# Tutorial problems for RMO 2019 : combinatorics continued

1) In how many ways can 5 men and 5 women be seated in a round table if no two women may be seated side by side?

2) Six generals propose locking a safe containing top secret with a number of different locks. Each general will be given keys to certain of these locks. How many locks are required and how many keys must each general have so that, unless at least four generals are present, the safe cannot be opened?

3) How many integers between 1000 and 9999 inclusive have distinct digits? Of these, how many are even numbers? How many consist entirely of odd digits?

4) In how many ways can 9 distinct objects be placed in 5 distinct boxes in such a way that 3 of these boxes would be occupied and 2 would be empty?

5) In how many permutations of the word AUROBIND do the vowels appear in the alphabetical order?

6) There is an unlimited supply of weights of integral numbers of grams. Using n or fewer weights, find the number of ways in which a weight of m grams can be obtained. Prove that there is a bijection of the set of all such ways on the set of increasing words of length $(n-1)$ or $(m+1)$ ordered letters.

7) How many distinct solutions are there of $x+y+z+w=10$ (a) in positive integers and (b) in non-negative integers?

8) A train with n passengers aboard makes m stops. In how many ways can the passengers distribute themselves among these m stops as alighting passengers? if we are concerned only with the number of alighting passengers at each stop, how would the answer be modified?

9) There are 16 books on a bookshelf. In how many ways can 6 of these books be selected if a selection must not include two neighbouring books?

10) Show that there are ${(n=5)} \choose 5$ distinct throws of a throw with n non-distinct dice.

11) Given n indistinguishable objects and n additional distinct objects —- also distinct from the earlier n objects — in how many ways can we choose n out of the 2n objects?

12) Establish the following relations:
12a) $B_{n+1}=\sum_{k=0}^{n}(B_{k}){n \choose k}$
12b) $\sum_{k}{p \choose k}{q \choose {n-k}}={{p+q} \choose n}$
12c) $S_{n+1}^{m} = \sum_{k=0}^{n}{n \choose k}S_{k}^{m-1}$
12d) $n^{p}=\sum_{k=0}^{n}{n \choose k}k! (S_{p}^{k})$

13) Prove the following identity for all real numbers x:
$x^{n}= \sum_{k=1}^{n}S_{n}^{k}[x]_{k}$

14) Express $x^{4}$ in terms of ${x \choose 4}$, ${x \choose 3}$, …by using the $S_{n}^{k}$‘s. Express ${x \choose 4}$ in terms of $x^{4}$, $x^{3}$, …by using the $s_{n}^{k}$‘s.

15) A circular loop is divided into p parts, p prime. In how many ways can we paint the loop with n colours if we do not distinguish between patterns which differ only by a rotation of the loop? Deduce Fermat’s Little theorem: $n^{p}-n$ is divisible by p if p is prime.

16) In problem 15, prove that $n^{p}-n$ is also divisible by 2p if $p \neq 2$. Where is the hypothesis that p is prime used in Problem 15 or in this problem?

17) How many equivalence relations are possible on an n-set?

18) The complete homogeneous symmetric function of n variables $\alpha_{1}$, $\alpha_{2}$, $\ldots$, $\alpha_{n}$ of degree r is defined as $h_{r}(\alpha_{1},\alpha_{2}, \alpha_{3}, \ldots, \alpha_{n})=\sum \alpha_{1}^{i_{1}}\alpha_{2}^{i_{2}}\ldots \alpha_{n}^{i_{n}}$ the summation being taken over all ordered partitions of r, where the parts are also allowed to be zero. How many terms are there in $h_{r}$?

Test yourself ! Improve your mettle in math !
Regards,
Nalin Pithwa.

# Pre RMO or PRMO problem set in elementary combinatorics

1) How many maps are there from an n-set to an m-set? How many of these are onto? How many are one-one? Under what conditions?

