Competition problems 2013

### 1 J

Given that there is exactly one way to write 2013 as a sum of two primes find the product of these primes.

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$4022$

### 2 J

Two circles with radius 1 intersect each other. The area of their intersection equals the sum of the areas of the two outer parts. What is the area of the intersection?

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$\frac23\pi$

### 3 J

We have several pins five of which are yellow, four red, three green, two blue, and the remaining one is orange. In how many ways can we place them in the given triangular grid (see diagram) so that no pair of pins with the same color is in the same row or column? The pins of the same color are considered indistinguishable.

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1

### 4 J

Find the smallest positive integer whose product of digits equals $600$.

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$3558$

### 5 J

Positive real numbers $a$ and $b$ satisfy What is the value of $ab+\frac{1}{ab}$?

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$33$

### 6 J

Luke has a six-digit number which satisfies the following conditions:

1. The number reads the same from left to right as from right to left.
2. It is divisible by 9.
3. After crossing out its first and last digit the only prime factor of the resulting number is 11.

What is Luke’s number?

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$513315$

### 7 J

Point $D$ lies on the diameter $AB$ of the semicircle $k$. A perpendicular to $AB$ through point $D$ intersects $k$ at point $C$. The lengths of the arcs $AC$ and $CB$ of $k$ are in the ratio $1:2$. Find $AD : DB$.

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$1 : 3$

### 8 J

Two spoilt brothers Jim and Tim got a bag of chocolate chips and split them evenly. Both of them eat from two to three chocolate chips a day. Jim finished his chips after fourteen days, Tim after exactly three weeks. How many chocolate chips were there in the bag?

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$84$

### 9 J

In how many ways can we read the word Náboj in the diagram?

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$16$

### 10 J

Inhabitants of an island are either liars or truth-tellers. Twelve of them sit around a table. They all claim to be truth-tellers and also claim that the person to their right is a liar. What is the maximum possible number of liars in such a group?

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$6$

### 11 J / 1 S

Jane has $11$ congruent square tiles six of which are red, three blue, and two green. In how many ways can she construct a $3 \times 3$ square from 9 of the 11 tiles such that the coloring of the square remains intact if we rotate the square by $90^\circ$ clockwise around its center? Two tiles of the same color are considered indistinguishable.

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$0$

### 12 J / 2 S

Two fifths of males and three fifths of females living on a certain island are married. What percentage of the island’s inhabitants is married?

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$48 \%=\frac{12}{25}$

### 13 J / 3 S

What is the maximum side length of an equilateral triangle which can be cut from a rectangular-shaped paper with dimensions $21 \times 29.7\,\text{cm}$?

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$14\sqrt3 = \frac{42}{\sqrt{3}}\,\text{cm}$

### 14 J / 4 S

Kate took a square piece of paper and repeatedly folded it in half until it was done four times. After every fold, she was left with an isosceles right triangle. How many squares can one see after unfolding the paper?

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$10$

### 15 J / 5 S

How many pentagons are there in the figure?

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$3^5=243$

### 16 J / 6 S

Joseph wanted to add two positive integers but accidentally wrote a zero digit to the end of one of them. The result he obtained was $3858$ instead of the correct $2013$. Find the value of the larger summand.

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$1808$

### 17 J / 7 S

What is the radius of the smallest circle which can cover a triangle with side lengths $3$, $5$, and $7$?

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$3.5$

### 18 J / 8 S

Lisa, Mary, Nancy and Susan have $100$ lollipops altogether. Each pair of them has at least $41$ lollipops. What is the minimum number of lollipops that Lisa can have?

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$12$

### 19 J / 9 S

The area of a rectangle $ABCD$ is $80$ and the length of its diagonal is $16$. Find the sine of the angle between its diagonals.

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$\frac58 = 0.625$

### 20 J / 10 S

What is the largest possible value of $a^b + c^d$, where $a$, $b$, $c$, and $d$ are distinct elements of the set $\{-7, -5, -4, -3, -2, -1\}$?

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$(-1)^{-4}+(-3)^{-2}=\frac{10}{9}$

### 21 J / 11 S

An angle of magnitude $110^\circ$ is placed in the coordinate plane at random. What is the probability that its legs form a graph of some function?

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$\frac{11}{18}$

### 22 J / 12 S

James drew a regular $100$-gon $A_1A_2 \dots A_{100}$ (numbered clockwise) on the last year’s Náboj (23rd March, 2012) and randomly placed a chip on one of the vertices. Every following day, he moved the chip clockwise by the number of vertices indicated by the number of the current vertex (from $A_3$ he would move it to $A_6$, from $A_{96}$ to $A_{92}$). Now the chip lies on $A_{100}$. What was the probability this would happen?

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$0.04=\frac1{25}$

### 23 J / 13 S

A positive integer is called differential if it can be written as a difference of squares of two integers. How many of the numbers $1,2, \dots, 2013$ are differential?

