rl-actors/Big-Math-RL-Verified-Subset huggingface dataset:
Problem: The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. 
Answer: 7
Problem: Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.
Answer: \sqrt{2}
Problem: Find all solutions  $x,y \in \mathbb{Z}$ ,  $x,y \geq 0$ , to the equation \[ 1 + 3^x = 2^y. \]
Answer: (x, y) = (0, 1) and (x, y) = (1, 2)
Problem: Let  $ P,\ Q,\ R$  be the points such that  $ BP: PC=CQ: QA=AR: RB=t: 1-t\ (0<t<1)$  for a triangle  $ ABC$ .
Denote  $ K$  by the area of the triangle with segments  $ AP,\ BQ,\ CR$  as side lengths and  $ L$  by triangle  $ ABC$ , find  $ \frac{K}{L}$  in terms of  $ t$ .
Answer: (1-t+t^2)^2
Problem: Let  $ABCDE$  be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$  and  $AB=BC=CD=DE$ . Find angle  $\angle DAE$ .
Answer: 30^	ext{o}
Problem: Let   $B_n$  be the set of all sequences of length  $n$ , consisting of zeros and ones. For every two sequences   $a,b \in B_n$  (not necessarily different) we define strings   $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$  and  $\delta_0\delta_1\delta_2 \dots \delta_n$  such that  $\varepsilon_0=\delta_0=0$  and   $$  \varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).  $$  . Let  $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$  . Find   $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $ .
. 
Answer: n \cdot 4^{n-1}
Problem: Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of  $\$ 370 $. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $ \ $180$ . On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?
Answer: 300
Problem: A *festive* number is a four-digit integer containing one of each of the digits  $0, 1, 2$ , and  $4$  in its decimal representation. How many festive numbers are there?
Answer: 18
Problem: A function  $f$  is defined on the complex numbers by  $f(z)=(a+bi)z,$  where  $a$  and  $b$  are positive numbers.  This function has the property that the image of each point in the complex plane is equidistant from that point and the origin.  Given that  $|a+bi|=8$  and that  $b^2=m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$ 
Answer: 259
Problem: Define a  $good~word$  as a sequence of letters that consists only of the letters  $A,$   $B,$  and  $C$   $-$  some of these letters may not appear in the sequence  $-$  and in which  $A$  is never immediately followed by  $B,$   $B$  is never immediately followed by  $C,$  and  $C$  is never immediately followed by  $A.$  How many seven-letter good words are there?
Answer: 192
Problem: Integers  $x_1,x_2,\cdots,x_{100}$  satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find  $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $ .
Answer: 0
Problem: How many positive integers which are less or equal with  $2013$  such that  $3$  or  $5$  divide the number.
Answer: 939
Problem: Find all polynomials  $f(x)$  with integer coefficients and leading coefficient equal to 1, for which  $f(0)=2010$  and for each irrational  $x$ ,  $f(x)$  is also irrational.
Answer: f(x) = x + 2010
Problem: Find the smallest positive integer  $n$  that is divisible by  $100$  and has exactly  $100$  divisors.
Answer: 162000
Problem: How many real numbers  $x$  are solutions to the equation  $|x - 2| - 4 =\frac{1}{|x - 3|}$  ?
Answer: 2
Problem: For any positive integers  $n$ , find all  $n$ -tuples of complex numbers  $(a_1,..., a_n)$  satisfying   $$ (x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n. $$  Proposed by USJL.
Answer:  a_1 = a_2 = \cdots = a_n
Problem: Evaluate the sum \[ 11^2 - 1^1 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2. \]
Answer: 2100
Problem: Point  $P_1$  is located  $600$  miles West of point  $P_2$ . At  $7:00\text{AM}$  a car departs from  $P_1$  and drives East at a speed of  $50$ mph. At  $8:00\text{AM}$  another car departs from  $P_2$  and drives West at a constant speed of  $x$  miles per hour. If the cars meet each other exactly halfway between  $P_1$  and  $P_2$ , what is the value of  $x$ ?
Answer: 60
Problem: Find the root that the following three polynomials have in common:
\begin{align*} & x^3+41x^2-49x-2009 
& x^3 + 5x^2-49x-245 
& x^3 + 39x^2 - 117x - 1435\end{align*}
Answer: 7
Problem: We are given a  $4\times4$  square, consisting of  $16$  squares with side length of  $1$ . Every  $1\times1$  square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals  $5$ . What is the greatest number of different numbers that can be used to cover the square?


