Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

Determine all sets of real numbers $x,y,z$ which satisfy the system of equations $$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$

Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?

If $x$ is a real number so $3^x=27x$, compute $\log_3 \left(\tfrac{3^{3^x}}{x^{3^3}}\right)$.

Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$. Compute the maximum possible value of $a$.

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$

A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?

We consider the following system with $q=2p$: \[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\] in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties: [b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$ [b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$ [b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$

[b]p1.[/b] Fun facts! We know that $1008^2-1007^2 = 1008+1007$ and $1009^2-1008^2 = 1009+1008$. Now compute the following: $$1010^2 - 1009^2 - 1.$$ [b]p2.[/b] Let $m$ be the smallest positive multiple of $2018$ such that the fraction $m/2019$ can be simplified. What is the number $m$? [b]p3.[/b] Given that $n$ satisfies the following equation $$n + 3n + 5n + 7n + 9n = 200,$$ find $n$. [b]p4.[/b] Grace and Somya each have a collection of coins worth a dollar. Both Grace and Somya have quarters, dimes, nickels and pennies. Serena then observes that Grace has the least number of coins possible to make one dollar and Somya has the most number of coins possible. If Grace has $G$ coins and Somya has $S$ coins, what is $G + S$? [b]p5.[/b] What is the ones digit of $2018^{2018}$? [b]p6.[/b] Kaitlyn plays a number game. Each time when Kaitlyn has a number, if it is even, she divides it by $2$, and if it is odd, she multiplies it by $5$ and adds $1$. Kaitlyn then takes the resulting number and continues the process until she reaches $1$. For example, if she begins with $3$, she finds the sequence of $6$ numbers to be $$3, 3 \cdot 5 + 1 = 16, 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1.$$ If Kaitlyn's starting number is $51$, how many numbers are in her sequence, including the starting number and the number $1$? [b]p7.[/b] Andrew likes both geometry and piano. His piano has $88$ keys, $x$ of which are white and $y$ of which are black. Each white key has area $3$ and each black key has area $11$. If the keys of his piano have combined area $880$, how many black keys does he have? [b]p8.[/b] A six-sided die contains the numbers $1$, $2$, $3$, $4$, $5$, and $6$ on its faces. If numbers on opposite faces of a die always sum to $7$, how many distinct dice are possible? (Two dice are considered the same if one can be rotated to obtain the other.) [b]p9.[/b] In $\vartriangle ABC$, $AB$ is $12$ and $AC$ is $15$. Alex draws the angle bisector of $BAC$, $AD$, such that $D$ is on $BC$. If $CD$ is $10$, then the area of $\vartriangle ABC$ can be expressed in the form $\frac{m \sqrt{n}}{p}$ where $m, p$ are relatively prime and $n$ is not divisible by the square of any prime. Find $m + n + p$. [b]p10.[/b] Find the smallest positive integer that leaves a remainder of $2$ when divided by $5$, a remainder of $3$ when divided by $6$, a remainder of $4$ when divided by $7$, and a remainder of $5$ when divided by $8$. [b]p11.[/b] Chris has a bag with $4$ marbles. Each minute, Chris randomly selects a marble out of the bag and flips a coin. If the coin comes up heads, Chris puts the marble back in the bag, while if the coin comes up tails, Chris sets the marble aside. What is the expected number of seconds it will take Chris to empty the bag? [b]p12.[/b] A real fixed point $x$ of a function $f(x)$ is a real number such that $f(x) = x$. Find the absolute value of the product of the real fixed points of the function $f(x) = x^4 + x - 16$. [b]p13.[/b] A triangle with angles $30^o$, $75^o$, $75^o$ is inscribed in a circle with radius $1$. The area of the triangle can be expressed as $\frac{a+\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p14.[/b] Dora and Charlotte are playing a game involving flipping coins. On a player's turn, she first chooses a probability of the coin landing heads between $\frac14$ and $\frac34$ , and the coin magically flips heads with that probability. The player then flips this coin until the coin lands heads, at which point her turn ends. The game ends the first time someone flips heads on an odd-numbered flip. The last player to flip the coin wins. If both players are playing optimally and Dora goes first, let the probability that Charlotte win the game be $\frac{a}{b}$ . Find $a \cdot b$. [b]p15.[/b] Jonny is trying to sort a list of numbers in ascending order by swapping pairs of numbers. For example, if he has the list $1$, $4$, $3$, $2$, Jonny would swap $2$ and $4$ to obtain $1$, $2$, $3$, $4$. If Jonny is given a random list of $400$ distinct numbers, let $x$ be the expected minimum number of swaps he needs. Compute $\left \lfloor \frac{x}{20} \right \rfloor$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Compute $\tan\left(\frac{\pi}{7}\right)\tan\left(\frac{2\pi}{7}\right)\tan\left(\frac{3\pi}{7}\right)$.