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$1510$

### 24 J / 14 S

Three non-overlapping regular convex polygons with side length $1$ have point $A$ in common and their union forms a (nonconvex) polygon $M$ which has $A$ in its interior. If one of the regular polygons is a square and another is a hexagon, find the perimeter of $M$.

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$16$

### 25 J / 15 S

Mike wrote the numbers from $1$ to $100$ in random order. What is the probability that for each $i = 1,\dots,50$ the number on the position $2i-1$ is smaller than the number on the position $2i$?

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$2^{-50}$

### 26 J / 16 S

Pink paint is made up of red and white paint in the ratio $1:1$, cyan paint is made up of blue and white in the ratio $1:2$. Karen intends to paint her room with paint which is made up of pink and cyan in the ratio $2:1$. She has already mixed three tins of blue and one tin of red paint. How many tins does she have to add to the mixture, provided there are only red and white tins left?

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$23$

### 27 J / 17 S

We have a hat with several white, gray, and black rabbits. When a magician starts taking them randomly out of the hat (without returning them), the probability of him taking a white rabbit before a gray one is $\frac34$. The probability of him taking a gray rabbit before a black one is $\frac34$ as well. What is the probability of him taking a white rabbit before a black one?

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$\frac{9}{10}$

### 28 J / 18 S

Positive integers $a$, $b$ satisfy $49a+99b = 2013$. Find the value of $a+b$.

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$37$

### 29 J / 19 S

In the corners of a square $PQRS$ with side length $6\,\text{cm}$ four smaller squares are placed with side lengths $2\,\text{cm}$. Let us denote their vertices $W$, $X$, $Y$, $Z$ like in the picture. A square $ABCD$ is constructed in such a way, that points $W$, $X$, $Y$, $Z$ lie inside the sides $AB$, $BC$, $CD$, $DA$ respectively. Find the largest possible distance between points $P$ and $D$.

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$6$

### 30 J / 20 S

We have twenty boxes of apples that altogether contain $129$ apples. We know that in some of the boxes there are exactly $4$ apples and all the remaining boxes contain exactly $x$ apples. Find all possible values of $x$.

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$11$, $53$

### 31 J / 21 S

Consider real numbers $a$, $b$ such that $a > b > 0$ and Find the value of the expression $\frac{a+b}{a-b}$.

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$\sqrt{\frac{2015}{2011}}$

### 32 J / 22 S

In how many ways can we fill in individual squares in the pictured heptomino piece with numbers 1 to 7 (each of which should be used once only) so that the sum of numbers in the bottom row is the same as the sum of numbers in the left column?

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$144=3 \cdot 2 \cdot 4!$

### 33 J / 23 S

Let $ABC$ be an acute triangle with $AB=4\pi$, $BC=4\pi+3$, $CA=4\pi+6$. Denote by $D$ the foot of the $A$-altitude. Find $CD-BD$.

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$12$

### 34 J / 24 S

Trams have the same intervals all day and in both directions. A pedestrian walked along the tram rails and observed that every 12 minutes he was overtaken by a tram, and every 4 minutes one tram coming in the opposite direction passed him. What is the time interval between two trams?

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$6$ minutes

### 35 J / 25 S

How many nondegenerate triangles can be formed by means of joining some point triplet from the picture?

Note: Points are aligned in the indicated grid.

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$148=\binom{11}{3}-17$

### 36 J / 26 S

Thomas got a box of chocolates with 30 sweets arranged in three rows by ten. He is eating the sweets one by one in such a way that the number of sweets in two rows always differs at most by one. In how many ways can he eat the whole box?

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$6^{10}\cdot (10!)^3$

### 37 J / 27 S

We call a positive integer with six digits doubling if the first three digits are the same (including the order) as the other three digits. For example the number $227227$ is doubling, but $135153$ is not doubling. How many six-digit doubling numbers are divisible by $2013$?

Note: The first digit of a positive integer cannot be $0$.

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$5$

### 38 J / 28 S

Suppose that we have $4\times 4$ chessboard and on each square we draw an arrow aiming right or aiming down at random. We put a robot on the upper left corner and let it move along the chessboard in the directions of the arrows. What is the probability that it will exit the chessboard from the square in the lower right corner?

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$\frac{5}{16}=\frac{\binom{6}{3}}{2^6}$

### 39 J / 29 S

Write in the basic form (i.e. as the fraction $\frac{a}{b}$, where $a$ and $b$ are positive integers with no common divisor).

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$\frac{70}{37}$

### 40 J / 30 S

Let $ABCD$ be a rectangle with side lengths $AB=30$ and $BC=20$. For how many points $X$ on its side $AB$ is the perimeter of the triangle $CDX$ an integer?

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$13$

### 41 J / 31 S

In what order should we place the rows $r_1, \dots, r_{11}$ of the table in the picture, if we want the newly created table to be symmetric with respect to the marked diagonal? It is sufficient to find one solution.

