Found problems: 34
2019 Indonesia Juniors, day 1
Actually, this is an MO I participated in :) but it's really hard to get problems from this year if you don't know some people.
P1. Let $f$ be a function satisfying $f(x + 1) + f(x - 1) = \sqrt{2} f(x)$, for all reals $x$. If $f(x - 1) = a$ and $f(x) = b$, determine the value of $f(x + 4)$.
[hide=Remarks]We found out that this is the modified version of a problem from LMNAS UGM 2008, Senior High School Level, on its First Round. This is also the same with Arthur Engel's "Problem Solving Strategies" Book, Example Problem E2.[/hide]
P2. The sequence of "Sanga" numbers is formed by the following procedure.
i. Pick a positive integer $n$.
ii. The first term of the sequence $(U_1)$ is $9n$.
iii. For $k \geq 2$, $U_k = U_{k-1} - 17$.
Sanga$[r]$ is the "Sanga" sequence whose smallest positive term is $r$.
As an example, for $n = 3$, the "Sanga" sequence which is formed is $27, 10, -7, -24, -41, \ldots.$ Since the smallest positive term of such sequence is $10$, for $n = 3$, the sequence formed is called Sanga$[10]$. For $n \leq 100$, determine the sum of all $n$ which makes the sequence Sanga$[4]$.
P3. The cube $ABCD.EFGH$ has an edge length of 6 cm. Point $R$ is on the extension of line (segment) $EH$ with $EH : ER = 1 : 2$, such that triangle $AFR$ cuts edge $GH$ at point $P$ and cuts edge $DH$ at $Q$. Determine the area of the region bounded by the quadrilateral $AFPQ$.
[url=https://artofproblemsolving.com/community/q1h2395046p19649729]P4[/url]. Ten skydivers are planning to form a circle formation when they are in the air by holding hands with both adjacent skydivers. If each person has 2 choices for the colour of his/her uniform to be worn, that is, red or white, determine the number of different colour formations that can be constructed.
P5. After pressing the start button, a game machine works according to the following procedure.
i. It picks 7 numbers randomly from 1 to 9 (these numbers are integers, not stated but corrected) without showing it on screen.
ii. It shows the product of the seven chosen numbes on screen.
iii. It shows a calculator menu (it does not function as a calculator) on screen and asks the player whether the sum of the seven chosen numbers is odd or even.
iv. Shows the seven chosen numbers and their sum and products.
v. Releases a prize if the guess of the player was correct or shows the message "Try again" on screen if the guess by the player was incorrect. (Although the player is not allowed to guess with those numbers, and the machine's procedures are started all over again.)
Kiki says that this game is really easy since the probability of winning is greater than $90$%. Explain, whether you agree with Kiki.
2006 Indonesia Juniors, day 1
p1. Given $N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}$. Determine the value of N.
p2. The triangle $ABC$ in the following picture is isosceles, with $AB = AC =90$ cm and $BC = 108$ cm. The points $P$ and $Q$ are located on $BC$, respectively such that $BP: PQ: QC = 1: 2: 1$. Points $S$ and $R$ are the midpoints of $AB$ and $AC$ respectively. From these two points draw a line perpendicular to $PR$ so that it intersects at $PR$ at points $M$ and $N$ respectively. Determine the length of $MN$.
[img]https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png[/img]
p3. If eight equilateral triangles with side $ 12$ cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron.
[img]https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png[/img]
p4. It is known that $a^2 + b^2 = 1$ and $x^2 + y^2 = 1$. Continue with the following algebraic process.
$(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...$
a. What relationship can be concluded between $ax + by$ and $1$?
b. Why?
p5. A set of questions consists of $3$ questions with a choice of answers True ($T$) or False ($F$), as well as $3$ multiple choice questions with answers $A, B, C$, or $D$. Someone answer all questions randomly. What is the probability that he is correct in only $2$ questions?
2005 Indonesia Juniors, day 1
p1. $A$ is a set of numbers. The set $A$ is closed to subtraction, meaning that the result of subtracting two numbers in $A$ will be
returns a number in $A$ as well. If it is known that two members of $A$ are $4$ and $9$, show that:
a. $0\in A$
b. $13 \in A$
c. $74 \in A$
d. Next, list all the members of the set $A$ .
p2. $(2, 0, 4, 1)$ is one of the solutions/answers of $x_1+x_2+x_3+x_4=7$. If all solutions belong on the set of not negative integers , specify as many possible solutions/answers from $x_1+x_2+x_3+x_4=7$
p3. Adi is an employee at a textile company on duty save data. One time Adi was asked by the company leadership to prepare data on production increases over five periods. After searched by Adi only found four data on the increase, namely $4\%$, $9\%$, $7\%$, and $5\%$. One more data, namely the $5$th data, was not found. Investigate increase of 5th data production, if Adi only remembers that the arithmetic mean and median of the five data are the same.
p4. Find all pairs of integers $(x,y)$ that satisfy the system of the following equations:
$$\left\{\begin{array}{l} x(y+1)=y^2-1 \\
y(x+1)=x^2-1
\end{array} \right. $$
p5. Given the following image. $ABCD$ is square, and $E$ is any point outside the square $ABCD$. Investigate whether the relationship $AE^2 + CE^2 = BE^2 +DE^2$ holds in the picture below.