2) Consider the letters of the word DELHI. Let us form new words, whether or not meaningful, using these letters. The *length* of a word is the number of letters in it, e.g., the length of “Delhi” is 5; the length of “Hill” is 4. Answer the following questions when (a) repetition of letters is not allowed and (b) repetition of the letters is allowed:
(i) How many words can be formed of length 1,2,3,4,…?
(ii) How many words in (i) will consist of all the letters?
(iii) How many words of the words in (i) will consist of 1,2,3,4, …specified letters?
(iv) How many of the words in (i) will consist of only 1,2,3,4,…letters?
(v) How many of the words in (i) will be in the alphabetical order of the letters?

3) Repeat Problem 2 with the word MISSISSIPPI.

4) Suppose there are 5 distinct boxes and we want to sort out 1,2,3,…,n objects into these boxes.
4i) In how many ways can this be done?
4ii) In how many of these situations would no box be empty?
4iii) In how many of the above would only 1,2,3,4 … specified boxes be occupied?
4iv) In how many would only 1,2,3,4…boxes be occupied?
4iv) If the objects are indistinguishable from one another, how would the answers to (i) to (iv) change, it at all?

If there is an added restriction that each box can hold only one object and no more, what will be the answers to (i) to (v)?

5) Repeat Problem 4 with 9 boxes.

6) Repeat Problem 4 with 5 non-distinct (=indistinguishable identical) boxes.

7) Repeat Problem 6 with 9 boxes.

8) How many 5-letter words of binary digits are there?

9) Ten teams participate in a tournament. The first team is awarded a gold medal, the second a silver medal, and the third a bronze medal. In how many ways can the medals be distributed?

10) The RBI prints currency notes in denominations of One Rupee, Two Rupees, Five Rupees, Ten Rupees, Twenty Rupees, Fifty Rupees, and One hundred rupees. In how many ways can it display 10 currency notes, not necessarily of different denominations? How many of these will have all denominations?

11) In how many ways can an employer distribute INR 100/- as Holiday Bonus to his 5 employees? No fraction of a rupee is allowed. Also, do not worry about question of equity and fairness!

12) The results of 20 chess games (win, lose, or draw) have to be predicted. How many different forecasts can contain exactly 15 correct results?

13) How many distinct results can we obtain from one throw of four dice? five dice? Can you generalize this?

14) In how many ways can 8 rooks be placed on a standard chess board so that no rook can attack another? How many if the rooks are labelled? How would the answer be modified if we remove the restriction that “no rook can attack another”?

15) Show that there are 7 partitions of the integer 5, and 33 partitions of the integer 9. How many of these have 4 parts ? How many have the largest part equal to 4? Experiment with other partitions and other numbers.

Cheers,
Nalin Pithwa.

# RMO and Pre RMO Geometry Tutorial Worksheet 1: Based on Geometric Refresher

1) Show that quadrilateral ABCD can be inscribed in a circle iff $\angle B$ and $\angle D$ are supplementary.

2) Prove that a parallelogram having perpendicular diagonals is a rhombus.

3) Prove that a parallelogram with equal diagonals is a rectangle.

4) Show that the diagonals of an isosceles trapezoid are equal.

5) A straight line cuts two concentric circles in points A, B, C and D in that order. AE and BF are parallel chords, one in each circle. If CG is perpendicular to BF and DH is perpendicular to AE, prove that $GF = HE$.

6) Construct triangle ABC, given angle A, side AC and the radius r of the inscribed circle. Justify your construction.

7) Let a triangle ABC be right angled at C. The internal bisectors of angle A and angle B meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.

8) Three circles $C_{1}, C_{2}, C_{3}$ with radii $r_{1}, r_{2}, r_{3}$, with $r_{1}. They are placed such that $C_{2}$ lies to the right of $C_{1}$ and touches it externally; $C_{3}$ lies to the right of $C_{2}$ and touches it externally. Further, there exist two straight lines each of which is a direct common tangent simultaneously to all the three circles. Find $r_{2}$ in terms of $r_{1}$ and $r_{3}$.

