Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \). Proposed by Soroush Behroozifar

[u]Round 1[/u] [b]p1.[/b] In order to make good salad dressing, Bob needs a $0.9\%$ salt solution. If soy sauce is $15\%$ salt, how much water, in mL, does Bob need to add to $3$ mL of pure soy sauce in order to have a good salad dressing? [b]p2.[/b] Alex the Geologist is buying a canteen before he ventures into the desert. The original cost of a canteen is $\$20$, but Alex has two coupons. One coupon is $\$3$ off and the other is $10\%$ off the entire remaining cost. Alex can use the coupons in any order. What is the least amount of money he could pay for the canteen? [b]p3.[/b] Steve and Yooni have six distinct teddy bears to split between them, including exactly $1$ blue teddy bear and $1$ green teddy bear. How many ways are there for the two to divide the teddy bears, if Steve gets the blue teddy bear and Yooni gets the green teddy bear? (The two do not necessarily have to get the same number of teddy bears, but each teddy bear must go to a person.) [u]Round 2[/u] [b]p4.[/b] In the currency of Mathamania, $5$ wampas are equal to $3$ kabobs and $10$ kabobs are equal to $2$ jambas. How many jambas are equal to twenty-five wampas? [b]p5.[/b] A sphere has a volume of $81\pi$. A new sphere with the same center is constructed with a radius that is $\frac13$ the radius of the original sphere. Find the volume, in terms of $\pi$, of the region between the two spheres. [b]p6.[/b] A frog is located at the origin. It makes four hops, each of which moves it either $1$ unit to the right or $1$ unit to the left. If it also ends at the origin, how many $4$-hop paths can it take? [u]Round 3[/u] [b]p7.[/b] Nick multiplies two consecutive positive integers to get $4^5 - 2^5$ . What is the smaller of the two numbers? [b]p8.[/b] In rectangle $ABCD$, $E$ is a point on segment $CD$ such that $\angle EBC = 30^o$ and $\angle AEB = 80^o$. Find $\angle EAB$, in degrees. [b]p9.[/b] Mary’s secret garden contains clones of Homer Simpson and WALL-E. A WALL-E clone has $4$ legs. Meanwhile, Homer Simpson clones are human and therefore have $2$ legs each. A Homer Simpson clone always has $5$ donuts, while a WALL-E clone has $2$. In Mary’s secret garden, there are $184$ donuts and $128$ legs. How many WALL-E clones are there? [u]Round 4[/u] [b]p10.[/b] Including Richie, there are $6$ students in a math club. Each day, Richie hangs out with a different group of club mates, each of whom gives him a dollar when he hangs out with them. How many dollars will Richie have by the time he has hung out with every possible group of club mates? [b]p11.[/b] There are seven boxes in a line: three empty, three holding $\$10$ each, and one holding the jackpot of $\$1, 000, 000$. From the left to the right, the boxes are numbered $1, 2, 3, 4, 5, 6$ and $7$, in that order. You are told the following: $\bullet$ No two adjacent boxes hold the same contents. $\bullet$ Box $4$ is empty. $\bullet$ There is one more $\$10$ prize to the right of the jackpot than there is to the left. Which box holds the jackpot? [b]p12.[/b] Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$. [u]Round 5[/u] [b]p13.[/b] Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$. Let M be the midpoint of $CD$, and $P$ be a point on $BM$ such that $BP = BC$. Find the area of $ABPD$. [b]p14.[/b] The number $19$ has the following properties: $\bullet$ It is a $2$-digit positive integer. $\bullet$ It is the two leading digits of a $4$-digit perfect square, because $1936 = 44^2$. How many numbers, including $19$, satisfy these two conditions? [b]p15.[/b] In a $3 \times 3$ grid, each unit square is colored either black or white. A coloring is considered “nice” if there is at most one white square in each row or column. What is the total number of nice colorings? Rotations and reflections of a coloring are considered distinct. (For example, in the three squares shown below, only the rightmost one has a nice coloring. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e6932c822bec77aa0b07c98d1789e58416b912.png[/img] PS. You should use hide for answers. Rest rounds have been posted [url=https://artofproblemsolving.com/community/c4h2786958p24498425]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

On regular hexagon $GOBEAR$ with side length $2$, bears are initially placed at $G, B, A$, forming an equilateral triangle. At time $t = 0$, all of them move clockwise along the sides of the hexagon at the same pace, stopping once they have each traveled $1$ unit. What is the total area swept out by the triangle formed by the three bears during their journey?

