Competition problems 2017

### 1 J

The windows on an old tram look like shown in the picture. All curves forming the round corners are arcs of a quarter circle with radius $10\,\text{cm}$. A portion of the sliding window is opened $10\,\text{cm}$ as you can see in the second picture. The height of the open section is $13\,\text{cm}$. What is the area of the opening in $\text{cm}^2$?

Show result

$130$

### 2 J

A rectangle is subdivided into nine smaller rectangles as shown in the picture. The number written inside a small rectangle denotes its perimeter. Find the perimeter of the large rectangle.

Show result

$42$

### 3 J

Peter deleted one digit from a four-digit prime number and obtained $630$. What was the prime number?

Show result

$6301$

### 4 J

Star architect Pegi wants to build a very modern pentagonal mansion on a rectangular lot of land having side lengths $35\, \mbox{m}$ and $25\, \mbox{m}$. The floor area of the mansion fits into the lot as can be seen in the picture (the dots on the boundary mark $5\,\text{m}$ distance). What fraction of the area of the lot does the floor area of the mansion cover?

Show result

$\frac{41}{70}$

### 5 J

A square grid of $16$ dots, as can be seen in the picture, contains the corners of nine $1\times 1$-squares, four $2\times 2$-squares, and one $3\times 3$-square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner?

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$4$

### 6 J

Find the digit at the units place of the sum of squares

Show result

$5$

### 7 J

Express the quotient as a fraction $\dfrac{a}{b}$ in lowest terms with positive integers $a$ and $b$.

Note: The dot denotes periodic decimal expansion, for example $0.\dot12\dot3 = 0.123123\ldots$.

Show result

$\frac{11}{12}$

### 8 J

Passau has a railway station in the form of a triangle. Anna, Boris, and Cathy observe the railway traffic in Linz, Regensburg, and Waldkirchen, respectively, on the rails coming from Passau. Anna counts $190$, Boris $208$ and Cathy $72$ incoming and outgoing trains in total. How many trains went from Linz to Regensburg or vice versa if no train starts, ends or reverses its direction in Passau?

Show result

$163$

### 9 J

Find all positive integers $x < 10{,}000$ such that $x$ is a fourth power of some even integer and one can permute the digits of $x$ in order to obtain a fourth power of some odd integer. The result of permuting may not begin with zero.

Show result

$256$

### 10 J

In parallelogram $ABCD$ a line through point $C$ meets side $AB$ in point $E$ such that $EB =\frac 15\, AE$. The line segment $CE$ intersects the diagonal $BD$ in point $F$. Find the ratio $BF : BD$.

Show result

$1 : 7$

### 11 J / 11 S

A big house consists of $100$ numbered flats. In every flat, there lives one person or there live two or three persons. The total number of people living in the flats No. 1 to No. 52 is $56$ and the total number of people living in the flats No. 51 to No. 100 is $150$. How many people live in this house?

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$200$

### 12 J / 12 S

In the first stage, Nicholas wrote the number 3 with a red pencil and 2 with a green pencil on a sheet of paper. In the following stages, he used the red pencil for writing the sum of the two numbers from the previous stage and the green pencil for their (positive) difference. What number did he write in red in the 2017th stage?

Show result

$3 \cdot 2^{1008}$

### 13 J / 13 S

Little Red Riding Hood finds herself at the entrance of the “Rectangular Forest”. Starting at point $A$, she has to reach point $B$ as fast as possible. One possibility is to walk along the edges of the woods which will be $140\,\text{m}$ in total. Of course, she knows that according to the triangle inequality a direct path would be shorter. Unfortunately, there is only a path which has the form of a zigzag with two right-angled turns as can be seen in the picture. If she knew that this way was shorter than $140\,\text{m}$, she would dare to take the shortcut. Find the length (in meters) of the zigzag through the forest!

Show result

$124$

### 14 J / 14 S

Eight-digit palindromes are numbers of the form $\overline{abcddcba}$ where $a$, $b$, $c$, and $d$ are not necessarily distinct digits. How many eight-digit palindromes have the property that we can delete some digits in such a way that the resulting number would be $2017$?

