Navvye/Olympiad-Problems-CSV huggingface dataset:
1981 All Soviet Union Mathematical Olympiad
There are $1000$ inhabitants in a settlement.

Every evening every inhabitant tells all his friends all the news he had heard the previous day.

Every news becomes finally known to every inhabitant.

Prove that it is possible to choose $90$ of inhabitants so, that if you tell them a news simultaneously, it will be known to everybody in $10$ days.
1981 All Soviet Union Mathematical Olympiad
It is known about real $a$ and $b$ that the inequality $a \cos x + b \cos (3x) > 1$ has no real solutions.

Prove that $|b|\le 1$.
1981 All Soviet Union Mathematical Olympiad
The points $K$ and $M$ are the centres of the $AB$ and $CD$ sides of the convex quadrangle $ABCD$.

The points $L$ and $M$ belong to two other sides and $KLMN$ is a rectangle.

Prove that $KLMN$ area is a half of $ABCD$ area.
1981 All Soviet Union Mathematical Olympiad
Find all the sequences of natural $k_n$ with two properties:

a)  $k_n \le  n \sqrt {n}$ for all $n$

b) $(k_n - k_m)$ is divisible by $(m-n)$  for all $m>n$
1981 All Soviet Union Mathematical Olympiad
Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?
1981 All Soviet Union Mathematical Olympiad
The quadrangles $AMBE, AHBT, BKXM$, and $CKXP$ are parallelograms.

Prove that the quadrangle $ABTE$ is also parallelogram.

(the vertices are mentioned counterclockwise)
1981 All Soviet Union Mathematical Olympiad
Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.
1981 All Soviet Union Mathematical Olympiad
Eighteen soccer teams have played $8$ tours of a one-round tournament.

Prove that there is a triple of teams, having not met each other yet.
1981 All Soviet Union Mathematical Olympiad
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the $ABC$ triangle.

$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$.

Prove that the perimeter $P$ of the $ABC$ triangle and the perimeter $p$ of the $A_1B_1C_1$ triangle,

satisfy inequality $\frac{P}{2} < p < \frac{3P}{4}$.
1981 All Soviet Union Mathematical Olympiad
Positive numbers $x,y$ satisfy equality $x^3+y^3=x-y$.

Prove that $x^2+y^2<1$.
1981 All Soviet Union Mathematical Olympiad
A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated $d$ units of length far from each other. The pupil draw angles precisely, but made relative error less than $p$ in the lengths of sides.  Prove that $d < 4p$.
1981 All Soviet Union Mathematical Olympiad
A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:
1981 All Soviet Union Mathematical Olympiad
Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.
1981 All Soviet Union Mathematical Olympiad
The natural numbers from $100$ to $999$ are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into $10$ piles according to the least significant digit. The first pile will contain cards with $0$ at the end, ... , the tenth -- with $9$. Then we shall gather $10$ piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one.

What will be the order of the cards in the obtained pile?
1981 All Soviet Union Mathematical Olympiad
Six points are marked inside the $3x4$ rectangle.

Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.
1981 All Soviet Union Mathematical Olympiad
a) Find the minimal value of the polynomial $P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$

b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.
1981 All Soviet Union Mathematical Olympiad
The segments $[AD], [BE]$ and $[CF]$ are the side edges of the right triangle prism. (the equilateral triangle is a base)

Find all the points in its base $ABC$, situated on the equal distances from the $(AE), (BF)$ and $(CD)$ lines.
1980 All Soviet Union Mathematical Olympiad
All the two-digit numbers from $19$ to $80$ are written in a line without spaces.

Is the obtained number $192021....7980$ divisible by $1980$?
1980 All Soviet Union Mathematical Olympiad
The vertical side of a square is divided onto $n$ segments. The sum of the segments with even numbers lengths equals to the sum of the segments with odd numbers lengths. $n-1$ lines parallel to the horizontal sides are drawn from the segments ends, and, thus, $n$ strips are obtained. The diagonal is drawn from the lower left corner to the upper right one. This diagonal divides every strip onto left and right parts.

Prove that the sum of the left parts of odd strips areas equals to the sum of the right parts of even strips areas.
1980 All Soviet Union Mathematical Olympiad
The load for the space station "Salute" is packed in containers. There are more than $35$ containers, and the total weight is $18$ metric tons. There are $7$ one-way transport spaceships "Progress", each able to bring $3$ metric tons to the station. It is known that they are able to take an arbitrary subset of $35$ containers. Prove that they are able to take all the load.
1980 All Soviet Union Mathematical Olympiad
The points $M$ and $P$ are the middles of $[BC]$ and $[CD]$  sides of a convex quadrangle $ABCD$.

