Navvye/Olympiad-Problems-CSV huggingface dataset:
1973 All Soviet Union Mathematical Olympiad
Fourteen coins are submitted to the judge. An expert knows, that the coins from number one to seven are false, and from $8$ to $14$ -- normal. The judge is sure only that all the true coins have the same weight and all the false coins weights equal each other, but are less then the weight of the true coins. The expert has the scales without weights.

a) The expert wants to prove, that the coins $1--7$ are false. How can he do it in three weighings?

b) How can he prove, that the coins $1--7$ are false and the coins $8--14$ are true in three weighings?
1973 All Soviet Union Mathematical Olympiad
Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.
1973 All Soviet Union Mathematical Olympiad
Given $n$ points, $n > 4$. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)
1973 All Soviet Union Mathematical Olympiad
Given an angle with the vertex $O$ and a circle touching its sides in the points $A$ and $B$. A ray is drawn from the point $A$ parallel to $[OB)$. It intersects with the circumference in the point $C$. The segment $[OC]$ intersects the circumference in the point $E$. The straight lines $(AE)$ and $(OB)$ intersect in the point $K$.  Prove that $|OK| = |KB|$.
1973 All Soviet Union Mathematical Olympiad
The real numbers $a,b,c$ satisfy the condition:

for all $x$, such that for $ -1 \le x \le 1$, the inequality$| ax^2 + bx + c | \le 1$ is held.

Prove that for the same $x$ , $| cx^2 + bx + a |  \le 2$.
1973 All Soviet Union Mathematical Olympiad
The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less.

What is the greatest possible number of the winner?
1973 All Soviet Union Mathematical Olympiad
The square polynomial $f(x)=ax^2+bx+c$ is of such a sort, that the equation $f(x)=x$ does not have real roots. Prove that the equation $f(f(x))=0$ does not have real roots also.
1973 All Soviet Union Mathematical Olympiad
$n$ squares of the infinite cross-lined sheet of paper are painted with black colour (others are white). Every move all the squares of the sheet change their colour simultaneously. The square gets the colour, that had the majority of three ones: the square itself, its neighbour from the right side and its neighbour from the upper side.

a) Prove that after the finite number of the moves all the black squares will disappear.

b) Prove that it will happen not later than on the $n$-th move
1973 All Soviet Union Mathematical Olympiad
Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are built on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.)

a) Prove that the circumferences outscribed around the outer triangles intersect in one point.

b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point
1973 All Soviet Union Mathematical Olympiad
$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$.
1973 All Soviet Union Mathematical Olympiad
The king have revised the chess-board $8\times 8$ having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared.

a) Give an example that the king could make $28$ steps parallel the sides of the board only.

b) Prove that he could not make less than $28$ such a steps.

c) What is the maximal and minimal length of the broken line if the side of a field is $1$?
1973 All Soviet Union Mathematical Olympiad
Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge  b \ge  c$). Prove that $b^2 \ge 2$.
1973 All Soviet Union Mathematical Olympiad
Given a convex $n$-angle with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-angle.
1973 All Soviet Union Mathematical Olympiad
Prove that for every positive $x_1, x_2, x_3, x_4, x_5$ holds inequality:

$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge  4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)$.
1973 All Soviet Union Mathematical Olympiad
Given $4$ points in three-dimensional space, not lying in one plane.

What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?
1972 All Soviet Union Mathematical Olympiad
Given a rectangle $ABCD$, points $M$ -- the middle of $[AD]$ side, $N$ -- the middle of $[BC]$ side. Let us take a point $P$ on the continuation of the $[DC]$ segment over the point $D$. Let us denote the point of intersection of lines $(PM)$ and $(AC)$ as $Q$. Prove that the angles $QNM$ and $MNP$ are equal.
1972 All Soviet Union Mathematical Olympiad
Given $50$ segments on the line. Prove that one of the following statements is valid:

1. Some $8$ segments have the common point.

2. Some $8$ segments do not intersect each other.
1972 All Soviet Union Mathematical Olympiad
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
1972 All Soviet Union Mathematical Olympiad
a) Let $a,n,m$ be natural numbers, $a > 1$.

Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$.

b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime.

Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.
1972 All Soviet Union Mathematical Olympiad
The triangle table is built according to the rule: You put the natural number $a>1$ in the upper row, and then you write under the number $k$ from the left side $k^2$, and from the right side -- $(k+1)$. For example, if $a = 2$, you get the table on the picture. Prove that all the numbers on each particular line are different.



2

/ \
/     \
4       3

/  \     /  \
16   5   9    4

/ \  / \  /\  / \
1972 All Soviet Union Mathematical Olympiad
Given several squares with the total area $1$.

Prove that you can pose them in the square of the area $2$ without any intersections.
1972 All Soviet Union Mathematical Olympiad
Let $O$ be the intersection point of the of the convex quadrangle $ABCD$ diagonals.