Find all real $x$ such that $0 < x < \pi $ and $\frac{8}{3 sin x - sin 3x} + 3 sin^2x \le 5$.

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

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.

[i](5 pts)[/i] For some positive integer $n$, let $P(x)$ be an $n$th degree polynomial with real coefficients. [i]Note: you may cite, without proof, the Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has a complex root.[/i] (a) [i](2 pts)[/i] Show that there is an integer $k \ge \frac{n}{2}$ and a sequence of non-constant polynomials with real coefficients $Q_1(x), Q_2(x), \dots, Q_k(x)$ such that \[ P(x) = \prod_{i = 1}^k Q_i(x). \] (b) [i](1 pt)[/i] If $n$ is odd, then show that $P(x)$ has a real root. (c) [i](2 pts)[/i] Let $a$ and $b$ be real numbers, and let $m$ be a positive integer. If $\zeta = a + bi$ is a nonreal root of $P(x)$ of multiplicity $m$, then show that $\overline{\zeta} = a - bi$ is a nonreal root of $P(x)$ of multiplicity $m$.

[b]p1.[/b] Replace $*$’s by an arithmetic operations (addition, subtraction, multiplication or division) to obtain true equality $$2*0*1*6*7=1.$$ [b]p2.[/b] The interval of length $88$ cm is divided into three unequal parts. The distance between middle points of the left and right parts is $46$ cm. Find the length of the middle part. [b]p3.[/b] A $5\times 6$ rectangle is drawn on a square grid. Paint some cells of the rectangle in such a way that every $3\times 2$ sub‐rectangle has exactly two cells painted. [b]p4.[/b] There are $8$ similar coins. $5$ of them are counterfeit. A detector can analyze any set of coins and show if there are counterfeit coins in this set. The detector neither determines which coins nare counterfeit nor how many counterfeit coins are there. How to run the detector twice to find for sure at least one counterfeit coin? [b]p5.[/b] There is a set of $20$ weights of masses $1, 2, 3,...$ and $20$ grams. Can one divide this set into three groups of equal total masses? [b]p6.[/b] Replace letters $A,B,C,D,E,F,G$ by the digits $0,1,...,9$ to get true equality $AB+CD=EF * EG$ (different letters correspond to different digits, same letter means the same digit, $AB$, $CD$, $EF$, and $EG$ are two‐digit numbers). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.

Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.

[b]p1.[/b] Prove that for every point inside a regular polygon, the average of the distances to the sides equals the radius of the inscribed circle. The distance to a side means the shortest distance from the point to the line obtained by extending the side. [b]p2.[/b] Suppose we are given positive (not necessarily distinct) integers $a_1, a_2,..., a_{1997}$ . Show that it is possible to choose some numbers from this list such that their sum is a multiple of $1997$. [b]p3.[/b] You have Blue blocks, Green blocks and Red blocks. Blue blocks and green blocks are $2$ inches thick. Red blocks are $1$ inch thick. In how many ways can you stack the blocks into a vertical column that is exactly $12$ inches high? (For example, for height $3$ there are $5$ ways: RRR, RG, GR, RB, BR.) [b]p4.[/b] There are $1997$ nonzero real numbers written on the blackboard. An operation consists of choosing any two of these numbers, $a$ and $b$, erasing them, and writing $a+b/2$ and $b-a/2$ instead of them. Prove that if a sequence of such operations is performed, one can never end up with the initial collection of numbers. [b]p5.[/b] An $m\times n$ checkerboard (m and n are positive integers) is covered by nonoverlapping tiles of sizes $2\times 2$ and $1\times 4$. One $2\times 2$ tile is removed and replaced by a $1\times 4$ tile. Is it possible to rearrange the tiles so that they cover the checkerboard? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$

Find all functions f : R-->R such that f (f (x) + y^2) = x −1 + (y + 1)f (y) holds for all real numbers x, y

Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

Let $n \geqslant 3$ be a positive integer. Furthermore, let $x_1, x_2,\ldots, x_n \in [0, 2]$ be real numbers subject to $x_1 + x_2 +\cdots + x_n = 5$. Prove the inequality$$x_1^2 + x_2^2 + \cdots + x_n^2 \leqslant 9.$$When does equality hold? [i](Walther Janous)[/i]