Answer: 3
Problem:  $101$  numbers are written on a blackboard:  $1^2, 2^2, 3^2, \cdots, 101^2$ . Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.
Answer: 1
Problem: The natural numbers from  $1$  to  $50$  are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
Answer: 25
Problem: Find all functions  $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$  such that for all  $x, y\in\mathbb{R}^+$ , 
\[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\]
( $\mathbb{R}^+$  denotes the set of positive real numbers.)

*(Proposed by Ivan Chan Guan Yu)*
Answer: f(x) = \frac{C}{x^2}
Problem: A square is divided into  $25$  unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?
Answer: 12
Problem: Suppose there are  $n$  points in a plane no three of which are collinear with the property that if we label these points as  $A_1,A_2,\ldots,A_n$  in any way whatsoever, the broken line  $A_1A_2\ldots A_n$  does not intersect itself. Find the maximum value of  $n$ .
Answer: 3
Problem: Let  $p$  be a prime number.  $T(x)$  is a polynomial with integer coefficients and degree from the set  $\{0,1,...,p-1\}$  and such that  $T(n) \equiv T(m)  (mod p)$  for some integers m and n implies that  $ m \equiv n (mod p)$ . Determine the maximum possible value of degree of  $T(x)$ 
Answer: p-2
Problem: Determine all triples  $(x, y, z)$  of positive integers satisfying  $x | (y + 1)$ ,  $y | (z + 1)$  and  $z | (x + 1)$ .
Answer:  (1, 1, 1), (1, 2, 1), (1, 1, 2), (1, 3, 2), (2, 1, 1), (2, 1, 3), (3, 1, 2), (2, 2, 2), (2, 2, 3), (2, 3, 2), (3, 2, 2)
Problem: Let  $s_1, s_2, \dots$  be an arithmetic progression of positive integers. Suppose that
\[ s_{s_1} = x+2, \quad s_{s_2} = x^2+18, \quad\text{and}\quad s_{s_3} = 2x^2+18. \]
Determine the value of  $x$ .
Answer: 4
Problem: Let  $ABCD$  be a cyclic quadrilateral such that the lines  $AB$  and  $CD$  intersects in  $K$ , let  $M$  and  $N$  be the midpoints of  $AC$  and  $CK$  respectively. Find the possible value(s) of  $\angle ADC$  if the quadrilateral  $MBND$  is cyclic.
Answer: 90^\circ
Problem: Boeotia is comprised of  $3$  islands which are home to  $2019$  towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country
Answer: \left(\frac{2019}{3}\right)^2
Problem: In a triangle  $ABC$ , the lengths of the sides are consecutive integers and median drawn from  $A$  is perpendicular to the bisector drawn from  $B$ . Find the lengths of the sides of triangle  $ABC$ .
Answer: 2, 3, 4
Problem: Determine all functions  $f: \mathbb{R} \to \mathbb{R}$  such that   $$  f(x^3) + f(y)^3 + f(z)^3 = 3xyz  $$  for all real numbers  $x$ ,  $y$  and  $z$  with  $x+y+z=0$ .
Answer: f(x) = x
Problem: Find all pairs  $(x,y)$  with  $x,y$  real, satisfying the equations   $$ \sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1 $$  
Answer: \left(\frac{1}{2}, -\frac{1}{2}\right), \left(-\frac{1}{2}, \frac{1}{2}\right)
Problem: Let  $ABC$  and  $DEF$  be two triangles, such that  $AB=DE=20$ ,  $BC=EF=13$ , and  $\angle A = \angle D$ . If  $AC-DF=10$ , determine the area of  $\triangle ABC$ .
Answer: 178.5
Problem: An integer is between  $0$  and  $999999$  (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be  $19$ ?
Answer: \frac{7623}{250000}
Problem: Find all pairs of prime numbers  $p, q$  such that the number  $pq + p - 6$  is also prime.
Answer:  (2, 3) and (3, 2)
Problem: Evan writes a random positive integer on a board: the integer  $k$  has probability  $2^{-k}$  of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote  $5,8,2,3,6,10,2$  in that order then his two sequences would be  $5,8,2,3,6,10$  and  $2,3,5,6,8,10.$ Find the probability that for all  $k \in \{ 1,4,34 \},$  that  $k$  was written, and  $k$  appears in the same position in both sequences.
Answer: 2^{-39}
