Let be a natural number $ n\ge 2. $ Prove that there exist exactly two subsets of the set $ \left\{ \left.\left(\begin{matrix} a& b\\-b& a \end{matrix}\right)\right| a,b\in\mathbb{R} \right\} $ that are closed under multiplication and their cardinal is $ n. $ [i]Marcel Tena[/i]

Point $M$ is the midpoint of side $BC$ of an acute—angled triangle $ABC$. The point $U$ is symmetric to the orthocenter $ABC$ relative to its circumcenter. The point $S$ inside triangle $ABC$ is such that $US = UM$. Prove that $SA + SB + SC + AM < AB + BC + CA$.

Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.

$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.

Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.

Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

In a one-round football tournament, three points were awarded for a victory. All the teams scored different numbers of points. If not three, but two points were given for a victory, then all teams would also have a different number of points, but each team's place would be different. What is the smallest number of teams for which this is possible? [i]Proposed by A. Zaslavsky[/i]

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

Let $G$ be a group and $A$ a nonempty subset of $G$. Let $AA=\{xy\mid x,y\in A\}$. [list=a] [*] Prove that if $G$ is finite, then $AA=A$ if and only if $|A|=|AA|$ and $e\in A$. [*] Give an example of a group $G$ and a nonempty subset $A$ of $G$ such that $AA\neq A$, $|AA|=|A|$ and $AA$ is a proper subgroup of $G$. [/list] [i]Mathematical Gazette - Robert Rogozsan[/i]

Circle $\omega_1$ with center $K$ of radius $4$ and circle $\omega_2$ of radius $6$ intersect at points $W$ and $U$. If the incenter of $\triangle KWU$ lies on circle $\omega_2$, find the length of $\overline{WU}$. (Note: The incenter of a triangle is the intersection of the angle bisectors of the angles of the triangle) [i]Proposed by Bradley Guo[/i]

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

Determine the largest positive integer $k$ for which the following statement is true: given $k$ distinct subsets of the set $\{1, 2, 3, \dots , 2023\}$, each with $1011$ elements, it is possible partition the subsets into two collections so that any two subsets in one same collection have some element in common.

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

We say the point $i$ in the permutation $\sigma$ [b]ongoing[/b] if for every $j<i$ we have $\sigma (j)<\sigma (i)$. [b]a)[/b] prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$. [b]b)[/b] prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.

A uniform circular disk of radius $R$ begins with a mass $M$; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is $I_\text{0}=\frac{1}{2}MR^2$. A hole is cut in the disk as shown in the diagram. In terms of the radius $R$ and the mass $M$ of the original disk, what is the moment of inertia of the resulting object about the axis shown? [asy] size(14cm); pair O=origin; pair A=O, B=(3,0), C=(6,0); real r_1=1, r_2=.5; pen my_fill_pen_1=gray(.8); pen my_fill_pen_2=white; pen my_fill_pen_3=gray(.7); pen my_circleline_draw_pen=black+1.5bp; //fill(); filldraw(circle(A,r_1),my_fill_pen_1,my_circleline_draw_pen); filldraw(circle(B,r_1),my_fill_pen_1,my_circleline_draw_pen); // Ellipse filldraw(yscale(.2)*circle(C,r_1),my_fill_pen_1,my_circleline_draw_pen); draw((C.x,C.y-.75)--(C.x,C.y-.2), dashed); draw(C--(C.x,C.y+1),dashed); label("axis of rotation",(C.x,C.y-.75),3*S); // small ellipse pair center_small_ellipse; center_small_ellipse=midpoint(C--(C.x+r_1,C.y)); //dot(center_small_ellipse); filldraw(yscale(.15)*circle(center_small_ellipse,r_1/2),white); pair center_elliptic_arc_arrow; real gr=(sqrt(5)-1)/2; center_elliptic_arc_arrow=(C.x,C.y+gr); //dot(center_elliptic_arc_arrow); draw(//shift((0*center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y-.2))* ( yscale(.2)* ( arc((center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y+2.4), .4,120,360+60)) ),Arrow); //dot(center_elliptic_arc_arrow); // lower_Half-Ellipse real downshift=1; pair C_prime=(C.x,C.y-downshift); path lower_Half_Ellipse=yscale(.2)*arc(C_prime,r_1,180,360); path upper_Half_Ellipse=yscale(.2)*arc(C,r_1,180,360); draw(lower_Half_Ellipse,my_circleline_draw_pen); //draw(upper_Half_Ellipse,red); // Why here ".2*downshift" instead of downshift seems to be not absolutely clean. filldraw(upper_Half_Ellipse--(C.x+r_1,C.y-.2*downshift)--reverse(lower_Half_Ellipse)--cycle,gray); //filldraw(shift(C-.1)*(circle((B+.5),.5)),my_fill_pen_2);// filldraw(circle((B+.5),.5),my_fill_pen_2);//shift(C-.1)* /* filldraw(//shift((C.x,C.y-.45))* yscale(.2)*circle((C.x,C.y-1),r_1),my_fill_pen_3,my_circleline_draw_pen); */ draw("$R$",A--dir(240),Arrow); draw("$R$",B--shift(B)*dir(240),Arrow); draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow); draw(scale(1)*"$\scriptstyle R/2$",(B+.5)--(B+1),.5*LeftSide,Arrow); [/asy] (A) $\text{(15/32)}MR^2$ (B) $\text{(13/32)}MR^2$ (C) $\text{(3/8)}MR^2$ (D) $\text{(9/32)}MR^2$ (E) $\text{(15/16)}MR^2$

Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$. a) Prove that $DE$ is parallel to $BC$. b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.

Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$: \[ f(xf(y)) \plus{} y \plus{} f(x) \equal{} f(x \plus{} f(y)) \plus{} yf(x)\]

A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$. [asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

Let $ P $ be a point on the side $ AB $ of the triangle $ ABC. $ The parallels through $ P $ of the medians $ AA_1,BB_1 $ intersect $ BC,AC $ at $ R,Q, $ respectively. Show that $ P, $ the middlepoint of $ RQ $ and the centroid of $ ABC $ are collinear.

Find the side length of the largest square that can be inscribed in the unit cube.

In a rectangular box are a number of rectangular blocks, not necessarily identical to one another. Each block has one of its dimensions reduced. Is it always possible to pack these blocks in a smaller rectangular box, with the sides of the blocks parallel to the sides of the box? [i](6 points)[/i]

$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of \[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]

Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.