[img]https://cdn.artofproblemsolving.com/attachments/2/5/a339b0e4df8407f97a4df9d7e1aa47283553c1.png[/img]
2018 Indonesia Juniors, day 1
The problems are really difficult to find online, so here are the problems.
P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form.
a) Determine the smallest possible value of $k$.
b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$.
c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.
I added this because my translation is a bit weird.
[hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana.
a) Tentukan bilangan $k$ terkecil yang mungkin.
b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$.
c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide]
P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs.
[hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description:
For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0).
Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve.
Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide]
Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$.
a) Draw the graph of the function $g \circ f$.
b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$.
P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$.
P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$.
P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.
2006 Indonesia Juniors, day 2
p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime.
p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of their respective origins.
[img]https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png[/img]
p3. The system of equations of four variables is given: $\left\{\begin{array}{l}
23x + 47y - 3z = 434 \\
47x - 23y - 4w = 183 \\
19z + 17w = 91
\end{array} \right. $
where $x, y, z$, and $w$ are positive integers.
Determine the value of $(13x - 14y)^3 - (15z + 16w)^3$
p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png[/img]
Initially the vehicle contains $ 3$ liters of fuel. After two hours, in the journey of fuel remains $ 1$ liter.
a. If in $ 1$ liter he can cover a distance of $32$ km, what is the distance taken as a whole? Explain why you answered like that?
b. After two hours of travel, is there any acceleration or deceleration? Explain your answer.
c. Determine what the average speed of the vehicle is.
p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below.
[img]https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png[/img]
He made a rule that the bottom four circles would be filled with positive numbers less than $10$ that can be taken from the numbers on the date of his birth, i.e. $26 \,\, - \,\, 12 \,\, - \,\,1961$ without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath.
a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain.
b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as $8$ circles. He no longer cares whether the numbers are repeated or not .
i. In order to get the smallest value in the top circle, how should the numbers be arranged?
ii. How many arrays are worth considering to produce the smallest value?
2015 Indonesia Juniors, day 2
p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$.
p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ?
p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements.
p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$.
p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.
2020 Indonesia Juniors, day 2
p1. Let $U_n$ be a sequence of numbers that satisfy:
$U_1=1$, $U_n=1+U_1U_2U_3...U_{n-1}$ for $n=2,3,...,2020$
Prove that $\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2$
p2. If $a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil$ , $b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor$, and $c=a-b$, then determine the value of $c$.
p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks $2$ fish, then the probability that the $2$ fish are of the same sex is $1/2$. Prove that the number of koi fish in the aquarium is a perfect square.
p4. A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are 3 bottle choices, namely
a. Bottle A
$\bullet$ Capacity: $5$ ml
$\bullet$ The price of one bottle is $10,000$ Rp
$\bullet$ If you buy the next bottle, you will get a $20\%$ discount, up to the $4$th purchase or if you buy $4$ bottles, get $ 1$ free bottle A
b. Bottle B
$\bullet$ Capacity: $8$ ml
$\bullet$ The price of one bottle is $15.000$ Rp
$\bullet$ If you buy $2$ : $20\%$ discount
$\bullet$ If you buy $3$ : Free $ 1$ bottle of B
c. Bottle C
$\bullet$ Capacity : $14$ ml
$\bullet$ Buy $ 1$ : $25.000$ Rp
$\bullet$ Buy $2$ : Free $ 1$ bottle of A
$\bullet$ Buy $3$ : Free $ 1$ bottle of B
If in one purchase, you can only buy a maximum of $4$ bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity).
p5. Two circles, let's say $L_1$ and $L_2$ have the same center, namely at point $O$. Radius of $L_1$ is $10$ cm and radius of $L_2$ is $5$ cm. The points $A, B, C, D, E, F$ lie on $L_1$ so the arcs $AB,BC,CD,DE,EF,FA$ are equal. The points $P, Q, R$ lie on $L_2$ so that the arcs $PQ,QR,RS$ are equal and $PA=PF=QB=QC=RD=RD$ . Determine the area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png[/img]
2009 Indonesia Juniors, day 2
p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained?
p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$
p3. A function $f$ is defined as in the following table.
[img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img]
Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$
p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$
p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?
2010 Indonesia Juniors, day 1
p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$.
p2. If $a, b$, and $c$ satisfy the system of equations
$$ \frac{ab}{a+b}=\frac12$$
$$\frac{bc}{b+c}=\frac13 $$
$$ \frac{ac}{a+c}=\frac17 $$
Determine the value of $(a- c)^b$.
p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of the quadrilateral $PMCN$.
[img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img]
p4. Given the rule of motion of a particle on a flat plane $xy$ as following:
$N: (m, n)\to (m + 1, n + 1)$
$T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers.
How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ?
p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.