Rectangle $ ABCD$ has $ AB \equal{} 4$ and $ BC \equal{} 3$. Segment $ EF$ is constructed through $ B$ so that $ EF$ is perpendicular to $ DB$, and $ A$ and $ C$ lie on $ DE$ and $ DF$, respectively. What is $ EF$? $ \textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ \frac {125}{12}\qquad \textbf{(D)}\ \frac {103}{9}\qquad \textbf{(E)}\ 12$

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$. [i](Russia) Fedor Ivlev[/i]

Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that: (i) the volume of $T_1$ is greater than that of $T_2$; (ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$.

Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$. [i]Proposed by Michael Tang[/i]

Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)? (T. Kazitsyna)

In the plane there are given the lines $\ell_1$, $\ell_2$, the circle $\mathcal{C}$ with its center on the line $\ell_1$ and a second circle $\mathcal{C}_1$ which is tangent to $\ell_1$, $\ell_2$ and $\mathcal{C}$. Find the locus of the tangent point between $\mathcal{C}$ and $\mathcal{C}_1$ while the center of $\mathcal{C}$ is variable on $\ell_1$. [i]Mircea Becheanu[/i]

Let $ABCD$ be a trapezium in which $AB$ is parallel to $CD$. The circles on $AD$ and $BC$ as diameters intersect at two distinct points $P$ and $Q$. Prove that the lines $PQ,AC,BD$ are concurrent.

The perpendicular bisector to side $AC$ of triangle $ABC$ intersects side $BC$ at point $M$ (see fig.). The bisector of angle $\angle AMB$ intersects the circumcircle of triangle $ABC$ at point $K$. Prove that the line passing through the centers of the inscribed circles triangles $AKM$ and $BKM$, perpendicular to the bisector of angle $\angle AKB$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/b53ec7df0643a90b835f142d99c417a2a1dd45.png[/img]

Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.

A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.

Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$. Let $X$ be the reflection of $A$ across the point $B$, and denote by $\gamma$ the circumcircle of triangle $BXC$. Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$. Prove that line $EX$ is tangent to the circle $\gamma$. [i]Proposed by David Stoner[/i]

In a circle with center at $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$ a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: $\textbf{(A) }\dfrac{3r^2}{16}\qquad \textbf{(B) }\dfrac{\pi r^2}{16}\qquad \textbf{(C) }\dfrac{\pi r^2\sqrt2}{8}\qquad \textbf{(D) }\dfrac{r^2\sqrt3}{32}\qquad \textbf{(E) }\dfrac{r^2\sqrt6}{48}$

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

Let $ABC$ be an acute triangle, with $AB<AC{}$ and let $D$ be a variable point on the side $AB{}$. The parallel to $D{}$ through $BC{}$ crosses $AC{}$ at $E{}$. The perpendicular bisector of $DE{}$ crosses $BC{}$ at $F{}$. The circles $(BDF)$ and $(CEF)$ cross again at $K{}$. Prove that the line $FK{}$ passes through a fixed point. [i]Proposed by Ana Boiangiu[/i]

On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear. (A. Zaslavsky)

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

$\triangle ABC$ has $AB=4$ and $AC=6$. Let point $D$ be on line $AB$ so that $A$ is between $B$ and $D$. Let the angle bisector of $\angle BAC$ intersect line $BC$ at $E$, and let the angle bisector of $\angle DAC$ intersect line $BC$ at $F$. Given that $AE=AF$, find the square of the circumcircle's radius' length.

Determine the type of triangle if the lengths of its sides $a, b, c$ satisfy the relation $$a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2$$

Let $ABC$ be an acute-angled scalene triangle. Inside the triangle, distinct points $A_1, B_1, C_1$ are chosen such that \[ \frac{AB_1}{CB_1} = \frac{AB}{CB} \quad \text{and} \quad \frac{AC_1}{BC_1} = \frac{AC}{BC}. \] Let $A_2, B_2, C_2$ be the reflections of $A_1, B_1, C_1$ across lines $BC, AC, AB$, respectively. These points satisfy the following conditions: [list] [*] The four points $A, A_2, B, C_2$ are concyclic. [*] The four points $A, A_2, B_2, C$ are concyclic. [*] The four points $B, B_2, C, C_2$ are concyclic. [*] The three points $A_2, B_2, C_2$ do not lie on the circumcircle of $\triangle ABC$. [/list] Prove that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that: (i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon; (ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane; (iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$. Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.

Let $ABCD$ be a trapezoid where $AB$ is parallel to $CD.$ Let $P$ be the intersection of diagonal $AC$ and diagonal $BD.$ If the area of triangle $PAB$ is $16,$ and the area of triangle $PCD$ is $25,$ find the area of the trapezoid.

