[b]p1.[/b] How many ways can we arrange the elements $\{1, 2, ..., n\}$ to a sequence $a_1, a_2, ..., a_n$ such that there is only exactly one $a_i$, $a_{i+1}$ such that $a_i > a_{i+1}$? [b]p2. [/b]How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter $2012$? [b]p3.[/b] Let $\phi$ be the Euler totient function, and let $S = \{x| \frac{x}{\phi (x)} = 3\}$. What is $\sum_{x\in S} \frac{1}{x}$? [b]p4.[/b] Denote $f(N)$ as the largest odd divisor of $N$. Compute $f(1) + f(2) + f(3) +... + f(29) + f(30)$. [b]p5.[/b] Triangle $ABC$ has base $AC$ equal to $218$ and altitude $100$. Squares $s_1, s_2, s_3, ...$ are drawn such that $s_1$ has a side on $AC$ and has one point each touching $AB$ and $BC$, and square $s_k$ has a side on square $s_{k-}1$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the area of these squares? [b]p6.[/b] Let $P$ be a parabola $6x^2 - 28x + 10$, and $F$ be the focus. A line $\ell$ passes through $F$ and intersects the parabola twice at points $P_1 = (2,-22)$, $P_2$. Tangents to the parabola with points at $P_1, P_2$ are then drawn, and intersect at a point $Q$. What is $m\angle P_1QP_2$? PS. You had better use hide for answers.

Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that \[\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.\]

a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that \[a^2+b^2+c^2+abc<8.\] b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have \[a^2+b^2+c^2+abc<d \quad\] where $a,b$ and $c$ are the side lengths of the triangle?

[b](i)[/b] Solve the equation: \[ \sin^3(x) + \sin^3\left( \frac{2 \pi}{3} + x\right) + \sin^3\left( \frac{4 \pi}{3} + x\right) + \frac{3}{4} \cos {2x} = 0.\] [b](ii)[/b] Supposing the solutions are in the form of arcs $AB$ with one end at the point $A$, the beginning of the arcs of the trigonometric circle, and $P$ a regular polygon inscribed in the circle with one vertex in $A$, find: 1) The subsets of arcs having the other end in $B$ in one of the vertices of the regular dodecagon. 2) Prove that no solution can have the end $B$ in one of the vertices of polygon $P$ whose number of sides is prime or having factors other than 2 or 3.

Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.

Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.

Let $ABC$ be a triangle and $\Omega$ its circumcircle. Let the internal angle bisectors of $\angle BAC, \angle ABC, \angle BCA$ intersect $BC,CA,AB$ on $D,E,F$, respectively. The perpedincular line to $EF$ through $D$ intersects $EF$ on $X$ and $AD$ intersects $EF$ on $Z$. The circle internally tangent to $\Omega$ and tangent to $AB,AC$ touches $\Omega$ on $Y$. Prove that $(XYZ)$ is tangent to $\Omega$.

$AD$, $BE$, and $CF$ are the bisectors of the interior angles of triangle $ABC$, with $D$, $E$, and $F$ lying on the perimeter. If angle $EDF$ is $90$ degrees, determine all possible values of angle $BAC$.

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points $m(P)$ where $P$ is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$?

Let triangle $ABC$ be right-angled at $A$. Let $D$ be the point on $AC$ such that $BD$ bisects angle $\angle ABC$. Prove that $BC - BD = 2AB$ if and only if $\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}$.

The quadrilateral $ABCD$ is described around a circle centered on $I$. Let the diagonals $AC$ and $BD$ intersect at point $E$. The perpendicular bisectors to the segments $AC$ and $BD$ intersect at the point $P$ lying inside the triangle $BEC$. The circumscribed circles of the triangles $APC$ and $BPD$ intersect at points $P$ and $Q$. Prove that $I$ lies on the line $PQ$. [i] Proposed by Tran Quang Hung [/i]

The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area. [asy] import graph; size(3cm); pair A = (0,0); pair temp = (1,0); pair B = rotate(45,A)*temp; pair C = rotate(90,B)*A; pair D = rotate(270,C)*B; pair E = rotate(270,D)*C; pair F = rotate(90,E)*D; pair G = rotate(270,F)*E; pair H = rotate(270,G)*F; pair I = rotate(90,H)*G; pair J = rotate(270,I)*H; pair K = rotate(270,J)*I; pair L = rotate(90,K)*J; draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle); [/asy]

A hexagon inscribed in a circle has three consecutive sides each of length $3$ and three consecutive sides each of length $5$. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length $3$ and the other with three sides each of length $5$, has length equal to $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $\text{(A)}\ 309 \qquad \text{(B)}\ 349 \qquad \text{(C)}\ 369 \qquad \text{(D)}\ 389\qquad \text{(E)}\ 409$

Find the area of the region bounded by the graphs $y=x^2$, $y=x$, and $x=2$.

Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.

Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$. Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.

Let $ ABC$ be an acute-angled triangle such that there exist points $ D,E,F$ on side $ BC,CA,AB$, respectively such that the inradii of triangle $ AEF,BDF,CDE$ are all equal to $ r_0$. If the inradii of triangle $ DEF$ and $ ABC$ are $ r$ and $ R$, respectively, prove that \[ r\plus{}r_0\equal{}R.\] [i]Soewono, Bandung[/i]

Let $ABCD$ be a parallelogram. Let $G,H$ be the feet of the altitudes from $A$ to $CD$ and $BC$ respectively. If $AD=15$, $AG=12$, and $AH=16$, find the length of $AB$. [i]2019 CCA Math Bonanza Lightning Round #2.4[/i]

On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $K, L$ and $M$ are selected, respectively, such that $AK = AM$ and $BK = BL$. If $\angle{MLB} = \angle{CAB}$, Prove that $ML = KI$, where $I$ is the incenter of triangle $CML$.

On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?

Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur. Proposed by: G.Galyapin

The side lengths of a triangle area $t$ form an arithmetic progression with difference $d$. Find the sides and angles of the triangle. Specifically, solve this problem for $d=1$ and $t=6$.

In an acute triangle $ABC$, its inscribed circle touches the sides $AB,BC$ at the points $C_1,A_1$ respectively. Let $M$ be the midpoint of the side $AC$, $N$ be the midpoint of the arc $ABC$ on the circumcircle of triangle $ABC$, and $P$ be the projection of $M$ on the segment $A_1C_1$. Prove that the points $P,N$ and the incenter $I$ of the triangle $ABC$ lie on the same line.

Let $ABC$ be an equilateral triangle with side 3. A circle $C_1$ is tangent to $AB$ and $AC$. A circle $C_2$, with a radius smaller than the radius of $C_1$, is tangent to $AB$ and $AC$ as well as externally tangent to $C_1$. Successively, for $n$ positive integer, the circle $C_{n+1}$, with a radius smaller than the radius of $C_n$, is tangent to $AB$ and $AC$ and is externally tangent to $C_n$. Determine the possible values for the radius of $C_1$ such that 4 circles from this sequence, but not 5, are contained on the interior of the triangle $ABC$.

A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.