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.

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]

Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], if for every $n$ positive integer, one have these two conditions I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number $a_n$ is the $n$-power of a positive integer. Find all the sequence(s) [i]alagoana[/i].

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$? $\textbf{(A)}~5$ $\textbf{(B)}~\frac{25}6$ $\textbf{(C)}~29$ $\textbf{(D)}~35$

If $a = 2b + c$, $b = 2c + d$, $2c = d + a -1$, $d = a - c$, what is $b$?

[b]p1.[/b] What is the last digit of $$1! + 2! + ... + 10!$$ where $n!$ is defined to equal $1 \cdot 2 \cdot ... \cdot n$? [b]p2.[/b] What pair of positive real numbers, $(x, y)$, satisfies $$x^2y^2 = 144$$ $$(x - y)^3 = 64?$$ [b]p3.[/b] Paul rolls a standard $6$-sided die, and records the results. What is the probability that he rolls a $1$ ten times before he rolls a $6$ twice? [b]p4.[/b] A train is approaching a $1$ kilometer long tunnel at a constant $40$ km/hr. It so happens that if Roger, who is inside, runs towards either end of the tunnel at a contant $10$ km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger? [b]p5.[/b] Let $ABC$ be a triangle with $A$ being a right angle. Let $w$ be a circle tangent to $\overline{AB}$ at $A$ and tangent to $\overline{BC}$ at some point $D$. Suppose $w$ intersects $\overline{AC}$ again at $E$ and that $\overline{CE} = 3$, $\overline{CD} = 6$. Compute $\overline{BD}$. [b]p6.[/b] In how many ways can $1000$ be written as a sum of consecutive integers? [b]p7.[/b] Let $ABC$ be an isosceles triangle with $\overline{AB} = \overline{AC} = 10$ and $\overline{BC} = 6$. Let $M$ be the midpoint of $\overline{AB}$, and let $\ell$ be the line through $A$ parallel to $\overline{BC}$. If $\ell$ intersects the circle through $A$, $C$ and $M$ at $D$, then what is the length of $\overline{AD}$? [b]p8.[/b] How many ordered triples of pairwise relatively prime, positive integers, $\{a, b, c\}$, have the property that $a + b$ is a multiple of $c$, $b + c$ is a multiple of $a$, and $a + c$ is a multiple of $b$? [b]p9.[/b] Consider a hexagon inscribed in a circle of radius $r$. If the hexagon has two sides of length $2$, two sides of length $7$, and two sides of length $11$, what is $r$? [b]p10.[/b] Evaluate $$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0} \frac{\left( (-1)^i + (-1)^j\right) \cos (i) \sin (j)}{i!j!} ,$$ where angles are measured in degrees, and $0!$ is defined to equal $1$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to replace stars with plusses or minusses in the following expression $$1 \star 2 \star 3 \star 4 \star 5 \star 6 \star 7 \star 8 \star 9 \star 10 = 0$$ so that to obtain a true equality?

For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ . A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove : a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$. b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.

Prove that if $F(x)$ and $G(x)$ are polynomials with coefficients $0$ and $1$ such that $$F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}$$ holds for some $n > 1$, then one of them can be represented in the form $$ (1 +x + x^2 +...+ x^{k-1}) T(x)$$ for some $k > 1$ where $T(x)$ is a polynomial with coefficients $0$ and $1$. (V Senderov, M Vialiy)

Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral. \[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]

$10$ distinct numbers are given. Professor Odd had calculated all possible products of $1$, $3$, $5$, $7$, $9$ numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of $2$, $4$, $6$, $8$, $10$ numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by $1$. Prove that one of $10$ given numbers is equal to $1$.

Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.

For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that \[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\] is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.

To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What number will occupy the lower right-hand square? \[ \begin{tabular}{|c|c|c|c|}\hline 1 & & 2 & \\ \hline 2 & 3 & & \\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular} \] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{cannot be determined}$

[u]Round 1[/u] [b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ? [b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$? [b]p3.[/b] Find the number of three-by-three grids such that $\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and $\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square. [u]Round 2 [/u] [b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$. [b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$. [b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$. [u]Round 3 [/u] [b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$? [b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on? [b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that $\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and $\bullet$ the only time she visits town $10$ is at the very end? One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$. [u]Round 4[/u] [b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$? [b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if 1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and 2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers. Find the sum of all primed integers. [b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions: (a) $P(2017)=2016$ and (b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.

Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where \[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\] is it true that $P (P (a)) = a$?

You’re given the complex number $\omega = e^{2i\pi/13} + e^{10i\pi/13} + e^{16i\pi/13} + e^{24i\pi/13}$, and told it’s a root of a unique monic cubic $x^3 +ax^2 +bx+c$, where $a, b, c$ are integers. Determine the value of $a^2 + b^2 + c^2$.

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$