Prove that the line drawn through the points of intersection of the medians of $AOB$ and $COD$ triangles is orthogonal to the line drawn through the points of intersection of the heights of $BOC$ and $AOD$ triangles.
1972 All Soviet Union Mathematical Olympiad
Each of the $9$ straight lines divides the given square onto two quadrangles with the areas related as $2:3$.

Prove that there exist three of them intersecting in one point
1972 All Soviet Union Mathematical Olympiad
The $7$-angle $A_1A_2A_3A_4A_5A_6A_7$ is inscribed in a circle.

Prove that if the centre of the circle is inside the $7$-angle, than the sum of $A_1,A_2$ and $A_3$ angles is less than $450$ degrees.
1972 All Soviet Union Mathematical Olympiad
A game for two.

One gives a digit and the second substitutes it instead of a star in the following difference:

$****  -  **** =   $

Then the first gives the next digit, and so on $8$ times.

The first wants to obtain the greatest possible difference, the second -- the least. Prove that:

1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour.

2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.
1972 All Soviet Union Mathematical Olympiad
Let $x,y$ be positive numbers, $s$ -- the least of $\{ x, (y+ 1/x), 1/y\}$.

What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?
1972 All Soviet Union Mathematical Olympiad
The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices.

Prove that $O$ is the centre of the outscribed circle.
1972 All Soviet Union Mathematical Olympiad
Is it possible to put the numbers $0,1$ or $2$ in the unit squares of the cross-lined paper $100x100$ in such a way, that every rectangle $3x4$ (and $4x3$) would contain three zeros, four ones and five twos?
1972 All Soviet Union Mathematical Olympiad
Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$.

Let $s$ be the greatest of the numbers $\big\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\big\}$.

What is the minimal possible $s$? What $x_i $correspond it?
1972 All Soviet Union Mathematical Olympiad
One-round hockey tournament is finished (each plays with each one time, the winner gets $2$ points, looser -- $0$, and $1$ point for draw). For arbitrary subgroup of teams there exists a team (may be from that subgroup) that has got an odd number of points in the games with the teams of the subgroup. Prove that there was even number of the participants.
1971 All Soviet Union Mathematical Olympiad
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).
1971 All Soviet Union Mathematical Olympiad
a) Given a triangle $A_1A_2A_3$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A3], B_3$ and $D_1$ on the side $[A_3A_1]$.  If you build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2$ and $A_3B_3C_3D_3$, the lines $(A_1C_1), (A_2C_2)$ and $(A_3C_3)$, will cross in one point $O$. Prove that if $|A_1B_1| = |A_2D_2|$ and $|A_2B_2|  = |A_3D_3|$, than $|A_3B_3|  = |A_1D_1|$.

b) Given a convex polygon $A_1A_2 ... A_n$ and the points $B_1$ and $D_2$ on the side $[A_1A_2], B_2$ and $D_3$ on the side $[A_2A_3], ...  B_n$ and $D_1$ on the side $[A_nA_1]$. Ifyou build parallelograms $A_1B_1C_1D_1, A_2B_2C_2D_2 ... , A_nB_nC_nD_n$, the lines $(A_1C_1), (A_2C_2), ..., (A_nC_n)$, will cross in one point $O$.  Prove that $|A_1B_1| \cdot  |A_2B_2|\cdot  ... \cdot  |A_nB_n|  = |A_1D_1|\cdot  |A_2D_2|\cdot  ...\cdot  |A_nD_n|$.
1971 All Soviet Union Mathematical Olympiad
a) A game for two.

The first player writes two rows of ten numbers each, the second under the first. He should provide the following property: if number $b$ is written under $a$, and $d$ -- under $c$, then $a + d = b + c$.

The second player has to determine all the numbers. He is allowed to ask the questions like "What number is written in the $x$ place in the $y$ row?"

What is the minimal number of the questions asked by the second player before he founds out all the numbers?

b) There was a table $m\times n$ on the blackboard with the property: if You chose two rows and two columns, then the sum of the numbers in the two opposite vertices of the rectangles formed by those lines equals the sum of the numbers in two another vertices. Some of the numbers are cleaned   but it is still possible to restore all the table.