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$8$

### 15 J / 15 S

For a positive integer $n$ the sum of its digits is denoted by $\operatorname S(n)$, and the product of its digits by $\operatorname P(n)$. How many positive integers $n$ are there having the property $n = \operatorname S(n) + \operatorname P(n)$?

Show result

$9$

### 16 J / 16 S

A factory owner hires $100$ employees. Each work team leader earns £ $5{,}000$ per month, each labourer £ $1{,}000$ per month and each part-time worker earns £ $50$ per month. In total, the factory owner pays £ $100{,}000$ monthly to his employees and there is at least one employee of each type. How many work team leaders are there in his factory?

Show result

$19$

### 17 J / 17 S

The mosaic below is composed of regular polygons. The hexagon and the dark grey triangle are inscribed in the same circle. If the area of a striped triangle is $17$, determine the area of the dark grey triangle.

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$51$

### 18 J / 18 S

In the course of renovating the train station of Passau, special paving for visually handicapped people is being built in. The shape of the paving is shown in the picture below. Unfortunately, only tiles of the size $1\times 2$ are available. In how many ways can the paving be filled with tiles? The tiles are indistinguishable and two pavings are considered to be different if at some spot tiles are in a different position.

Show result

$15$

### 19 J / 19 S

Michael picked a positive integer $n$. Then he chose a (positive) divisor of $n$, multiplied it by $4$, and subtracted this result from $n$, getting $2017$. Find all numbers that Michael could have picked.

Show result

$2021$, $10085$

### 20 J / 20 S

Grisha and Vanechka are very good friends, so whenever they sit next to each other they begin chatting. Five students (including Grisha and Vanechka) want to have a constructive discussion, so they want to take five chairs around a round table in such a way that Grisha and Vanechka are not neighbours. In how many ways can they do it? Arrangements that differ by rotations are different.

Show result

$60$

### 21 J / 21 S

In the figure, $AB$ is a diameter of a circle with center $M$. The two points $D$ and $C$ are on the circle in such a way that $AC\perp DM$ and $\angle MAC = 56^\circ$. Find the size of the acute angle between the lines $AC$ and $BD$ in degrees.

Show result

$73^\circ$

### 22 J / 22 S

An object is built from unit squares by successively composing its copies as indicated in the figure below. What is the length of the thick boundary of the object in the stage 6?

Show result

$488$

### 23 J / 23 S

All the $2017$ seats around a very large round table are occupied by superheroes and villains. The superheroes always tell the truth, whereas the villains always lie. Each person sitting at the table reported that he or she was sitting between a superhero and a villain. For unknown reasons, exactly one superhero made a mistake. How many superheroes are there?

Show result

$1345$

### 24 J / 24 S

Find all positive real numbers $x$ such that

Show result

$\sqrt[2016]{2017}$

### 25 J / 25 S

Leo wants to colour the edges of a regular dodecahedron in a special way: He chooses a felt pen of one colour, starts at one vertex of the dodecahedron and moves along connected edges without lifting his pen and without colouring any edge twice until he wants to or is forced to stop. Then he takes another felt pen and begins colouring some connected uncoloured edges. He continues this procedure using one colour at a time until every edge of the dodecahedron is coloured exactly once. What is the minimum number of colours which can be used?

Note: Regular dodecahedron is a regular solid with twelve pentagonal faces as shown in the picture:

Show result

$10$

### 26 J / 26 S

Landscape gardener Joe designed a new gravel path around a lake. The triangle $ABC$ marks the middle of the path, its side lengths are $a = 80\,\text{m}$, $b = 100\,\text{m}$ and $c = 120 \,\text{m}$. The borders of the path have distance $1\,\text{m}$ from this triangle, as can be seen in the following picture. How many $\text{m}^3$ of fine loose gravel must Joe order if the covering with gravel should be of height $4\,\text{cm}$ in average?

Show result

$24$

### 27 J / 27 S

Find all four-digit squares of integers such that the first two digits are equal and the last two digits are also equal.

