $$\frac{\cos(\beta-\gamma)}{cos\alpha}+\frac{\cos(\gamma-\alpha)}{\cos \beta}+\frac{\cos(\alpha-\beta)}{\cos \gamma} \geq \frac{3}{2}$$

Bob has picked positive integer $1<N<100$. Alice tells him some integer, and Bob replies with the remainder of division of this integer by $N$. What is the smallest number of integers which Alice should tell Bob to determine $N$ for sure?

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

For a given real number $a$ and a positive integer $n$, prove that: i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that \[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\] ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$. [i]Liang Yengde[/i]

Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.

Let $f(n)=\varphi(n^3)^{-1}$, where $\varphi(n)$ denotes the number of positive integers not greater than $n$ that are relatively prime to $n$. Suppose \[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \] where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Lewis Chen[/i]

Consider the $4(N-1)$ squares on the boundary of an $N$ by $N$ array of squares. We wish to insert in these squares $4 (N-1)$ consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number. Is this possible? Consider the cases (a) $N = 3$ (b) $N = 4$ (c) $N = 5$ (VG Boltyanskiy, Moscow)

Let $ABCD$ be a convex quadrilateral and $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$,and $O$ their intersection point.Point $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$ such that $ MN \cap AC ={E},MN \cap BD={F}$.Prove that $OE \cdot QF= OF\cdot PE $

Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Let $ABC$ be an acute triangle and $H$ be its orthocenter. Let $E$ be the foot of the altitude from $C$ to $AB$, $F$ be the foot of the altitude from $B$ to $AC$. Let $G \neq H$ be the intersection of the circles $(AEF)$ and $(BHC)$. Prove that $AG$ bisects $BC$. [i]Proposed by Kang Taeyoung, South Korea[/i]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?

Determine all positive integers $n{}$ which can be expressed as $d_1+d_2+d_3$ where $d_1,d_2,d_3$ are distinct positive divisors of $n{}$.

Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.

Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk? $\textbf{(A)}\ 2.0 \qquad \textbf{(B)}\ 2.2\qquad \textbf{(C)}\ 2.8 \qquad \textbf{(D)}\ 3.4 \qquad \textbf{(E)}\ 4.4$

Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$. Given that the product $abcd = 2021$, $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$.

Let $M$ be the (tridiagonal) $10\times10$ matrix $$M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}$$Show that $M$ has exactly nine positive real eigenvalues (counted with multiplicities).

On the radius $OA$ of a certain circle, as on the diameter, a circle is constructed. A ray is drawn from the center $O$, intersecting the larger and smaller circles at points $B$ and $C$, respectively. Show that the lengths of arcs $AB$ and $AC$ are equal.

What is the remainder when \[\sum_{k=0}^{100}10^k\] is divided by $9$?

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]

Let $n$ be a fixed positive integer. To any choice of real numbers satisfying \[0\le x_{i}\le 1,\quad i=1,2,\ldots, n,\] there corresponds the sum \[\sum_{1\le i<j\le n}|x_{i}-x_{j}|.\] Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$.

Let $n,k$ be integers greater than $1$, $n<2^k$. Prove that there exist $2k$ integers none of which are divisible by $n$, such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by $n$.