Circles $Q_1$ and $Q_2$ are tangent to each other externally at $T$. Points $A$ and $E$ are on $Q_1$, lines $AB$ and $DE$ are tangent to $Q_2$ at $B$ and $D$, respectively, lines $AE$ and $BD$ meet at point $P$. Prove that (1) $\frac{AB}{AT}=\frac{ED}{ET}$; (2) $\angle ATP + \angle ETP = 180^{\circ}$. [asy]import graph; size(5.97cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(10); pointfontpen=fp; real xmin=-6,xmax=5.94,ymin=-3.19,ymax=3.43; pair Q_1=(-2.5,-0.5), T=(-1.5,-0.5), Q_2=(0.5,-0.5), A=(-2.09,0.41), B=(-0.42,1.28), D=(-0.2,-2.37), P=(-0.52,2.96); D(CR(Q_1,1)); D(CR(Q_2,2)); D(A--B); D((-3.13,-1.27)--D); D(P--(-3.13,-1.27)); D(P--D); D(T--(-3.13,-1.27)); D(T--A); D(T--P); D(Q_1); MP("Q_1",(-2.46,-0.44),NE*lsf); D(T); MP("T",(-1.46,-0.44),NE*lsf); D(Q_2); MP("Q_2",(0.54,-0.44),NE*lsf); D(A); MP("A",(-2.22,0.58),NE*lsf); D(B); MP("B",(-0.35,1.45),NE*lsf); D((-3.13,-1.27)); MP("E",(-3.52,-1.62),NE*lsf); D(D); MP("D",(-0.17,-2.31),NE*lsf); D(P); MP("P",(-0.47,3.02),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Two circles have radius $9$, and one circle has radius $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown. The sine of the base angle of the triangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/c34ff6bcaf6f07e6ba81a7d256e15a61f0e1fa.png[/img]

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube. [i]Dinu Serbanescu[/i]

Horizontal line $m$ passes the center of circle $\odot O$. Line $l\perp m$, $l$ and $m$ intersect at $M$, and $M$ is on the right side of $O$. Three points $A,B,C$ ($B$ is in the middle) lie on line $l$, which are outside the circle, above line $m$. $AP,BQ,CR$ are tangent to $\odot O$ at $P,Q,R$. Prove: [b](a)[/b] If $l$ is tangent to $\odot O$, then $AB\cdot CR+BC\cdot AP=AC\cdot BQ$. [b](b)[/b] If $l$ and $\odot O$ intersect, then $AB\cdot CR+BC\cdot AP<AC\cdot BQ$. [b](c)[/b] If $l$ and $\odot O$ are apart, then $AB\cdot CR+BC\cdot AP>AC\cdot BQ$.

Let $\omega$ be the circumcircle of an acute angled tirangle $ABC.$ The line tangent to $\omega$ at $A$ intersects the line $BC$ at the point $T.$ Let the midpoint of segment $AT$ be $N,$ and the centroid of $\triangle ABC$ be the point $G.$ The other tangent line drawn from $N$ to $\omega$ intersects $\omega$ at the point $L.$ The line $LG$ meets $\omega$ at $S\neq L.$ Prove that $AS\parallel BC.$

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of $$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$ 18. An $8$ by $8$ grid of numbers obeys the following pattern: 1) The first row and first column consist of all $1$s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$. What is the number in the 8th row and 8th column?

Let $\omega$ be the circumcircle of $\Delta ABC$, where $|AB|=|AC|$. Let $D$ be any point on the minor arc $AC$. Let $E$ be the reflection of point $B$ in line $AD$. Let $F$ be the intersection of $\omega$ and line $BE$ and Let $K$ be the intersection of line $AC$ and the tangent at $F$. If line $AB$ intersects line $FD$ at $L$, Show that $K,L,E$ are collinear points

It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$

Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic. [i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]

Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

Let $N_{11}$ be the answer to problem 11. Concave heptagon $HOPKINS$, where $180^\circ<\angle HOP<270^\circ$, has area $N_{11}$, and $HP=NI\sqrt{24}$. Suppose that $HONS$ and $OPKI$ are congruent squares. Compute the common area of each of these squares.

In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

Point $D$ on side $BC$ of acute triangle ABC is such that $AB=AD$. The circumcircle of triangle $ABD$ intersects side $AC$ at points $A$ and $K$. Line $DK$ intersects the perpendicular drawn from $B$ on $AC$, at the point $L$. Prove that $CL= BC$

In the acute triangle $ABC$ on $AC$ point $P$ is chosen such that $2AP=BC$. Points $X$ and $Y$ are symmetric to $P$ wrt $A$ and $C$ respectively. It turned out that $BX=BY$. Find angle $C$.

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

Parallelogram $JHMT$ satisfies $JH=11$ and $HM=6,$ and point $P$ lies on $\overline{MT}$ such that $JP$ is an altitude of $JHMT.$ The circumcircles of $\triangle{HMP}$ and $\triangle{JMT}$ intersect at the point $Q\neq M.$ Let $A$ be the point lying on $\overline{JH}$ and the circumcircle of $\triangle{JMT}.$ If $MQ=10,$ then the perimeter of $\triangle{JAM}$ can be expressed in the form $\sqrt{a}+\tfrac{b}{c},$ where $a, \ b,$ and $c$ are positive integers, $a$ is not divisible by the square of any prime, and $b$ and $c$ are relatively prime. Find $a+b+c.$

Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds. a) Prove that: \[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)} \] b) Prove that \[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1) \] c) Prove the statement of problem. d) Prove Van der Waerden's Theorem.

Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively. $M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$. Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$. Show that: $DK=DL$