Problem: Sherlock and Mycroft are playing Battleship on a  $4\times4$  grid. Mycroft hides a single  $3\times1$  cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
Answer: 4
Problem: Let  $ABCD$  be a rectangle with area  $1$ , and let  $E$  lie on side  $CD$ . What is the area of the triangle formed by the centroids of triangles  $ABE$ ,  $BCE$ , and  $ADE$ ?
Answer: \frac{1}{18}
Problem: For each positive integer  $n$ , let  $s(n)$  be the sum of the squares of the digits of  $n$ . For example,  $s(15)=1^2+5^2=26$ . Determine all integers  $n\geq 1$  such that  $s(n)=n$ .
Answer: 1
Problem: If  $x$ ,  $y$ ,  $z$  are nonnegative integers satisfying the equation below, then compute  $x+y+z$ .
\[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\]
Answer: 6
Problem: A non-equilateral triangle  $\triangle ABC$   of perimeter  $12$  is inscribed in circle  $\omega$  .Points  $P$  and  $Q$  are arc midpoints of arcs  $ABC$  and  $ACB$  , respectively. Tangent to  $\omega$  at  $A$  intersects line  $PQ$  at  $R$ .
It turns out that the midpoint of segment  $AR$  lies on line  $BC$  . Find the length of the segment  $BC$ .
Answer: 4
Problem: Let  $ n$  be a natural number. A cube of edge  $ n$  may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for  $ n$ .
Answer:  n = 13
Problem: Consider a curve  $C$  on the  $x$ - $y$  plane expressed by  $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$ .
 For a constant  $t>0$ , let the line  $l$  pass through the point  $P(t,\ 0)$  and is perpendicular to the  $x$ -axis,intersects with the curve  $C$  at  $Q$ . Denote by  $S_1$  the area of the figure bounded by the curve  $C$ , the  $x$ -axis, the  $y$ -axis and the line  $l$ , and denote by  $S_2$  the area of  $\triangle{OPQ}$ . Find  $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$ 
Answer: \frac{1}{2}
Problem: Let  $n$  be a positive integer. There are  $n$  soldiers stationed on the  $n$ th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of  $n$ , required to eliminate all the soldiers?
Answer: \lceil \log_2(n) \rceil
Problem: Andrew flips a fair coin  $5$  times, and counts the number of heads that appear. Beth flips a fair coin  $6$  times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
Answer: 0.5
Problem: An equilateral triangle of side  $n$  is divided into equilateral triangles of side  $1$ .  Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
Answer: n(n+1)
Problem: Consider  $\triangle \natural\flat\sharp$ .  Let  $\flat\sharp$ ,  $\sharp\natural$  and  $\natural\flat$  be the answers to problems  $4$ ,  $5$ , and  $6$ , respectively.  If the incircle of  $\triangle \natural\flat\sharp$  touches  $\natural\flat$  at  $\odot$ , find  $\flat\odot$ .
Answer: 2.5
Problem:  $ABCDE$  is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments  $AC$  and  $BD$ ?
Answer: 72^	extcirc}
Problem: In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?
Answer: 72
Problem: Find the largest value of  $a^b$  such that the positive integers  $a,b>1$  satisfy   $$ a^bb^a+a^b+b^a=5329 $$  
Answer: 64
Problem: Find the sum of all prime numbers  $p$  which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct)  $a$ ,  $b$  and  $c$ .
Answer: 719
Problem: Three palaces, each rotating on a duck leg, make a full round in  $30$ ,  $50$ , and  $70$  days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?
Answer: 525
Problem: Determine the smallest possible value of the expression   $$ \frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a} $$   where  $a,b,c \in \mathbb{R}$  satisfy  $a+b+c = -1$  and  $abc \leqslant -3$ 
Answer: 3
Problem: Let  $\operatorname{cif}(x)$  denote the sum of the digits of the number  $x$  in the decimal system. Put  $a_1=1997^{1996^{1997}}$ , and  $a_{n+1}=\operatorname{cif}(a_n)$  for every  $n>0$ . Find  $\lim_{n\to\infty}a_n$ .
Answer: 1
Problem: Solve the following equation for real values of  $x$ :

