Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$? [asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label("$C_4$", D); label("$C_1$", (-1.375, 0)); label("$C_2$", (1.375,0)); label("$\frac{1}{2}$", (0, -.125)); label("$C_3$", (-0.4, -0.4)); label("$1$", (-.85, 0.70)); label("$1$", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy] $\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

An infinite fairytale is a book with pages numbered $1,2,3,\ldots$ where all natural numbers appear. An author wants to write an infinite fairytale such that a new dwarf is introduced on each page. Afterward, the page contains several discussions between groups of at least two of the already introduced dwarfs. The publisher wants to make the book more exciting and thus requests the following condition: Every infinite set of dwarfs contains a group of at least two dwarfs, who formed a discussion group at some point as well as a group of the same size for which this is not true. Can the author fulfill this condition?

The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $ 6$. How many two-digit numbers have this property? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 19$

A right triangle has sides of integer length. One side has length 11. What is the area of the triangle?

Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Let $x$ and $y$ be non-negative real numbers that sum to $ 1$. Compute the number of ordered pairs $(a, b)$ with $a, b \in \{0, 1, 2, 3, 4\}$ such that the expression $x^ay^b + y^ax^b$ has maximum value $2^{1-a-b}$ .

International Mathematical Olympiad National Selection Test Malaysia 2020 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. Annie asks his brother four questions, "What is $20$ plus $20$? What is $20$ minus $20$? What is $20$ times $20$? What is $20$ divided by $20$?". His brother adds the answers to these four questions, and then takes the (positive) square root of the result. What number does he get? p2. A broken watch moves slower than a regular watch. In every $7$ hours, the broken watch lags behind a regular watch by $10$ minutes. In one week, how many hours does the broken watch lags behind a regular watch? p3. Given a square $ABCD$. A point $P$ is chosen outside the square so that triangle $BCP$ is equilateral. Find $\angle APC$, in degrees. p4. Hussein throws 4 dice simultaneously, and then adds the number of dots facing up on all $4$ dice. How many possible sums can Hussein get? Note: For example, he can get sum $14$, by throwing $4$, $6$, $3$, and $ 1$. Assume these are regular dice, with $1$ to $6$ dots on the faces. p5. Mrs. Sheila says, "I have $5$ children. They were born one by one every $3$ years. The age of my oldest child is $7$ times the age of my youngest child." What is the age of her third child? [b]Part B [/b](2 points each) p6. The number $N$ is the smallest positive integer with the sum of its digits equal to $2020$. What is the first (leftmost) digit of $N$? p7. 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. p8. 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$$\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.$$ p9. 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 $-1 - 2 - 3 + 4$. p10. How many $3$-digit numbers have its sum of digits equal to $4$? [b]Part C[/b] (3 points each) p11. 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. p12. 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. p13. 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 $A/B$. p14. This sequence lists the perfect squares in increasing order:$$0, 1, 4, 9, 16, ... ,a, 10^8, b, ...$$Determine the value of $b - a$. p15. Determine the last digit of $5^5 + 6^6 + 7^7 + 8^8 + 9^9$ [b]Part D[/b] (4 points each) p16. Find the sum of all integers between $-\sqrt{1442}$ and $\sqrt{2020}$. p17. 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 $k$? p18. 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 $PD$? p19. What is the smallest positive multiple of $225$ that can be written using digits $0$ and $ 1$ only? p20. Given positive integers $a, b$, and $c$ with $a + b + c = 20$. Determine the number of possible integer values for $\frac{a + b}{c}$. PS. Problems 6-20 were also used in [url=https://artofproblemsolving.com/community/c4h2675966p23194287]Juniors [/url]as 1-15. Problems 11-20 were also used in Seniors 1-10.

Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers? [i]Proposed by Jacob Weiner[/i]

Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to $1$. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a [i]dominant pair[/i] if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number dominant pairs. [i]Proposed by Nathan Ramesh

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?

Alice and Bob are racing. Alice runs at a rate of $2\text{ m/s}$. Bob starts $10\text{ m}$ ahead of Alice and runs at a rate of $1.5\text{ m/s}$. How many seconds after the race starts will Alice pass Bob?

Point $E$ is the midpoint of the base $AD$ of the trapezoid $ABCD$. Segments $BD$ and $CE$ intersect at point $F$. It is known that $AF$ is perpendicular to $BD$. Prove that $BC = FC$.

Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

On a straight road, points $1, 2, \ldots, n$ are marked. The distance between any two adjacent points is 1. A "placement" refers to the arrangement of $n$ cars, numbered with the same numbers, at the marked points (there can be multiple cars at one point). The "distance" between two placements is defined as the minimum total length of sections that need to be paved so that cars from the first placement can drive on the asphalt, forming the second one (cars can change places on the road). Prove that for any $\alpha<1$, there exists an integer number $n$ for which there are $100^n$ placements, the pairwise distances between which are greater than $\alpha n$. [i]Proposed by Ilya Bogdanov[/i]

There are $2000$ components in a circuit, every two of which were initially joined by a wire. The hooligans Vasya and Petya cut the wires one after another. Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three. The hooligan who cuts the last wire from some component loses. Who has the winning strategy?

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.