Cheers,

Nalin Pithwa

# Basic Geometry: Refresher for pre-RMO, RMO and IITJEE foundation maths

1) Assume that in a $\bigtriangleup$ ABC and $\bigtriangleup RST$, we know that $AB=RS$, $AC=RT$, $BC=ST$. Prove that $\bigtriangleup ABC \cong \bigtriangleup RST$ without using the SSS congruence criterion.

2) Let $\bigtriangleup ABC$ be isosceles with base BC. Then, $\angle B = \angle C$. Also, the median from vertex A, the bisector of $\angle A$, and the altitude from vertex A are all the same line. Prove this.

3) If two triangles have equal hypotenuses and an arm of one of the triangles equals an arm of the other, then the triangles are congruent. Prove.

4) An exterior angle of a triangle equals the sum of the two remote interior angles. Also, the sum of all three interior angles of a triangle is 180 degrees.

5) Find a formula for the interior angles of an n-gon.

6) Prove that the opposite sides of a parallelogram are equal.

7) In a quadrilateral ABCD, suppose that AB=CD and AD=BC. Then, prove that ABCD is a parallelogram.

8) In a quadrilateral ABCD, suppose that AB=CD and AB is parallel to CD. Then, prove that ABCD is a parallelogram.

9) Prove that a quadrilateral is a parallelogram iff its diagonals bisect each other.

10) Given a line segment BC, the locus of all points equidistant from B and C is the perpendicular bisector of the segment. Prove.

11) Corollary to problem 10 above: The diagonals of a rhombus are perpendicular. Prove.

12) Let AX be the bisector of $\angle A$ in $\triangle ABC$. Then, prove $\frac{BX}{XC} = \frac{AB}{AC}$. In other words, X divides BC into pieces proportional to the lengths of the nearer sides of the triangle. Prove.

13) Suppose that in $\triangle ABC$, the median from vertex A and the bisector of $\angle A$ are the same line. Show that $AB=AC$.

14) Prove that there is exactly one circle through any three given non collinear points.

15) An inscribed angle in a circle is equal in degrees to one half its subtended arc. Equivalently, the arc subtended by an inscribed angle is measured by twice the angle. Prove.

16) Corollary to above problem 15: Opposite angles of an inscribed quadrilateral are supplementary. Prove this.

17) Another corollary to above problem 15: The angle between two secants drawn to a circle from an exterior point is equal in degrees to half the difference of the two subtended arcs. Prove this.

18) A third corollary to above problem 15: The angle between two chords that intersect in the interior of a circle is equal in degrees to half the sum of the two subtended arcs. Prove this.

19) Theorem (Pythagoras): If a right triangle has arms of lengths a and b and its hypotenuse has length c, then $a^{2}+b^{2}=c^{2}$. Prove this.

20) Corollary to above theorem: Given a triangle ABC, the angle at vertex C is a right angle iff side AB is a diameter of the circumcircle. Prove this.

21) Theorem: The angle between a chord and the tangent at one of its endpoints is equal in degrees to half the subtended arc. Prove.

22) Corollary to problem 21: The angle between a secant and a tangent meeting at a point outside a circle is equal in degrees to half the difference of the subtended arcs.

23) Fix an integer, $n \geq 3$. Given a circle, how should n points on this circle be chosen so as to maximize the area of the corresponding n-gon?

24) Theorem: Given $\bigtriangleup ABC$ and $\bigtriangleup XYZ$, suppose that $\angle A = \angle X$ and $\angle B= \angle Y$. Then, prove that $\angle C = \angle Z$ and so $\bigtriangleup ABC \sim \bigtriangleup XYZ$. Prove this theorem.

25) Theorem: If $\bigtriangleup ABC \sim \bigtriangleup XYZ$, then the lengths of the corresponding sides of these two triangles are proportional. Prove.

26) The following lemma is important to prove the above theorem: Let U and V be points on sides AB and AC of $\bigtriangleup ABC$. Then, UV is parallel to BC if and only if $\frac{AU}{AB} = \frac{AV}{AC}$. You will have to prove this lemma as a part of the above proof.