    \[
    2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
    \]
Answer: x = 0 \text{ or } 1
Problem: Let be a real number  $ a\in \left[ 2+\sqrt 2,4 \right] . $  Find  $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $ 
Answer: 1
Problem: Consider all  $6$ -digit numbers of the form  $abccba$  where  $b$  is odd. Determine the number of all such  $6$ -digit numbers that are divisible by  $7$ .
Answer: 70
Problem: Let  $a_{10} = 10$ , and for each integer  $n >10$  let  $a_n = 100a_{n - 1} + n$ . Find the least  $n > 10$  such that  $a_n$  is a multiple of  $99$ .
Answer: 45
Problem: While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers  $a$ ,  $b$ , and  $c$ , and recalled that their product is  $24$ , but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than  $25$  with fewer than  $6$  divisors. Help James by computing  $a+b+c$ .
Answer: 10
Problem: Let  $a$  and  $b$  be given positive real numbers, with  $a<b.$  If two points are selected at random from a straight line segment of length  $b,$  what is the probability that the distance between them is at least  $a?$ 
Answer: \frac{(b - a)^2}{b^2}
Problem: Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in *minutes*, between consecutive runners' finishes?
Answer: 30
Problem: Find  $f(2)$  given that  $f$  is a real-valued function that satisfies the equation  $$ 4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1. $$  
Answer: \frac{19}{12}
Problem: To any triangle with side lengths  $a,b,c$  and the corresponding angles  $\alpha, \beta, \gamma$  (measured in radians), the 6-tuple  $(a,b,c,\alpha, \beta, \gamma)$  is assigned. Find the minimum possible number  $n$  of distinct terms in the 6-tuple assigned to a scalene triangle.

Answer: 4
Problem: Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
Answer: 2
Problem: Let n be a non-negative integer. Define the *decimal digit product* \(D(n)\) inductively as follows:

- If \(n\) has a single decimal digit, then let \(D(n) = n\).

- Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\).

Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0.

Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\).
Answer: 0
Problem: Suppose that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .
Answer: 2022
Problem: Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs  $3$  RM ,each mango costs  $4$  RM , and each papaya costs  $5$  RM . He bought at least one of each fruit, and paid exactly  $50$  RM. What is the maximum number of fruits that he could have bought?
Answer: 15
Problem: Path graph  $U$  is given, and a blindfolded player is standing on one of its vertices. The vertices of  $U$  are labeled with positive integers between 1 and  $n$ , not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after  $k$  steps are the  $k$  degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?
Answer:  n
Problem: Consider  $13$  marbles that are labeled with positive integers such that the product of all  $13$  integers is  $360$ . Moor randomly picks up  $5$  marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
Answer: 24
Problem: In  $10\times 10$  square we choose  $n$  cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of  $n$ ?
Answer: 50
Problem: Determine all positive integers  $n$  so that both  $20n$  and  $5n + 275$  are perfect squares.