What is the minimal possible number of the remaining numbers?
1971 All Soviet Union Mathematical Olympiad
Given an unite square and some circles inside. Radius of each circle is less than $0.001$, and there is no couple of points belonging to the different circles with the distance between them $0.001$ exactly. Prove that the area, covered by the circles is not greater than $0.34$.
1971 All Soviet Union Mathematical Olympiad
The volumes of the water containing in each of three big enough containers are integers. You are allowed only to relocate some times from one container to another the same volume of the water, that the destination already contains. Prove that you are able to discharge one of the containers.
1971 All Soviet Union Mathematical Olympiad
Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2  -p_2q_1)<0$,

then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.
1971 All Soviet Union Mathematical Olympiad
The projections of the body on two planes are circles. Prove that they have the same radius.
1971 All Soviet Union Mathematical Olympiad
Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.
1971 All Soviet Union Mathematical Olympiad
a) Prove that the line dividing the triangle onto two polygons with equal perimeters and equal areas passes through the centre of the inscribed circle.

b) Prove the same statement for the arbitrary polygon outscribed around the circle.

c) Prove that all the lines halving its perimeter and area simultaneously, intersect in one point.
1971 All Soviet Union Mathematical Olympiad
Given $25$ different positive numbers. Prove that you can choose two of them such, that none of the other numbers equals neither to the sum nor to the difference between the chosen numbers.
1971 All Soviet Union Mathematical Olympiad
a) The vertex $A_1$ of the right $12$-angle (dodecagon) $A_1A_2...A_{12}$ is marked with "$-$" and all the rest $--$ with "$+$". You are allowed to change the sign simultaneously in the $6$ vertices in succession. Prove that is impossible to obtain dodecagon with $A_2$ marked with "$-$" and the rest of the vertices $--$ with "$+$".

b) Prove the same statement if it is allowed to change the signs not in six, but in four vertices in succession.

c) Prove the same statement if it is allowed to change the signs in three vertices in succession.
1971 All Soviet Union Mathematical Olympiad
$N$ unit squares on the infinite sheet of cross-lined paper are painted with black colour.

Prove that you can cut out the finite number of square pieces and satisfy two conditions all the black squares are contained in those pieces the area of black squares is not less than $1/5$ and not greater than $4/5$ of every piece area.
1971 All Soviet Union Mathematical Olympiad
A cube with the edge of length $n$ is divided onto $n^3$ unit ones.

Let us choose some of them and draw three lines parallel to the edges through their centres.

What is the least possible number of the chosen small cubes necessary to make those lines cross all the smaller cubes?

a) Find the answer for the small $n$ ($n = 2,3,4$).

b) Try to find the answer for $n = 10$.

c) If You can not solve the general problem, try to estimate that value from the upper and lower side.

d) Note, that You can reformulate the problem in such a way:

Consider all the triples $(x_1,x_2,x_3)$, where $x_i$ can be one of the integers $1,2,...,n$.

What is the minimal number of the triples necessary to provide the property:

for each of the triples there exist the chosen one, that differs only in one coordinate.

Try to find the answer for the situation with more than three coordinates, for example, with four.
1971 All Soviet Union Mathematical Olympiad
a) Consider the function $f(x,y) = x^2 + xy + y^2$.

Prove that for the every point $(x,y)$ there exist such integers $(m,n)$, that $f((x-m),(y-n)) \le 1/2$.

b) Let us denote with $g(x,y)$ the least possible value of the $f((x-m),(y-n))$ for all the integers $m,n$. The statement a) was equal to the fact $g(x,y) \le  1/2$.

Prove that in fact, $g(x,y) \le  1/3$.

Find all the points $(x,y)$, where $g(x,y)=1/3$.

c) Consider function $f_a(x,y) = x^2 + axy + y^2$ ($0 \le  a \le  2$).

Find any $c$ such that $g_a(x,y) \le c$.

Try to obtain the closest estimation.
1971 All Soviet Union Mathematical Olympiad
A switch has two inputs $1, 2$ and two outputs $1, 2$. It either connects $1$ to $1$ and $2$ to $2$, or $1$ to $2$ and $2$ to 1. If you have three inputs $1, 2, 3$ and three outputs $1, 2, 3$, then you can use three switches, the first across $1$ and $2$, then the second across $2$ and $3$, and finally the third across $1$ and $2$. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for $4$ inputs, so that by suitably setting the switches the output can be any permutation of the inputs?
2014 Argentine National Olympiad, Level 3
$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$. Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.
2014 Argentine National Olympiad, Level 3
Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called good if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
2014 Argentine National Olympiad, Level 3
Two circumferences of radius $1$ that do not intersect, $c_1$ and $c_2$, are placed inside an angle whose vertex is $O$. $c_1$ is tangent to one of the rays of the angle, while $c_2$ is tangent to the other ray. One of the common internal tangents of $c_1$ and $c_2$ passes through $O$, and the other one intersects the rays of the angle at points $A$ and $B$, with $AO=BO$. Find the distance of point $A$ to the line $OB$.
2014 Argentine National Olympiad, Level 3
Consider the following $50$-term sums:



$S=\frac{1}{1\cdot 2}+\frac{1}{3\cdot 4}+...+\frac{1}{99\cdot 100}$,



$T=\frac{1}{51\cdot 100}+\frac{1}{52\cdot 99}+...+\frac{1}{99\cdot 52}+\frac{1}{100\cdot 51}$.



