A [i]cuboctahedron[/i], shown below, is a polyhedron with 8 equilateral triangle faces and 6 square faces. Each edge has the same length and each of the 24 vertices borders 2 squares and 2 triangles. An \textit{octahedron} is a regular polyhedron with 6 vertices and 8 equilateral triangle faces. Compute the sum of the volumes of an octahedron with side length 5 and a cuboctahedron with side length 5. [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi82LzBmNjM1OTM2M2ExYTQzOTFhODEwODkwM2FiYmM1MTljOGQzNmJhLmpwZw==&rn=Q3Vib2N0YWhlZHJvbi5qcGc=[/img] [i]2016 CCA Math Bonanza Team #7[/i]

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.

[u]Down the Infinite Corridor[/u] Consider an isosceles triangle $T$ with base $10$ and height $12$. Define a sequence $\omega_1$, $\omega_2$,$...$of circles such that $\omega_1$ is the incircle of $T$ and $\omega_{i+1}$ is tangent to $\omega_i$ and both legs of the isosceles triangle for $i > 1$. [b]p1.[/b] Find the radius of $\omega_1$. [b]p2.[/b] Find the ratio of the radius of $\omega_{i+1}$ to the radius of $\omega_i$. [b]p3.[/b] Find the total area contained in all the circles.

The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$. The circle $s$ with diameter $AH$ ($H$ is the orthocenter of $ABC$) meets $\omega$ for the second time at point $P$. Restore the triangle $ABC$ if the points $A$, $P$, $W$ are given.

Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.

In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to $\textbf {(A) } a \qquad \textbf {(B) } RQ \qquad \textbf {(C) } k \qquad \textbf {(D) } \frac{h+k}{2} \qquad \textbf {(E) } h$

Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$. For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$

A castaway is building a rectangular raft $ABCD$. He fixes a mast perpendicular to the raft, with ropes passing from the top of the mast (point $S$ in the figure) to the four corners of the raft. The rope $SA$ measures $8$ meters, the rope $SB$ measures $2$ meters and the rope $SC$ measures $14$ meters. Compute the length of the rope $SD$. [asy] size(250); // Coordinates for the parallelogram ABCD pair A = (0, 0); pair B = (8, 0); pair C = (10, 5); pair D = (2, 5); // Position of point S (outside the parallelogram) pair S = (5, 8); pair T = (5, 3); // Draw the parallelogram ABCD filldraw(A--B--C--D--cycle, lightgray, black); // Draw the ropes from point S to each corner of the parallelogram draw(S--A, blue); draw(S--B, blue); draw(S--C, blue); draw(S--D, blue); draw(S--T, black); // Mark the points dot(A); dot(B); dot(C); dot(D); dot(S); dot(T); // Label the points label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, NW); label("S", S, N); [/asy]

Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar? (D. Shnol)

Let $ \triangle{ABC}$ be a triangle with $ AB\not\equal{}AC$. The incircle with centre $ I$ touches $ BC$, $ CA$, $ AB$ at $ D$, $ E$, $ F$, respectively. Furthermore let $ M$ the midpoint of $ EF$ and $ AD$ intersect the incircle at $ P\not\equal{}D$. Show that $ PMID$ ist cyclic.

Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$

We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle?

Show that $ \rho (f)$ changes continously over $ f$. It means for every bijection $ f: S^1\rightarrow S^1$ and $ \epsilon > 0$ there is $ \delta > 0$ such that if $ g: S^1\rightarrow S^1$ is a bijection such that $ \parallel{}f \minus{} g\parallel{} < \delta$ then $ |\rho(f) \minus{} \rho(g)| < \epsilon$. Note that $ \rho(f)$ is the rotatation number of $ f$ and $ \parallel{}f \minus{} g\parallel{} \equal{} \sup\{|f(x) \minus{} g(x)| | x\in S^1\}$.

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

In making Euclidean constructions in geometry it is permitted to use a ruler and a pair of compasses. In the constructions considered in this question no compasses are permitted, but the ruler is assumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the rule. Then the distance between the parallel lines is equal to the breadth of the ruler. Carry through the following constructions with such a ruler. Construct: [b]a)[/b] The bisector of a given angle. [b]b)[/b] The midpoint of a given rectilinear line segment. [b]c)[/b] The center of a circle through three given non-collinear points. [b]d)[/b] A line through a given point parallel to a given line.

Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.

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}$

Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.

IMONST = [b]I[/b]nternational [b]M[/b]athematical [b]O[/b]lympiad [b]N[/b]ational [b]S[/b]election [b]T[/b]est Malaysia 2020 Round 1 Juniors Time: 2.5 hours [hide=Rules] $\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer. $\bullet$ No mark is deducted for a wrong answer. $\bullet$ The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.[/hide] [b]Part A[/b] (1 point each) p1. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p2. At a food stall, the price of $16$ banana fritters is $k$ RM , and the price of $k$ banana fritters is $ 1$ RM . What is the price of one banana fritter, in sen? Note: $1$ RM is equal to $100$ sen. p3. Given a trapezium $ABCD$ with $AD \parallel$ to $BC$, and $\angle A = \angle B = 90^o$. It is known that the area of the trapezium is 3 times the area of $\vartriangle ABD$. Find $$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p4. Each $\vartriangle$ symbol in the expression below can be substituted either with $+$ or $-$: $$\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.$$ How many possible values are there for the resulting arithmetic expression? Note: One possible value is $-2$, which equals $-1 - 2 - 3 + 4$. p5. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part B[/b] (2 points each) p6. Find the value of $$+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020$$ where the sign alternates between $+$ and $-$ after every three numbers. p7. If Natalie cuts a round pizza with $4$ straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p8. Given a square with area $ A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $ B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $A/B$. p9. This sequence lists the perfect squares in increasing order: $$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$ Determine the value of $b - a$. p10. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$. [b]Part C[/b] (3 points each) p11. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p12. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in $3$ months if I work alone". Wahib says, "I can paint this building in $2$ months if I work alone". Wahub says, "I can paint this building in $k$ months if I work alone". If they work together, they can finish painting the building in $1$ month only. What is $k$? p13. Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17$, $PB = 15$, and $PC = 6$. What is the length of $PD$? p14. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p15. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. [b]Part D[/b] (4 points each) p16. If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$. p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by $5$ hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football? p18. Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle? p19. A perfect square ends with the same two digits. How many possible values of this digit are there? p20. Find the sum of all integers $n$ that fulfill the equation $2^n(6 - n) = 8n$.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties: $(\text{i})$ $K$ contains three points $A,B,C$, that are the vertices of an acute-angled triangle $(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$. Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$.