(A perfect square is a number which can be expressed as  $k^2$ , where  $k$  is an integer.)
Answer: 125
Problem: The sum  $$ \sum_{n=0}^{2016\cdot2017^2}2018^n $$  can be represented uniquely in the form  $\sum_{i=0}^{\infty}a_i\cdot2017^i$  for nonnegative integers  $a_i$  less than  $2017$ . Compute  $a_0+a_1$ .
Answer: 0
Problem: Find all quadruples  $(a, b, c, d)$  of non-negative integers such that  $ab =2(1 + cd)$  and there exists a non-degenerate triangle with sides of length  $a - c$ ,  $b - d$ , and  $c + d$ .
Answer:  (1, 2, 0, 1) or (2, 1, 1, 0)
Problem: Let  $O$  be the center of the circle  $\omega$  circumscribed around the acute-angled triangle  $\vartriangle ABC$ , and  $W$  be the midpoint of the arc  $BC$  of the circle   $\omega$ , which does not contain the point  $A$ , and  $H$  be the point of intersection of the heights of the triangle   $\vartriangle ABC$ . Find the angle  $\angle BAC$ , if  $WO = WH$ .
Answer: \angle BAC = 60^\circ
Problem: Let  $q(n)$  denote the sum of the digits of a natural number  $n$ . Determine  $q(q(q(2000^{2000})))$ .
Answer: 4
Problem: Let  $(F_n)_{n\in{N^*}}$  be the Fibonacci sequence defined by $F_1=1$ ,  $F_2=1$ ,  $F_{n+1}=F_n+F_{n-1}$  for every  $n\geq{2}$ . Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]
Answer: 2
Problem: Angle bisectors from vertices  $B$  and  $C$  and the perpendicular bisector of side  $BC$  are drawn in a non-isosceles triangle  $ABC$ . Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.

(Yu. Blinkov) 
Answer: The original triangle ABC is restored.
Problem: Find the largest positive integer  $k{}$  for which there exists a convex polyhedron  $\mathcal{P}$  with 2022 edges, which satisfies the following properties:


- The degrees of the vertices of  $\mathcal{P}$  don’t differ by more than one, and
- It is possible to colour the edges of  $\mathcal{P}$  with  $k{}$  colours such that for every colour  $c{}$ , and every pair of vertices  $(v_1, v_2)$  of  $\mathcal{P}$ , there is a monochromatic path between  $v_1$  and  $v_2$  in the colour  $c{}$ .
Answer: k = 2
Problem: A natural number is a *palindrome* when one obtains the same number when writing its digits in reverse order. For example,  $481184$ ,  $131$  and  $2$  are palindromes.

Determine all pairs  $(m,n)$  of positive integers such that  $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$  is a palindrome.
Answer:  (m, n) such that m = n and m, n \leq 9
Problem: Find prime numbers  $p , q , r$   such that  $p+q^2+r^3=200$ . Give all the possibilities.

Remember that the number  $1$  is not prime.
Answer:  (167, 5, 2), (71, 11, 2), (23, 13, 2), (71, 2, 5)
Problem: Let  $x$  be a number such that  $x +\frac{1}{x}=-1$ . Determine the value of   $x^{1994} +\frac{1}{x^{1994}}$ .
Answer: -1
Problem: Introduce a standard scalar product in  $\mathbb{R}^4.$  Let  $V$  be a partial vector space in  $\mathbb{R}^4$  produced by  $\left(
\begin{array}{c}
1  
-1  
-1  
1 
\end{array}
\right),\left(
 \begin{array}{c}
1  -1  
1  
-1 
\end{array}
\right).$ Find a pair of base of orthogonal complement  $W$  for  $V$  in  $\mathbb{R}^4.$ 
Answer:  \left\{ \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} \right\}
Problem: Find all ordered pairs  $ (m,n)$  where  $ m$  and  $ n$  are positive integers such that  $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$  is an integer.
Answer: (1, 2), (1, 3), (2, 1), (2, 2), (2, 5), (3, 1), (3, 5), (5, 2), (5, 3)