27) Special case of above lemma: Let U and V be the midpoints of sides AB and AC, respectively in $\bigtriangleup ABC$. Then, UV is parallel to BC and $UV = \frac{1}{2}BC$.

28) Suppose that the sides of $\bigtriangleup ABC$ are proportional to the corresponding sides of $\bigtriangleup XYZ$. Then, $\bigtriangleup ABC \sim \bigtriangleup XYZ$.

29) Given $\bigtriangleup ABC$ and $\bigtriangleup XYZ$, assume that $\angle X = \angle A$ and that $\frac{XY}{AB} = \frac{XZ}{AC}$. Then, $\bigtriangleup ABC \sim \bigtriangleup XYZ$.

30) Consider a non-trivial plane geometry question now: Let P be a point outside of parallelogram ABCD and $\angle PAB = \angle PCB$. Prove that $\angle APD = \angle CPB$.

31) Given a circle and a point P not on the circle, choose an arbitrary line through P, meeting the circle at points X and Y. Then, the quantity $PX.PY$ depends only on the point P and is independent of the choice of the line through P.

32) You can given an alternative proof of Pythagoras’s theorem based on the following lemma: Suppose $\bigtriangleup ABC$ is a right triangle with hypotenuse AB and let CP be the altitude drawn to the hypotenuse. Then, $\bigtriangleup ACP \sim \bigtriangleup ABC \sim \bigtriangleup CBP$. Prove both the lemma and based on it produce an alternative proof of Pythagorean theorem.

33) Prove the following: The three perpendicular bisectors of the sides of a triangle are concurrent at the circumcenter of the triangle.

34) Prove the law of sines.

35) Let R and K denote the circumradius and area of $\bigtriangleup ABC$, respectively and let a, b and c denote the side lengths, as usual. Then, $4KR = abc$.

36) Theorem: The three medians of an arbitrary triangle are concurrent at a point that lies two thirds of the way along each median from the vertex of the triangle toward the midpoint of the opposite side.

37) Time to ponder: Prove: Suppose that in $\bigtriangleup ABC$, medians BY and CZ have equal lengths. Prove that $AB=AC$.

38) If the circumcenter and the centroid of a triangle coincide, then the triangle must be equilateral. Prove this fact.

39) Assume that $\bigtriangleup ABC$ is not equilateral and let G and O be its centroid and circumcentre respectively. Let H be the point on the Euler line GO that lies on the opposite side of G from O and such that $HG = 2GO$. Then, prove that all the three altitudes of $\bigtriangleup ABC$ pass through H.

40) Prove the following basic fact about pedal triangles: The pedal triangles of each of the four triangles determined by an orthic quadruple are all the same.

41) Prove the following theorem: Given any triangle, all of the following points lie on a common circle: the three feet of the altitudes, the three midpoints of the sides, and the three Euler points. Furthermore, each of the line segments joining an Euler point to the midpoint of the opposite side is a diameter of this circle.

42) Prove the following theorem and its corollary: Let R be the circumradius of triangle ABC. Then, the distance from each Euler point of $\bigtriangleup ABC$ to the midpoint of the opposite side is R, and the radius of the nine-point circle of $\bigtriangleup ABC$ is $R/2$. The corollary says: Suppose $\bigtriangleup ABC$ is not a right angled triangle and let H be its orthocentre. Then, $\bigtriangleup ABC$, $\bigtriangleup HBC$, $\bigtriangleup AHC$, and $\bigtriangleup ABH$ have equal circumradii.

43) Prove the law of cosines.

44) Prove Heron’s formula.

45) Express the circumradius R of $\bigtriangleup ABC$ in terms of the lengths of the sides.

46) Prove that the three angle bisectors of a triangle are concurrent at a point I, equidistant from the sides of the triangle. If we denote the by r the distance from I to each of the sides, then the circle of radius r centered at I is the unique circle inscribed in the given triangle. Note that in order to prove this, the following elementary lemma is required to be proved: The bisector of angle ABC is the locus of points P in the interior of the angle that are equidistant from the sides of the triangle.