It is known that $|AM| + |AP| = a$.  Prove that the $ABCD$ area is less than $\frac{a^2}{2}$.
1980 All Soviet Union Mathematical Olympiad
Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?
1980 All Soviet Union Mathematical Olympiad
Given a point $E$ on the diameter $AC$ of the certain circle.

Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.
1980 All Soviet Union Mathematical Olympiad
There are several settlements on the bank of the Big Round Lake. Some of them are connected with the regular direct ship lines. Two settlements are connected if and only if two next (counterclockwise) to each ones are not connected.

Prove that You can move from the arbitrary settlement to another arbitrary settlement, having used not more than three ships.
1980 All Soviet Union Mathematical Olympiad
The six-digit decimal number contains six different non-zero digits and is divisible by $37$.

Prove that having transposed its digits you can obtain at least $23$ more numbers divisible by $37$
1980 All Soviet Union Mathematical Olympiad
Find real solutions of the system :



$\sin x + 2 \sin (x+y+z) = 0$

$\sin y + 3 \sin (x+y+z) = 0$

$\sin z + 4 \sin (x+y+z) = 0$
1980 All Soviet Union Mathematical Olympiad
Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum.  Prove that the sum of all vectors equals to the zero vector.
1980 All Soviet Union Mathematical Olympiad
Let us denote with $S(n)$ the sum of all the digits of $n$.

a) Is there such an $n$ that $n+S(n)=1980$?

b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.
1980 All Soviet Union Mathematical Olympiad
Some squares of the infinite sheet of cross-lined paper are red. Each $2\times 3$ rectangle (of $6$ squares) contains exactly two red squares. How many red squares can be in the $9\times 11$ rectangle of $99$ squares?
1980 All Soviet Union Mathematical Olympiad
An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that

a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time.

b) If nobody had the immunity on the first day, the epidemic influenza will stop some day.
1980 All Soviet Union Mathematical Olympiad
Let us denote with $P(n)$ the product of all the digits of $n$.

Consider the sequence $n_{k+1} = n_k + P(n_k)$.

Can it be unbounded for some $n_1$?
1980 All Soviet Union Mathematical Olympiad
Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ - the middle of the $[AP]$ segment. Find the angles of $DEC$ triangle.
1980 All Soviet Union Mathematical Olympiad
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$).

Let $p=4(x+y+z), s=2(xy+yz+zx)$ and $d=\sqrt{x^2+y^2+z^2}$ be

its perimeter, surface area and diagonal length, respectively.

Prove that $x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 -  \frac{s}{2}}\right )$ and $z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 -  \frac{s}{2}}\right )$
1980 All Soviet Union Mathematical Olympiad
The $A$ set consists of integers only. Its minimal element is $1$ and its maximal element is $100$. Every element of $A$ except $1$ equals to the sum of two (may be equal) numbers being contained in $A$.

What is the least possible number of elements in $A$?
1980 All Soviet Union Mathematical Olympiad
Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$.



($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)
1980 All Soviet Union Mathematical Olympiad
The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$.

Prove that the cosine of the angle between $(AC)$ and $(BD)$ lines is less than $|CD|/|AB|$.
1980 All Soviet Union Mathematical Olympiad
The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$.

a) Prove that the sequence $x_i$ has limit.

b) Can this limit be irrational if we have started with the rational number?

c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.
1979 All Soviet Union Mathematical Olympiad
What is the least possible relation of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?
1979 All Soviet Union Mathematical Olympiad
A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$.  Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.
1979 All Soviet Union Mathematical Olympiad
Every member of a certain parliament has not more than $3$ enemies.

Prove that it is possible to divide it onto two subparliaments so, that everyone will have not more than one enemy in his subparliament. ($A$ is the enemy of $B$ if and only if $B$ is the enemy of $A$.)
1979 All Soviet Union Mathematical Olympiad
Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain :

a) $1/5$,

b) an arbitrary rational number between $0$ and $1$.
1979 All Soviet Union Mathematical Olympiad
For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$.

Prove that for every $n$, it also satisfies $x_1+x_2/2+x_3/3+...+x_n/n\le 3$.
1979 All Soviet Union Mathematical Olympiad
Given some points in the plane. For some pairs $A,B$ the vector $AB$ is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.
1979 All Soviet Union Mathematical Olympiad
What is the least possible number of the checkers being required

a) for the $8\times 8$ chess-board,

b) for the $n\times n$ chess-board,

to provide the property:

Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker ?
1979 All Soviet Union Mathematical Olympiad
Find $x$ and $y$ ($a$ and $b$ parameters):

$$\frac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a$$$$\frac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b$$
1979 All Soviet Union Mathematical Olympiad
Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.
1979 All Soviet Union Mathematical Olympiad
Prove that for the arbitrary numbers $x_1, x_2, ... , x_n$ from the $[0,1]$ segment

$(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)$.
1979 All Soviet Union Mathematical Olympiad
Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments.

Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$.
1979 All Soviet Union Mathematical Olympiad
Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it.

Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$.

Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2, A_2A_4$ and $l_3, ... , A_{i-1}A_{i+1}$ and $l_i, ... , A_{1977}A_{1979}$ and $l_{1978}, A_{1978}A_1$ and $l_{1979}, A_{1979}A_2$ and $l_1$
1979 All Soviet Union Mathematical Olympiad
The finite sequence $a_1, a_2, ... , a_n$ of ones and zeroes should satisfy a condition:

for every $k$ from $0$ to $(n-1)$ the sum a_1a_{k+1} + a_2a_{k+2} + ... + a_{n-k}a_n should be odd.

a) Build such a sequence for $n=25$.

b) Prove that there exists such a sequence for some $n > 1000$.
1979 All Soviet Union Mathematical Olympiad
The convex quadrangle is divided by its diagonals onto four triangles. The circles inscribed in those triangles are equal.

Prove that the given quadrangle is a diamond.
1979 All Soviet Union Mathematical Olympiad
Given $n$ points (in sequence)$ A_1, A_2, ... , A_n$ on a line. All the segments $A_1A_2, A_2A_3, ... A_{n-1}A_n$ are shorter than $1$. We need to mark $(k-1)$ points so that the difference of every two segments, with the ends in the marked points, is shorter than $1$.  Prove that it is possible

a) for $k=3$,

b) for every $k$ less than $(n-1)$.
1978 All Soviet Union Mathematical Olympiad
Let $a_n$ be the closest to $\sqrt n$ integer.

Find the sum $1/a_1 + 1/a_2 + ... + 1/a_{1980}$.
1978 All Soviet Union Mathematical Olympiad
Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. the angle $CBM$ equals to $CDM$. Prove that the angle $ACD$ equals to $BCM$.
1978 All Soviet Union Mathematical Olympiad
Prove that there is no $m$ such that ($1978^m - 1$) is divisible by ($1000^m - 1$).
1978 All Soviet Union Mathematical Olympiad
Given a finite set $K_0$ of points (in the plane or space).

The sequence of sets $K_1, K_2, ... , K_n, ...$ is build according to the rule: we take all the points of $K_i$, add all the symmetric points with respect to all its points, and, thus obtain $K_{i+1}$.

a) Let $K_0$ consist of two points $A$ and $B$ with the distance $1$ unit between them. For what $n$ the set $K_n$ contains the point that is $1000$ units far from $A$?

b) Let $K_0$ consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing $K_n. K_0$ below is the set of the unit volume tetrahedron vertices.

c) How many faces contain the minimal convex polyhedron containing $K_1$?

d) What is the volume of the above mentioned polyhedron?

e) What is the volume of the minimal convex polyhedron containing $K_n$?
1978 All Soviet Union Mathematical Olympiad
Given two heaps of checkers. the bigger contains $m$ checkers, the smaller -- $n$ ($m>n$). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins.

a) Prove that if $m > 2n$, than the first can always win.

b) Find all $x$ such that if $m > xn$, than the first can always win.
1978 All Soviet Union Mathematical Olympiad
Prove that there exists such an infinite sequence $\{x_i\}$, that

for all $m$ and all $k$ ($m\ne k$) holds the inequality $|x_m-x_k|>1/|m-k|$.
1978 All Soviet Union Mathematical Olympiad
Let $f(x) = x^2 + x + 1$.

Prove that for every natural $m>1$ the numbers $m, f(m), f(f(m)), ...$ are relatively prime.
1978 All Soviet Union Mathematical Olympiad
Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)
1978 All Soviet Union Mathematical Olympiad
Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, having got the pair of cards with $(a,b)$ and $(b,c)$ produces new card with $(a,c)$. All the automates return the initial cards also. Suppose there was $(5,19)$ card initially. Is it possible to obtain

a) $(1,50)$?

b) $(1,100)$?

c) Suppose there was $(a,b)$ card initially $(a<b)$. We want to obtain $(1,n)$ card. For what $n$ is it possible?
1978 All Soviet Union Mathematical Olympiad
Given a circle with radius $R$ and inscribed $n$-angle with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.
1978 All Soviet Union Mathematical Olympiad
The checker is standing on the corner field of a $n\times n$ chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts.

a) Prove that for even $n$ the first can always win, and if $n$ is odd, than the second can always win.

b) Who wins if the checker stands initially on the neighbour to the corner field?
1978 All Soviet Union Mathematical Olympiad
Given $n$ nonintersecting segments in the plane. Not a pair of those belong to the same straight line. We want to add several segments, connecting the ends of given ones, to obtain one nonselfintersecting broken line. Is it always possible?
1978 All Soviet Union Mathematical Olympiad
Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$.