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in the reverse order (eventually complemented by a circular shift), i.e. $r_{11},r_{10},r_9,\dots,r_1$ or $r_{10},r_9,\dots,r_1,r_{11}$, …, $r_1,r_{11},r_{10},\dots, r_2$

### 42 J / 32 S

For every positive integer $n$ let Find the smallest integer $k\geq 2$ such that $a_2\cdot a_3 \cdots a_k>4$.

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$254$

### 43 J / 33 S

Barbara cut the pizza into $n$ equal slices and then she labeled them with numbers $1,2,\dots,n$ (she used each number exactly once). The numbering had the property that between each two slices with consecutive numbers ($i$ and $i+1$) there was always the same number of other slices. Then came the fatty Paul and ate almost the whole pizza, only the three neighboring slices with numbers 11, 4, and 17 (in this exact order) remained. How many slices did the pizza have?

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$20$

### 44 J / 34 S

In one of the lecture halls at Matfyz the seats are arranged in a rectangular grid. During the lecture of analysis there were exactly 11 boys in each row and exactly 3 girls in each column. Moreover, two seats were empty. What is the smallest possible number of the seats in the lecture hall?

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$144$

### 45 J / 35 S

Circle $k$ with radius $3$ is internally tangent to circle $l$ with radius 4 at point $T$. Find the greatest possible area of triangle $TKL$, where $K \in k$ and $L \in l$.

Show result

$9\sqrt3= \frac{27}{\sqrt{3}}$

### 46 J / 36 S

Alice and Bob are playing the following game: At the beginning they have a set of numbers $\{0, 1, \dots, 1024\}$. First Alice removes any $2^9$ elements, then Bob removes any of the remaining $2^8$ elements, then Alice removes $2^7$ elements and so on until at last Bob removes one element and there are exactly two numbers remaining. The game ends then and Alice pays to Bob the absolute value of the difference of these two numbers in Czech crowns. How many crowns will Bob get if they both play in the best way they can?

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$32$

### 47 J / 37 S

Some students of Matfyz did not pass the exams and were expelled. They all started to study at SOU (some other university). It had the following consequences:

1. The number of students of Matfyz decreased by one sixth.
2. The number of students of SOU increased by one third.
3. The average IQ on both schools increased by 2%.

How many times was the average IQ at Matfyz higher than average IQ at SOU?

Show result

$\frac65 = 1.2$-times

### 48 J / 38 S

A regular tetradecagon $A_1A_2\dots A_{14}$ is inscribed in a circle $k$ of radius 1. How large is the part of the disc circumscribed by the circle $k$, which lies inside the angle $\angle A_1A_4A_{14}$?

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$\frac{\pi}{14}$

### 49 J / 39 S

Bill and Carol saw the 24-element set $\{1, 2, \ldots, 24\}$. Bill wrote down all its 12-element subsets whose sum of the elements was even. Carol wrote down all its 12-element subsets whose sum of the elements was odd. Who wrote more subsets and by how many?

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Bill, $\binom{12}6 = 924$ subsets more

### 50 J / 40 S

Jack thought of three distinct positive integers $a$, $b$, $c$ such that the sum of two of them was 800. When he wrote numbers $a$, $b$, $c$, $a+b-c$, $a+c-b$, $b+c-a$ and $a+b+c$ on a sheet of paper, he realized that all of them were primes. Determine the difference between the largest and the smallest number on Jack’s paper.

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$1594$

### 51 J / 41 S

Helen stained two randomly chosen places on a one-meter bar. Thereupon Alex came and shattered the bar, just as well randomly, into $2013$ pieces. What is the probability of both stains being now on the same piece?

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$\frac1{1007}$

### 52 J / 42 S

How many ten-digit positive integers that are made up of $0,1,\dots ,9$ (i.e. each digit is used exactly once) are multiples of $11 111$?

Note: The first digit of a positive integer cannot be $0$.

Show result

$3456 = 2^5\cdot 5! - 2^4\cdot 4!$

### 53 J / 43 S

A polynomial $P(x)$ of degree $2013$ with real coefficients fulfills for $n=0,1,\dots,2013$ the relation $P(n)=3^n$. Evaluate $P(2014)$.

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$3^{2014}-2^{2014}$

### 54 J / 44 S

Inside an isosceles triangle $ABC$ fulfilling $AB=AC$ and $\angle BAC=99.4^\circ$, a point $D$ is given such that $AD=DB$ and $\angle BAD=19.7^\circ$. Compute $\angle BDC$.

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$149.1^\circ$

### 55 J / 45 S

Find the largest positive integer not ending with a zero such that removal of one of its “inner” digits produces its divisor.

Note: An “inner” digit denotes any digit except for the first and the last one.

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$180625$

### 56 J / 46 S

Three mutually distinct real numbers $a$, $b$, $c$ satisfy Compute the product $abc$.

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$3$

In a scalene triangle $ABC$, there is an altitude of the same length as one median and another altitude of the same length as another median. In what ratio are the lengths of the third altitude and the third median?
$\frac{2}{7}$