Show result

$7744$

### 28 J / 28 S

At a fun fair there is a lottery drawing with the following rules: a participant has to choose one of four indistinguishable boxes and afterwards he draws one ball out of the chosen box. If this ball is white, the participant is a winner, if it is black, he loses. For example, if the distribution of the balls in the four boxes is where each pair $(w, b)$ represents a box with $w$ white and $b$ black balls inside, then a participant wins with a probability of $\frac{5}{8}$. Every $1000^{\text{th}}$ participant receives a super-joker: he or she is allowed to redistribute all the balls among the boxes arbitrarily, putting at least one ball in each box. Then the boxes are mixed again and the participant can choose a box and draw one ball. Johanna is a lucky girl and has won the super-joker. What is the largest probability for a win she can achieve by using an appropriate distribution of the balls given in the example?

Show result

$\frac{51}{58}$

### 29 J / 29 S

A bus, a truck, and a motorcycle move at constant speeds and pass a stationary observer in this order at equal time intervals. They pass another observer farther down the road at the same equal time intervals, but in different order: this time the order is bus, motorcycle, truck. Find the speed of the bus in $\text{km/h}$, if the speed of the truck is $60\,\text{km/h}$, and the speed of the motorcycle is $120\,\text{km/h}$.

Show result

$80$

### 30 J / 30 S

Find all ways to fill all gaps in the statement below with positive integers in order to make the statement true: “In this statement, $\boxed{\phantom{00}}\,\%$ of digits is greater than $4$, $\boxed{\phantom{00}}\,\%$ of digits is less than $5$, and $\boxed{\phantom{00}}\,\%$ of digits is equal to either $4$ or $5$”.

Show result

$50$, $50$, $60$

### 31 J / 31 S

Paul’s cattle grazes on a triangular meadow $ABC$. Since his spotted cows and those without spots did not go well together, Paul built a twenty-meter long fence perpendicular to the side $AC$, starting at point $P$ on this side and ending in $B$, dividing the meadow into two right-angled triangles. However, this was soon met with protests from the spotted cows (grazing in the part with point $A$), which pointed out that $AP : PC = 2 : 7$ and demanded a fairer division. Consequently, Paul replaced the fence with a new one, parallel to the old one and with ends on the borders of the meadow, but dividing the meadow into two parts of equal area. What was the length of the new fence in meters?

Show result

$30 \sqrt{\frac27}$

### 32 J / 32 S

There were five (not necessarily distinct) real numbers written on a blackboard. For every pair of these numbers, Wendy calculated their sum and wrote the ten results on the blackboard, erasing the initial numbers. Determine all possible values of the product of the erased numbers.

Show result

$-144$

### 33 J / 33 S

Write $333$ as the sum of squares of (arbitrarily many) distinct positive odd integers.

Show result

$3^2 + 5^2 + 7^2 + 9^2 + 13^2$

### 34 J / 34 S

Ellen picked three real numbers $a$, $b$, $c$ and defined the operation $\odot$ as As an exercise, she computed that $1 \odot 2 = 3$ and $2 \odot 3 = 4$. After further investigation, she noticed that there was a non-zero real number $u$ such that $z \odot u = z$ for every real $z$. What was the value of $u$?

Show result

$4$

### 35 J / 35 S

Three circles of radius $r$ and one circle of radius $1$ touch each other and a straight line as shown in the picture. Find $r$.

Show result

$\frac{1+\sqrt{5}}{2}$

### 36 J / 36 S

On an old concrete wall, someone had sprayed five (not necessarily distinct) real numbers, the sum of which was $20$. For each pair of these numbers, Harry computed their sum and rounded it down (i.e. took the greatest non-exceeding nearest integer), thus obtaining ten integers. Finally, he added together all these new numbers. What was the smallest possible value of the sum that Harry could get?

Show result

$72$

### 37 J / 37 S

For a composite positive integer $n$, let $\xi(n)$ denote the sum of the least three divisors of $n$, and $\vartheta(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$ such that $\vartheta(n)=(\xi(n))^4$.

Note: By a divisor we mean a positive, not necessarily proper divisor.

