The function $ 4x^2\minus{}12x\minus{}1$: $ \textbf{(A)}\ \text{always increases as }x\text{ increases}\\ \textbf{(B)}\ \text{always decreases as }x\text{ decreases to 1} \\ \textbf{(C)}\ \text{cannot equal 0} \\ \textbf{(D)}\ \text{has a maximum value when }x\text{ is negative} \\ \textbf{(E)}\ \text{has a minimum value of \minus{}10}$

Given any prime $p \ge 3$. Show that for all sufficient large positive integer $x$, at least one of $x+1,x+2,\cdots,x+\frac{p+3}{2}$ has a prime divisor greater than $p$.

Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB\equal{}x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively. (a) Find the locus of $ P$ as $ S$ moves on $ AB$. (b) Find the maximum and minimum lengths of $ SO$.

Let $ABC$ be a triangle with $CA>BC>AB$. Let $O$ and $H$ be the circumcentre and orthocentre of triangle $ABC$ respectively. Denote by $D$ and $E$ the midpoints of the arcs $AB$ and $AC$ of the circumcircle of triangle $ABC$ not containing the opposite vertices. Let $D'$ be the reflection of $D$ about $AB$ and $E'$ the reflection of $E$ about $AC$. Prove that $O,H,D',E'$ are concylic if and only if $A,D',E'$ are collinear.

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

The incircle of triangle $ABC$ touches its sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. The point $B_2$ is symmetric to $B_1$ with respect to line $A_1C_1$, lines $BB_2$ and $AC$ meet in point $B_3$. points $A_3$ and $C_3$ may be defined analogously. Prove that points $A_3, B_3$ and $C_3$ lie on a line, which passes through the circumcentre of a triangle $ABC$. [b] proposed by L. Emelyanov[/b]

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.

Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}$. Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$, makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]

Show that for any three positive integers $a,m,n$ such that $m$ divides $n$, there exists an integer $k$ such that $gcd(a,m) = gcd(a+km,n)$ .

If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?

Let $C$ be the left vertex of the ellipse $\frac{x^2}8+\frac{y^2}4 = 1$ in the Cartesian Plane. For some real number $k$, the line $y=kx+1$ meets the ellipse at two distinct points $A, B$. (i) Compute the maximum of $|CA|+|CB|$. (ii) Let the line $y=kx+1$ meet the $x$ and $y$ axes at $M$ and $N$, respectively. If the intersection of the perpendicular bisector of $MN$ and the circle with diameter $MN$ lies inside the given ellipse, compute the range of possible values of $k$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 12)[/i]

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$

Let be given a trapezoidal $ABCD$ with the based edges $BC = 3$ cm, $DA = 6$ cm ($AD // BC$). Then the length of the line $EF$ ($E \in AB , F \in CD$ and $EF // AD$) through the common point $M$ of $AC$ and $BD$ is (A) $3,5$ cm (B): $4$ cm (C) $4,5$ cm (D) $5$ cm (E) None of the above

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

Let $n \in \mathbb N$, $n \geq 2$. (a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \] (b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \] [i]Ion Savu[/i]

Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect square, while $bn$ is a perfect cube". Determine if the statement of Baron’s theorem is correct.

a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$ $$(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2) $$ $$(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)$$ there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We construct the table $n\times n$, with the numbers ($0 < i,j \le n$) $$(a_i + b_j)/(p_i + q_j)$$ in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).

In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$, not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$.

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

What is the smallest prime number $p$ such that $p^3+4p^2+4p$ has exactly $30$ positive divisors ?