47) Given a triangle with area K, semiperimeter s, and inradius r, prove that $rs=K$. Use this to express r in terms of the lengths of the sides of the triangle.

Please be aware that the above set of questions is almost like almost like a necessary set of pre-requisites for RMO geometry. You have to master the basics first.

Regards,

Nalin Pithwa.

# Eight digit bank identification number and other problems of elementary number theory

Question 1:

Consider the eight-digit bank identification number $a_{1}a_{2}\ldots a_{8}$, which is followed by a ninth check digit $a_{9}$ chosen to satisfy the congruence

$a_{9} \equiv 7a_{1} + 3a_{2} + 9a_{3} + 7a_{4} + 3a_{5} + 9a_{6} + 7a_{7} + 3a_{8} {\pmod {10}}$

(a) Obtain the check digits that should be appended to the two numbers 55382006 and 81372439.

(b) The bank identification number $237a_{4}18538$ has an illegitimate fourth digit. Determine the value of the obscured digit.

Question 2:

(a) Find an integer having the remainders 1,2,5,5 when divided by 2, 3, 6, 12 respectively (Yih-hing, died 717)

(b) Find an integer having the remainders 2,3,4,5 when divided by 3,4,5,6 respectively (Bhaskara, born 1114)

(c) Find an integer having remainders 3,11,15 when divided by 10, 13, 17, respectively (Regiomontanus, 1436-1476)

Question 3:

Question 3:

Let $t_{n}$ denote the nth triangular number. For which values of n does $t_{n}$ divide $t_{1}^{2} + t_{2}^{2} + \ldots + t_{n}^{2}$

Hint: Because $t_{1}^{2}+t_{2}^{2}+ \ldots + t_{n}^{2} = t_{n}(3n^{3}+12n^{2}+13n+2)/30$, it suffices to determine those n satisfying $3n^{3}+12n^{2}+13n+2 \equiv 0 {\pmod {2.3.5}}$

Question 4:

Find the solutions of the system of congruences:

$3x + 4y \equiv 5 {\pmod {13}}$
$2x + 5y \equiv 7 {\pmod {13}}$

Question 5:

Obtain the two incongruent solutions modulo 210 of the system

$2x \equiv 3 {\pmod 5}$
$4x \equiv 2 {\pmod 6}$
$3x \equiv 2 {\pmod 7}$

Question 6:

Use Fermat’s Little Theorem to verify that 17 divides $11^{104}+1$

Question 7:

(a) If $gcd(a,35)=1$, show that $a^{12} \equiv {\pmod {35}}$. Hint: From Fermat’s Little Theorem, $a^{6} \equiv 1 {\pmod 7}$ and $a^{4} \equiv 1 {\pmod 5}$

(b) If $gcd(a,42) =1$, show that $168=3.7.8$ divides $a^{6}-1$
(c) If $gcd(a,133)=gcd(b,133)=1$, show that $133| a^{18} - b^{18}$

Question 8:

Show that $561|2^{561}-1$ and $561|3^{561}-3$. Do there exist infinitely many composite numbers n with the property that $n|2^{n}-2$ and $n|3^{n}-3$?

Question 9:

Prove that any integer of the form $n = (6k+1)(12k+1)(18k+1)$ is an absolute pseudoprime if all three factors are prime; hence, $1729=7.13.19$ is an absolute pseudoprime.

Question 10:

Prove that the quadratic congruence $x^{2}+1 \equiv 0 {\pmod p}$, where p is an odd prime, has a solution if and only if $p \equiv {pmod 4}$.

Note: By quadratic congruence is meant a congruence of the form $ax^{2}+bx+c \equiv 0 {\pmod n}$ with $a \equiv 0 {\pmod n}$. This is the content of the above proof.

More later,
Nalin Pithwa.