Show result

$864$

### 38 J / 38 S

On a $3 \times 3$ board, a mouse sits in the bottom left square. Its goal is to get to a piece of cheese in the top right square using only moves between adjacent squares. In how many ways can we fill some (possibly none) of the free squares with obstacles so that the mouse is still able to reach the cheese?

Show result

$51$

### 39 J / 39 S

Among all the pairs $(x, y)$ of real numbers satisfying let $(x_0, y_0)$ be the one with minimal $x_0$. Find $y_0$.

Show result

$-3$

### 40 J / 40 S

Let $\triangle ABC$ be a right-angled triangle with right angle at $C$. Let $D$ and $E$ be points on $AB$ with $D$ between $A$ and $E$ such that $CD$ and $CE$ trisect $\angle ACB$. If $DE:BE = 8:15$, find $AC : BC$.

Show result

$\frac{4\sqrt3}{11}$

### 41 J / 41 S

Mike plays the following game: His task is to find an integer between $1$ and $N$ (inclusive). In each turn, he picks an integer from this interval; if it is the correct one, the game ends, otherwise he is told if his choice was too large or too small. However, if the picked number is too large, he has to pay £ 1, and if it is too small, he pays £ 2 (and he pays nothing if his choice is correct). What is the largest integer $N$ such that Mike can always finish the game provided that he can spend at most £ 10?

Show result

$232$

### 42 J / 42 S

Positive integers $a$, $b$, $c$ satisfy $a \geq b \geq c$ and Find all possible values of $a$.

Show result

$134$

### 43 J / 43 S

An alien spacecraft has the shape of a perfect ball of radius $R$, supported by three parallel vertical straight legs of length $1\,\text{af}$ (alien fathom) and negligible width. The lower ends of the legs form an equilateral triangle of side length $9\,\text{af}$ and when the spacecraft rests on a flat surface, the lowest point of the ball precisely touches the surface. What is the radius $R$ (in $\text{af}$)?

Show result

$14$

### 44 J / 44 S

Four brothers, Allan, Bert, Charlie, and Daniel, had collected a large amount of hazelnuts in a forest. At night, Allan woke up with the irrestible urge to eat, so he decided to eat some of them. After counting the hazelnuts, he found out that if one hazelnut was removed, the rest could have been divided into four equal parts, so he discarded the extra hazelnut and ate his share (one fourth of the rest); then he returned to bed. Later that night, Bert woke up because of hunger, too, and found out that there was again an extra nut, which he removed and ate one fourth of the remaining nuts. Until morning, exactly the same happened to Charile and Daniel. When all brothers met in the morning, they found out that the number of the remaining nuts was still such that it was divisible by four after removing one nut. What was the minimal number of hazelnuts the brothers could have collected?

Show result

$1021$

### 45 J / 45 S

We call a positive integer convenient if all its prime divisors are among $2$, $3$, and $7$. How many convenient numbers are among $1000, 1001, \dots, 2000$?

Show result

$19$

### 46 J / 46 S

Emperor Decimus has banned using the digit $0$ (introduced by his predecessor Nullus) and ordered the digit $\mathrm D$, representing the quantity $10$, to be used instead, thereby founding the Decimus notation. Each positive integer still has a unique representation, e.g. To smoothen the transfer to the new system, a list of all integers from $1$ to $\mathrm{DDD}$ (inclusive) has been written down. How many occurrences of the new digit $\mathrm D$ are there in the list? Multiple occurrences in a single number are counted with multiplicity, so e.g. $\mathrm{DD}$ is counted as two $\mathrm D$’s.

Show result

$321$

### 47 J / 47 S

Square $ABCD$ has inscribed circle $\omega$, which touches the square at points $W$, $X$, $Y$, $Z$ lying on sides $AB$, $BC$, $CD$, $DA$ respectively. Let $E$ be an inner point of the (shorter) arc of $\omega$ between $W$ and $X$ and $F$ the intersection of $BC$ and $EY$. Provided that $EF = 5$ and $EY = 7$, determine the area of triangle $FYC$.

Show result

$21$

### 48 J / 48 S

In the cryptogram different letters stand for different digits. Furthermore, we are given $\operatorname{S}(W\!E)=11$, $\operatorname{S}(LIKE)=23$, and $\operatorname{S}(N\!ABOJ)=19$, where $\operatorname{S}(n)$ denotes the sum of digits of a positive integer $n$. None of the three numbers may start with zero. Find the five-digit number $N\!ABOJ$.

