Found problems: 4275
Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers?
[i]Proposed by Finland.[/i]
Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.
[i]Proposed by Jozsef Pelikan, Hungary[/i]
Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$. Find the value of
$$\frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}.$$
Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$.
[i]Remus Nicoara[/i]
Consider the polynomial f(x) = cx(x - 2) where $c$ is a positive real number. For any $n \in Z^+$, the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers.
1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once.
2. Prove that $c\ge 1$.
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $\big(a_1, a_2, a_3, a_4\big)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial
$$q(x) = c_3x^3+c_2x^2+c_1x+c_0$$such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
$(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11$
Find all polynomials $p(x)$ with real coefficients that have the following property: there exists a polynomial $q(x)$ with real coefficients such that $$p(1) + p(2) + p(3) +\dots + p(n) = p(n)q(n)$$ for all positive integers $n$.
Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$.
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial
$f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where
$i \in {1,2,3,...,8}$.
Find the possible values of $a_1$.
A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\] where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$. Determine, with proof, $b_{32}$.
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
We have $n$ points in the plane, no three on a line.
We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.
Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.
Show that the following sum is independent of the structure of points and only depends on $n$ :
\[ \sum_{i=3}^n (-1)^i c_i \]
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P(\tfrac72)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Justify your answer whether $A=\left(
\begin{array}{ccc}
-4 & -1& -1 \\
1 & -2& 1 \\
0 & 0& -3
\end{array}
\right)$ is similar to $B=\left(
\begin{array}{ccc}
-2 & 1& 0 \\
-1 & -4& 1 \\
0 & 0& -3
\end{array}
\right),\ A,\ B\in{M(\mathbb{C})}$ or not.
[u]Round 9[/u]
[b]p25.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of pathsMaisy can take to reach the point $(x, y)$. The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, can be written as ${2k \choose k} - j$ for a minimum positive integer k and corresponding positive integer $j$ . Find $k + j$ .
[b]p26.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past B to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE =\sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. The value of $EF^2$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are integers, c is squarefree, and $gcd(a,b,d) = 1$. Find $a +b +c +d$.
[b]p27.[/b] Find the number of trailing zeroes at the end of $$\sum^{2021}_{i=1}(2021^i -1) = (2021^1 -1)...(2021^{2021}-1).$$
[u]Round 10[/u]
[b]p28.[/b] Points $A, B, C, P$, and $D$ lie on circle ω in that order. Let $AC$ and $BD$ intersect at $I$ . Given that
$PI = PC = PD$, $\angle DAB = 137^o$, and $\angle ABC = 109^o$, find the measure of $\angle BIC$ in degrees.
[b]p29.[/b] Find the sum of all positive integers $n < 2021$ such that when ${d_1,d_2,... ,d_k}$ are the positive
integer factors of $n$, then $$\left( \sum^{k}_{i=1}d_i \right) \left( \sum^{k}_{i=1} \frac{1}{d_i} \right)= r^2$$ for some rational number $r$ .
[b]p30.[/b] Let $a, b, c, d$ and $e$ be positive real numbers. Define the function $f (x, y) = \frac{x}{y}+\frac{y}{x}$ for all positive real numbers. Given that $f (a,b) = 7$, $f (b,c) = 5$, $f (c,d) = 3$, and $f (d,e) = 2$, find the sum of all possible values of $f (e,a)$.
[u]Round 11[/u]
[b]p31.[/b] There exist $100$ (not necessarily distinct) complex numbers $r_1, r_2,..., r_{100}$ such that for any positive integer $1 \le k \le 100$, we have that $P(r_k ) = 0$ where the polynomial $P$ is defined as $$P(x) =
\sum^{101}_{i=1}i \cdot x^{101-i} = x^{100} +2x^{99} +3x^{98} +...+99x^2 +100x +101.$$
Find the value of $$\prod^{100}_{j=1} (r^2_j+1) = (r^2_1 +1)(r^2_2 +1)...(r^2_{100} +1).$$
[b]p32.[/b] Let $BT$ be the diameter of a circle $\omega_1$, and $AT$ be a tangent of $\omega_1$. Line $AB$ intersects $\omega_1$ at $C$, and $\vartriangle ACT$ has circumcircle $\omega_2$. Points $P$ and $S$ exist such that $PA$ and $PC$ are tangent to $\omega_2$ and $SB = BT = 20$. Given that $AT = 15$, the length of $PS$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $b$, and $c$ are positive integers, $b$ is squarefree, and $gcd(a,b) = 1$. Find $a +b +c$.
[b]p33.[/b] There are a hundred students in math team. Each pair of students are either mutually friends or mutually enemies. It is given that if any three students are chosen, then they are not all mutually friends. The maximum possible number of ways to choose four students such that it is possible to label them $A, B, C$, and $D$ such that $A$ and $B$ are friends, $B$ and $C$ are friends, $C$ and $D$ are friends, and D and A are friends can be expressed as $n^4$. Find $n$.
[u]Round 12[/u]
[b]p34.[/b] Let $\{p_i\}$ be the prime numbers, such that $p_1 = 2, p_2 = 3, p_3 = 5, ...$ For each $i$ , let $q_i$ be the nearest perfect square to $p_i$ . Estimate $\sum^{2021}_{i=1}|p_i=q_i |$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 30 \cdot \max - \left(0,1-5 \cdot \left| \log_{10} \frac{A}{E} \right| \right)\right \rfloor.$
[b]p35.[/b] Estimate the number of digits of $(2021!)^{2021}$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 15 \cdot \max \left(0,2- \cdot \left| \log_{10} \frac{A}{E} \right| \right) \right \rfloor.$
[b]p36.[/b] Pick a positive integer between$ 1$ and $1000$, inclusive. If your answer is $E$ and a quarter of the mean of all the responses to this problem is $A$, your score will be $$ \lfloor \max \left(0,30- |A-E|, 2-|E-1000| \right) \rfloor.$$ Note that if you pick $1000$, you will automatically get $2$ points.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
Find out wich of the following polynomials are irreducible.
a) $t^4+1$ over $\mathbb{R}$;
b) $t^4+1$ over $\mathbb{Q}$;
c) $t^3-7t^2+3t+3$ over $\mathbb{Q}$;
d) $t^4+7$ over $\mathbb{Z}_{17}$;
e) $t^3-5$ over $\mathbb{Z}_{11}$;
f) $t^6+7$ over $\mathbb{Q}(i)$.