Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$

An isosceles trapezoid $PQRS$, with $\overline{PQ} = \overline{QR}= \overline{RS}$ and $\angle PQR = 120^o$, is inscribed in the graph of $y = x^2$ such that $QR$ is parallel to the $x$-axis and $R$ is in the first quadrant. The $x$-coordinate of point $R$ can be written as $\frac{\sqrt{A}}{B}$ for positive integers $A$ and $B$ such that $A$ is square-free. Find $1000A +B$.

Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$. [list=a] [*]Prove that $\triangle ALE\sim\triangle AEB$. [*]Prove that $FK=FL$. [/list]

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.

The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.

The coordinates of three vertices of a parallelogram are $A(1, 1)$, $B(2, 4)$, and $C(-5, 1)$. Compute the area of the parallelogram.

In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.

Point $O$ is chosen in a triangle $ABC$ such that ${d_a},{d_b},{d_c}$ are distance from point $O$ to sides $BC,CA,AB$, respectively. Find position of point $O$ so that product ${d_a} \cdot {d_b} \cdot {d_c}$ becomes maximum.

Let $I$ be the incenter of triangle $ABC$. The excircle with center $I_A$ touches the side $BC$ at point $A'$. The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$. Prove that $\angle B \leq 60^\circ$.

International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Primary 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. Faris has six cubes on his table. The cubes have a total volume of $2021$ cm$^3$. Five of the cubes have side lengths $5$ cm, $5$ cm, $6$ cm, $6$ cm, and $11$ cm. What is the side length of the sixth cube (in cm)? p2. What is the sum of the first $200$ even positive integers? p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than $10$. If we add any two of the numbers on the paper, then the result is never $10$. What is the number that Anushri writes down for certain? p4. If the time now is $10.00$ AM, what is the time $1,000$ hours from now? Note: Enter the answer in a $12$-hour system, without minutes and AM/PM. For example, if the answer is $9.00$ PM, just enter $9$. p5. Aminah owns a car worth $10,000$ RM. She sells it to Neesha at a $10\%$ profit. Neesha sells the car back to Aminah at a $10\%$ loss. How much money did Aminah make from the two transactions, in RM? [b]Part B[/b] (2 points each) p6. Alvin takes 250 small cubes of side length $1$ cm and glues them together to make a cuboid of size $5$ cm  $\times 5$ cm  $\times 10$ cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin? p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is $2021$. How many possible integers could have been selected by Cikgu Emma? p8. A three-digit number is called [i]superb[/i] if the first digit is equal to the sum of the other two digits. For example, $431$ and $909$ are superb numbers. How many superb numbers are there? p9. Given positive integers $a, b, c$, and $d$ that satisfy the equation $4a = 5b =6c = 7d$. What is the smallest possible value of $ b$? p10. Find the smallest positive integer n such that the digit sum of n is divisible by $5$, and the digit sum of $n + 1$ is also divisible by $5$. Note: The digit sum of $1440$ is $1 + 4 + 4 + 0 = 9$. [b]Part C[/b] (3 points each) p11. Adam draws $7$ circles on a paper, with radii $ 1$ cm, $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm, and $7$ cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm$^2$) between the total area of the red circles and the total area of the blue circles? p12. The number $2021$ has a special property that the sum of any two neighboring digits in the number is a prime number ($2 + 0 = 2$, $0 + 2 = 2$, and $2 + 1 = 3$ are all prime numbers). Among numbers from $2021$ to $2041$, how many of them have this property? p13. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of $14$ wings, $24$ heads, and $62$ legs. How many goldshes are there inside Clarissa's shop? p14. A positive integer $n$ is called [i]special [/i] if $n$ is divisible by $4$, $n+1$ is divisible by $5$, and $n + 2$ is divisible by $6$. How many special integers smaller than $1000$ are there? p15. Suppose that this decade begins on $ 1$ January $2020$ (which is a Wednesday) and the next decade begins on $ 1$ January $2030$. How many Wednesdays are there in this decade? [b]Part D[/b] (4 points each) p16. Given an isosceles triangle $ABC$ with $AB = AC$. Let D be a point on $AB$ such that $CD$ is the bisector of $\angle ACB$. If $CB = CD$, what is $\angle ADC$, in degrees? p17. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length $21$ cm. p18. Haz marks $k$ points on the circumference of a circle. He connects every point to every other point with straight lines. If there are $210$ lines formed, what is $k$? p19. What is the smallest positive multiple of $24$ that can be written using digits $4$ and $5$ only? p20. In a mathematical competition, there are $2021$ participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than $40\%$ of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded? PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors [/url]as 1-10.

A circle rolls along a side of an equilateral triangle. The radius of the circle is equal to the height of the triangle. Prove that the measure of the arc intercepted by the sides of the triangle on this circle is equal to $60^o$ at all times.

In a plane, we are given line $\ell$, two points $A$ and $B$ neither of which lies on line $\ell$, and the reflection $A_1$ of point $A$ across line $\ell$. Using only a straightedge, construct the reflection $B_1$ of point $B$ across line $\ell$. Prove that your construction works. Note: “Using only a straightedge” means that you can perform only the following operations: (a) Given two points, you can construct the line through them. (b) Given two intersecting lines, you can construct their intersection point. (c) You can select (mark) points in the plane that lie on or off objects already drawn in the plane. (The only facts you can use about these points are which lines they are on or not on.)

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.

Let $ABC$ be an acute triangle with $\angle ACB = 45^o$, $G$ is the point of intersection of the medians, and $O$ is the center of the circumscribed circle. If $OG =1$ and $OG \parallel BC$, find the length of $BC$.

Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear

Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?

We say that a real number $V$ is [i]good[/i] if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V\mid V\ \text{is good}\}$.

Let $MATH$ be a square with $MA = 1$. Point $B$ lies on $AT$ such that $\angle MBT = 3.5 \angle BMT$. What is the area of $\vartriangle BMT$?

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle? [asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy] $ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 $

Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.

The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.