Show result

$60{,}724$

### 49 J / 49 S

Find all positive integers $n$ with the property that the sum of all non-trivial divisors is $63$.

Note: A non-trivial divisor $d$ of $n$ fulfils $1 < d < n$.

Show result

$56$, $76$, $122$

### 50 J / 50 S

Bob has a cool car with square-shaped rear wheels (the front wheels are the standard round ones). Such a car would usually be rather unpleasant to drive, but Bob has installed very good shock absorbers for the rear wheels so that the car stays in a fixed position parallel to the surface while driving on a flat road. The side length of the rear wheel is $40\,\text{cm}$ and its axle is fixed with respect to the car in the horizontal direction. What is the radius of the front wheel (in cm) provided that when the car is moving with constant forward speed, the axle of the rear wheel is exactly half of the time lower and half of the time higher from the surface than the axle of the front wheel?

Show result

$10 \sqrt{7}$

### 51 J / 51 S

The City of the Future has the shape of a regular $2017$-gon. In vertices of the city there are $2017$ metro stations, labelled $1, 2, \ldots, 2017$ counterclockwise on the city map. There are two metro lines: side and diagonal. The side line provides direct connection from station $a$ to $b$ (but not in the opposite direction) if and only if $a-b+1$ is divisible by $2017$ and one such ride lasts $1$ minute. The diagonal line provides direct connection from station $a$ to $b$ if and only if $2b-2a+1$ is divisible by $2017$ and one ride lasts $15$ minutes. Ferdinand is an avid metro traveller and starting from station $1$ he wants to get to the station $n$ with the following property: the shortest possible time needed for getting to this station is the longest among all the stations. Find values $n$ of all possible destinations of Ferdinand.

Show result

$1984$, $1985$

### 52 J / 52 S

Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of $5$. Determine, how many of the $2017$ integers $f(1), f(2), \dots, f(2017)$ have the units digit equal to $1$.

Show result

$202$

### 53 J / 53 S

The picture shows two (indirectly) similar hexangular figures having some sides of length $a$, $b$, $c$, $d$ and $A$, $B$, $C$, $D$, respectively. If these two figures are put together as in the picture, we get a new hexangular figure being similar to each of the two smaller ones (directly similar to the right-hand one). Find the ratio $A : a$.

Show result

$\sqrt{\frac{1+\sqrt 5}{2}}$

### 54 J / 54 S

Determine all pairs of positive integers $(a,b)$ such that all the roots of both the equations are also positive integers.

Show result

$(2,2)$, $(6,6)$, $(7,8)$, $(8,7)$

### 55 J / 55 S

Triangle $ABC$ with $AB = 3$, $BC = 7$, and $AC = 5$ is inscribed in circle $\omega$. The bisector of angle $\angle BAC$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Determine the length of $AF$.

Show result

$\frac{30}{\sqrt{19}}$

### 56 J / 56 S

Determine the number of ordered triples $(x, y, z)$, where $x$, $y$, $z$ are non-negative integers smaller than $2017$ such that is divisible by $2017$.

Show result

$2017^2 + 1 = 4{,}068{,}290$

### 57 J / 57 S

Jane and Thomas play a game with a fair dice, which has two faces painted red, two green, and two blue. They alternately roll the dice until one of them has seen all three colors during his/her turns, this player being declared the winner. With what probability does Jane win the game, provided that she is the first one to roll the dice?

Show result

$81/140$

### 58 J / 58 S

Given the equation find the largest integer $n$ for which there exists an integer solution $(k, n)$.

Show result

$104$

The Quarter-Pizzeria delivers pizza in special pentagonal boxes, which are suitable both for a quarter of a big pizza and for three quarters of a small pizza (as shown in the picture). What is the radius of a small pizza (in $\text{cm}$) if the radius of a big pizza is $30\,\text{cm}$?
$5(1+\sqrt{7}-\sqrt{2\sqrt{7}-4})$