A fudgeflake is a planar fractal figure with a $120^{\circ}$ rotational symmetry such that three identical fudgeflakes in the same orientation fit together without gaps to form a larger fudgeflake with its orientation $30^{\circ}$ clockwise of the smaller fudgeflakes' orientation, as shown below. If the distance between the centers of the original three fudgeflakes is $1$, what is the area of one of those three fudgeflakes? Justify your answer. [asy] defaultpen(linewidth(.7)); string s = "00d08c8520612022202288272220065886,00e0708768822888788866683,01006c8765,01206b88227606,01208c858768588616678868160027,017068870228728868822872272220600278886,0190988886872272882228166060,02209486868506,02304b282022852282022828888828228166060066002000668666,02304d8882858688886166702,023050,0230637222282272220786,0240918681,02505f,0260908527222027285886852728522820766,02905b81,03904422888888766686882288888607,03908c58588685876668688228882078688228822220258587685886166702,049053852282000027888666,0490fa852061112222282282000702202220065868,04a0ae868822888888866602768688228888860728228822820228586888766702,04a0de2821666868822888500602,04b0ac5812022882227200070,04c04a8220228822272012882876882288227606,04c0da288282220228886882288227600660020060668,04c0f2,04e0fa88868588616668688228882078688228822220786,05e0ba878605,05e0d287688872006,05e102786720,05f0b88220228822816606066860,06002786882288886888666868886166600668666868887,0600748207,0600ce88616,0600ff86816,0610258222282220228887882288228202285868886668688228858688228822827066,062023223,0620cd88227600,0620fe8527222027686,064074555555555555555555555555555555555552882288888876668688866606060276866686882288878886668668522888868527282822816660222228588688861668678886166600668666868886660602782228,0650c985877,0680b0865282888882822202288878886668678282858768522882822200602788860660606,06a0fa88886070220,06c0b48830,0700fe8527222206702,0790fb867888666868822888788228722722720128288768886668688866600668666027686882288886888606786882288888607,0870b9822202288222022888886058,0880428582288868886668688228828768822882282166606060602,088070816,08903e7870,08a06c877,08b02a82858868886067222006,08b03b867202285272220786,08c06b8528220676,09003572,0900745555555555555555555555555555555555555555555555555555555555555555555555555558207,0920668858888285272220786,09606c,09812e88228166,09a128868827,09b12682202288222766060061,09c0648867,09c094868672020228160,09d04b86787228828868886668688866616006686605820228,09d0e38207,09e0908282822202288888828282220228886888681666868522888888285228202282886685886066886888606606666678868160027,09f0478765,0a10468822220276866606,0a411e828886882288227606060602,0a90428888681288227600,0aa046,0aa091,0b111f8282886788766866852288586882288228216660606,0b2122,0b30438861,0b5042858868886668688228882078688228822760600660,0b805f28868686868686850633,0be09d,0c3127868681020222022882276066660,0c40de86788228728868886668688866606066860678688228886888681666868527285886872272882228166602,0c504b88688810600220222027686,0c60da86816,0c70a8,0c80d98527222027686,0c90b08886888666868822888886058228822820228287688868166768886816606006786667021,0cb046827222025888685272850,0cc11f8702288888605822882281660602,0d00d588886070220,0d60d98527222070070,0d811a86868850607,0d90b8786783,0da0268768822882876888666868886661600668666868886816611,0da107272858868886166786888616660066870202,0db0b586816,0dc02527201288227600,0dd0b48527222027686,0e301e8861,0e501d88888605822882222027686,0e50b0888860702,0eb0b48527222206702,0f401d88868886166702,0f40b1867888666868822888788228822202786,0f50298822882888860652288227600,0fc02d,1010b98765,10206f82222282220228888882828222006588606600660020066660660066868,10207288228888685866686888681636160706768166686882288878886706786882288868822885228166,10303e28216668688228887685886166702,1030b8220258527227,10502b87,10503a288282220065886066006,10602958882078688866683,10904e8527285886882288227600,10a0b127282887685272858122816,1160ad86885,11706327683,1170ab8216,11904f877,11905f87022872812220605886,119097882886888666868886661600658,11b04e858868886166783"; string[] k = split(s,","); for(string str:k) { string a = substr(str,0,3),b=substr(str,3,3),c=substr(str,6); real x=hex(a),y=hex(b); for (int i=0;i<length(c);++i) { int next = hex(substr(c,i,1)); real yI=(int)((next-next%3)/3),xI=next%3; --xI; --yI; draw((x,-y)--(x+xI,-(y+yI))); x+=xI; y+=yI; } }[/asy]