Express $\frac{S}{T}$ as an irreducible fraction.
2014 Argentine National Olympiad, Level 3
An integer $n \geq 3$ is called special if it does not divide $\left ( n-1 \right )!\left ( 1+\frac{1}{2}+\cdot \cdot \cdot +\frac{1}{n-1} \right )$. Find all special numbers $n$ such that $10 \leq n \leq 100$.
2014 Argentine National Olympiad, Level 3
Determine whether there exists positive integers $a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}$ such that all sums $ a_{i}+a_{j}$, where 1 $\leq i < j \leq k$, are unique, and among those sums, there are $1000$ consecutive integers.
2017 Australian MO
Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:

(a) $P(2017)=2016$ and

(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.
2017 Australian MO
Let $ABCDE$ be a regular pentagon with center $M$. A point $P\neq M$ is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$.

Prove that $AR$ and $QR$ are of the same length.
2017 Australian MO
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table.

Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.
2017 Australian MO
Find all pairs $(a,b)$ of non-negative integers such that $2017^a=b^6-32b+1$.
1989 Federal Competition For Advanced Students
Natural numbers $ a \le b \le c \le d$ satisfy $ a+b+c+d=30$. Find the maximum value of the product $ P=abcd.$
1989 Federal Competition For Advanced Students
If $ a$ and $ b$ are nonnegative real numbers with $ a^2+b^2=4$, show that: $ \frac{ab}{a+b+2} \le \sqrt{2}-1$ and determine when equality occurs.
1989 Federal Competition For Advanced Students
Let $ a$ be a real number. Prove that if the equation $ x^2-ax+a=0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2+x_2^2 \ge 2(x_1+x_2).$
1989 Federal Competition For Advanced Students
Prove that for any triangle each exradius is less than four times the circumradius.
1991 Federal Competition For Advanced Students
Suppose that $ a,b,$ and $ \sqrt[3]{a}+\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.
1991 Federal Competition For Advanced Students
Solve in real numbers the equation:



$ \frac{1}{x}+\frac{1}{x+2}-\frac{1}{x+4}-\frac{1}{x+6}-\frac{1}{x+8}-\frac{1}{x+10}+\frac{1}{x+12}+\frac{1}{x+14}=0.$
1991 Federal Competition For Advanced Students
Find the number of squares in the sequence given by $ a_0=91$ and $ a_{n+1}=10a_n+(-1)^n$ for $ n \ge 0.$
1991 Federal Competition For Advanced Students
Let $ AB$ be a chord of a circle $ k$ of radius $ r$, with $ AB=c$.

$ (a)$ Construct the triangle $ ABC$ with $ C$ on $ k$ in which a median from $ A$ or $ B$ is of a given length $ d.$

$ (b)$ For which $ c$ and $ d$ is this triangle unique?
2002 Federal Competition For Advanced Students, Part 1
Determine all integers $a$ and $b$ such that

$$(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}$$

is a perfect square.
2002 Federal Competition For Advanced Students, Part 1
Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that

$$\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.$$

When does equality occur?
2002 Federal Competition For Advanced Students, Part 1
Let $f(x)=\frac{9^x}{9^x+3}$. Compute $\sum_{k} f \biggl( \frac{k}{2002} \biggr)$, where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$.
2002 Federal Competition For Advanced Students, Part 1
Let $A,C, P$ be three distinct points in the plane. Construct all parallelograms $ABCD$ such that point $P$ lies on the bisector of angle $DAB$ and $\angle APD = 90^\circ$.
2003 Federal Competition For Advanced Students, Part 1
Find all triples of prime numbers $(p, q, r)$ such that $p^q + p^r$ is a perfect square.
2003 Federal Competition For Advanced Students, Part 1
Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.
2003 Federal Competition For Advanced Students, Part 1
Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system

$$a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.$$
2003 Federal Competition For Advanced Students, Part 1
In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.
2004 Federal Competition For Advanced Students, Part 1
Find all quadruples $(a, b, c, d)$ of real numbers such that

$$a + bcd = b + cda = c + dab = d + abc.$$
2004 Federal Competition For Advanced Students, Part 1
A convex hexagon $ABCDEF$ with $AB = BC = a, CD = DE = b, EF = FA = c$ is inscribed in a circle. Show that this hexagon has three (pairwise disjoint) pairs of mutually perpendicular diagonals.
2004 Federal Competition For Advanced Students, Part 1
For natural numbers $a, b$, define $Z(a,b)=\frac{(3a)!\cdot (4b)!}{a!^4 \cdot b!^3}$.



(a) Prove that $Z(a, b)$ is an integer for $a \leq b$.



(b) Prove that for each natural number $b$ there are infinitely many natural numbers a such that $Z(a, b)$ is not an integer.[/list]