[b]p1.[/b] If Bob takes $6$ hours to build $4$ houses, how many hours will he take to build $ 12$ houses? [b]p2.[/b] Compute the value of $\frac12+ \frac16+ \frac{1}{12} + \frac{1}{20}$. [b]p3.[/b] Given a line $2x + 5y = 170$, find the sum of its $x$- and $y$-intercepts. [b]p4.[/b] In some future year, BmMT will be held on Saturday, November $19$th. In that year, what day of the week will April Fool’s Day (April $1$st) be? [b]p5.[/b] We distribute $78$ penguins among $10$ people in such a way that no person has the same number of penguins and each person has at least one penguin. If Mr. Popper (one of the $10$ people) wants to take as many penguins as possible, what is the largest number of penguins that Mr. Popper can take? [b]p6.[/b] A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? [b]p7. [/b]Alice, Bob, Cara, David, Eve, Fred, and Grace are sitting in a row. Alice and Bob like to pass notes to each other. However, anyone sitting between Alice and Bob can read the notes they pass. How many ways are there for the students to sit if Eve wants to be able to read Alice and Bob’s notes, assuming reflections are distinct? [b]p8.[/b] The pages of a book are consecutively numbered from $1$ through $480$. How many times does the digit $8$ appear in this numbering? [b]p9.[/b] A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length $2$, what is the area of the flower? [b]p10.[/b] There are two non-consecutive positive integers $a, b$ such that $a^2 - b^2 = 291$. Find $a$ and $b$. [b]p11.[/b] Let $ABC$ be an equilateral triangle. Let $P, Q, R$ be the midpoints of the sides $BC$, $CA$ and $AB$ respectively. Suppose the area of triangle $PQR$ is $1$. Among the $6$ points $A, B, C, P, Q, R$, how many distinct triangles with area $1$ have vertices from that set of $6$ points? [b]p12.[/b] A positive integer is said to be binary-emulating if its base three representation consists of only $0$s and $1$s. Determine the sum of the first $15$ binary-emulating numbers. [b]p13.[/b] Professor $X$ can choose to assign homework problems from a set of problems labeled $ 1$ to $30$, inclusive. No two problems in his assignment can share a common divisor greater than $ 1$. What is the maximum number of problems that Professor $X$ can assign? [b]p14.[/b] Trapezoid $ABCD$ has legs (non-parallel sides) $BC$ and $DA$ of length $5$ and $6$ respectively, and there exists a point $X$ on $CD$ such that $\angle XBC = \angle XAD = \angle AXB = 90^o$ . Find the area of trapezoid $ABCD$. [b]p15.[/b] Alice and Bob play a game of Berkeley Ball, in which the first person to win four rounds is the winner. No round can end in a draw. How many distinct games can be played in which Alice is the winner? (Two games are said to be identical if either player wins/loses rounds in the same order in both games.) [b]p16.[/b] Let $ABC$ be a triangle and M be the midpoint of $BC$. If $AB = AM = 5$ and $BC = 12$, what is the area of triangle $ABC$? [b]p17. [/b] A positive integer $n$ is called good if it can be written as $5x+ 8y = n$ for positive integers $x, y$. Given that $42$, $43$, $44$, $45$ and $46$ are good, what is the largest n that is not good? [b]p18.[/b] Below is a $ 7 \times 7$ square with each of its unit squares labeled $1$ to $49$ in order. We call a square contained in the figure [i]good [/i] if the sum of the numbers inside it is odd. For example, the entire square is [i]good [/i] because it has an odd sum of $1225$. Determine the number of [i]good [/i] squares in the figure. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [hide][img]https://cdn.artofproblemsolving.com/attachments/9/2/1039c3319ae1eab7102433694acc20fb995ebb.png[/hide] [b]p19.[/b] A circle of integer radius $ r$ has a chord $PQ$ of length $8$. There is a point $X$ on chord $PQ$ such that $\overline{PX} = 2$ and $\overline{XQ} = 6$. Call a chord $AB$ euphonic if it contains $X$ and both $\overline{AX}$ and $\overline{XB}$ are integers. What is the minimal possible integer $ r$ such that there exist $6$ euphonic chords for $X$? [b]p20.[/b] On planet [i]Silly-Math[/i], two individuals may play a game where they write the number $324000$ on a whiteboard and take turns dividing the number by prime powers – numbers of the form $p^k$ for some prime $p$ and positive integer $k$. Divisions are only legal if the resulting number is an integer. The last player to make a move wins. Determine what number the first player should select to divide $324000$ by in order to ensure a win. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.

Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$, such that $AX = AB$. Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$. Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$. (Theresia Eisenkölbl)

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$

Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.

In a quadrilateral $ABCD$ let $A', B', C'$ and $D'$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Denote by $S(X, YZ)$ the plane which passes through the point $X$ and is perpendicular to the line $YZ.$ Prove that if $A', B', C'$ and $D'$ don't lie in a plane, then four planes $S(A, C'D'), S(B, A'D'), S(C, A'B')$ and $S(D, B'C')$ pass through a common point.

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is: $ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.

Let $G$ be the centroid of the triangle $ABC$ in which the angle at $C$ is obtuse and $AD$ and $CF$ be the medians from $A$ and $C$ respectively onto the sides $BC$ and $AB$. If the points $\ B,\ D, \ G$ and $\ F$ are concyclic, show that $\dfrac{AC}{BC} \geq \sqrt{2}$. If further $P$ is a point on the line $BG$ extended such that $AGCP$ is a parallelogram, show that triangle $ABC$ and $GAP$ are similar.

Let $\omega_1, \omega_2$ be two circles with different radii, and let $H$ be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from $X$ to $\omega_1$ touch $\omega_1$ at $P, Q$, and the tangents from $X$ to $\omega_2$ touch $\omega_2$ at $R, S$. If $PR$ passes through $H$ and is not a common tangent line of $\omega_1, \omega_2$, prove that $QS$ also passes through $H$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes. (N. Belukhov)

Quadrilateral $ABCD$ is inscribed into a circle with center $O$. The bisectors of its angles form a cyclic quadrilateral with circumcenter $I$, and its external bisectors form a cyclic quadrilateral with circumcenter $J$. Prove that $O$ is the midpoint of $IJ$.

Let $ABC$ be a triangle where $AC > AB$ and $I$ is its incentre. Let $M$ be the midpoint of arc $BAC$. Let $MI$ meet $AB$ at $T$. Let the incircle of $ABC$ meet $BC$ at $D$. Show that $B,I,C,T$ are concyclic if and only if $CD =3BD$.

$\triangle ABC$ is given with $|AB|=7, |BC|=12$, and $|CA|=13$. Let $D$ be a point on $[BC]$ such that $|BD|=5$. Let $r_1$ and $r_2$ be the inradii of $\triangle ABD$ and $\triangle ACD$, respectively. What is $r_1/r_2$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{13}{12} \qquad \textbf{(C)}\ \frac{7}{5} \qquad \textbf{(D)}\ \frac{3}{2} \qquad \textbf{(E)}\ \text{None}$

Let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $AP$ and $AQ$. Let $T$ be another point of intersection of the circles $\Gamma_1$ and $\Gamma_2$. Let $Q_1$ be another point of intersection of the circle $\Gamma_1$ and the line $AQ$, and $P_1$ the other point of intersection of the circle $\Gamma_2$ and the line $AP$. The circle $\Gamma_3$ passes through the points $T$, $P$ and $P_1$ and the circle $\Gamma_4$ passes through the points $T$, $Q$ and $Q_1$. Prove that the line containing the common chord of the circles $\Gamma_3$ and $\Gamma_4$ passes through$A$.

Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$. Prove that the lines $OM$ and $AB$ are perpendicular.

Let $K$ denote the circumference of a circular disk of radius $1$, and let $k$ denote a circular arc that joins two points $a,b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disk into two parts of equal area. Prove that the length of $k$ exceeds $2.$

The plane angles at vertex $D$ of the pyramid $ABCD$ are equal to $\alpha$,$\beta$ and $\gamma$ ($\angle CDB = a$). An arbitrary point $M$ is taken on edge $CB$. A ball is inscribed in each of the pyramids $ABDM$ and $ACDM$. Let us draw through $D$ a plane distinct from $BCD$, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment $AM$ at point $P$. What is $\angle ADP$ equal to?