Prove that $(x_1+x_2+...+x_n)( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n})\le \frac{(a+b)^2}{4ab}n^2$
1978 All Soviet Union Mathematical Olympiad
Given a simple number $p>3$.

Consider the set $M$ of the pairs $(x,y)$ with the integer coordinates in the plane such that $0 \le x < p, 0 \le y < p$.

Prove that it is possible to mark $p$ points of $M$ such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points.
1978 All Soviet Union Mathematical Olympiad
Prove that for every tetrahedron there exist two planes such that the projection areas on those planes relation is not less than $\sqrt 2$.
1978 All Soviet Union Mathematical Olympiad
Given $a_1, a_2, ... , a_n$. Define $b_k = \frac{a_1 + a_2 + ... + a_k}{k}$ for $1 \le  k\le n$.

Let $C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2$,

$D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2$.



Prove that $C \le D \le 2C$.
1978 All Soviet Union Mathematical Olympiad
Consider a sequence $x_n=(1+\sqrt2+\sqrt3)^n$.

Each member can be represented as  $x_n=qn+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$, where $q_n, r_n, s_n, t_n$ are integers.

Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
1977 All Soviet Union Mathematical Olympiad
Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call "special" a couple of line's segments if the one's continuation intersects another. Prove that there is even number of special pairs.
1977 All Soviet Union Mathematical Olympiad
Given several points, not all lying on one straight line. Some number is assigned to every point. It is known, that if a straight line contains two or more points, than the sum of the assigned to those points equals zero. Prove that all the numbers equal to zero.
1977 All Soviet Union Mathematical Olympiad
a) Given a circle with two inscribed triangles $T_1$ and $T_2$. The vertices of $T_1$ are the middles of the arcs with the ends in the vertices of $T_2$. Consider a hexagon -- the intersection of $T_1$ and $T_2$. Prove that its main diagonals are parallel to $T_1$ sides and are intersecting in one point.

b) The segment, that connects the middles of the arcs $AB$ and $AC$ of the circle outscribed around the $ABC$ triangle, intersects $[AB]$ and $[AC]$ sides in $D$ and $K$ points.  Prove that the points $A,D,K$ and $O$ -- the centre of the circle -- are the vertices of a diamond.
1977 All Soviet Union Mathematical Olympiad
Several black and white checkers (tokens?) are standing along the circumference. Two men remove checkers in turn. The first removes all the black ones that had at least one white neighbour, and the second -- all the white ones that had at least one black neighbour. They stop when all the checkers are of the same colour.

a) Let there be $40$ checkers initially. Is it possible that after two moves of each man there will remain only one (checker)?

b) Let there be $1000$ checkers initially. What is the minimal possible number of moves to reach the position when there will remain only one (checker)?
1977 All Soviet Union Mathematical Olympiad
Given infinite sequence $a_n$. It is known that the limit of $b_n=a_{n+1}-a_n/2$ equals zero.

Prove that the limit of $a_n$ equals zero.
1977 All Soviet Union Mathematical Olympiad
There are direct routes from every city of a certain country to every other city. The prices are known in advance. Two tourists (they do not necessary start from one city) have decided to visit all the cities, using only direct travel lines. The first always chooses the cheapest ticket to the city, he has never been before (if there are several -- he chooses arbitrary destination among the cheapests). The second -- the most expensive (they do not return to the first city).

Prove that the first will spend not more money for the tickets, than the second.
1977 All Soviet Union Mathematical Olympiad
Every vertex of a convex polyhedron belongs to three edges. It is possible to outscribe a circle around all its faces.

Prove that the polyhedron can be inscribed in a sphere.
1977 All Soviet Union Mathematical Olympiad
The polynomial $x^{10} + ?x^9 + ?x^8 + ... + ?x + 1$ is written on the blackboard. Two players substitute (real) numbers instead of one of the question marks in turn. ($9$ turns total.) The first wins if the polynomial will have no real roots.

Who wins?