User:Ddk001: Difference between revisions

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Revision as of 17:47, 1 January 2024

See if you can solve these: 1. There is one and only one perfect square in the form

$(p^2+1)(q^2+1)-((pq)^2-pq+1)$

Find that perfect square.

2. Suppose there is complex values $x_1, x_2,$ and $x_3$ that satisfy

$(x_i-\sqrt[3]{13})((x_i-\sqrt[3]{53})(x_i-\sqrt[3]{103})=\frac{1}{3}$

Find $x_{1}^3+x_{2}^3+x_{2}^3$ .

3. Suppose

$x \equiv 2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6 \pmod{7!}$

Find the remainder when $\min{x}$ is divided by 1000.

4. Suppose $f(x)$ is a $10000000010$ -degrees polynomial. The Fundamental Theorem of Algebra tells us that there are $10000000010$ roots, say $r_1, r_2, \dots, r_{10000000010}$ . Suppose all integers $n$ ranging from $-1$ to $10000000008$ satisfies $f(n)=n$ . Also, suppose that

$(2+r_1)(2+r_2) \dots (2+r_{10000000010})=m!$

for an integer $m$ . If $p$ is the minimum possible value of

$(1+r_1)(1+r_2) \dots (1+r_{10000000010})$ .

Find the number of factors of the prime $999999937$ in $p$ .

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+See if you can solve these:
+. There is one and only one perfect square in the form
+<math>(p^2+1)(q^2+1)-((pq)^2-pq+1)</math>
+Find that perfect square.
+. Suppose there is complex values <math>x_1, x_2,</math> and <math>x_3</math> that satisfy
+<math>(x_i-\sqrt[3]{13})((x_i-\sqrt[3]{53})(x_i-\sqrt[3]{103})=\frac{1}{3}</math>
+Find <math>x_{1}^3+x_{2}^3+x_{2}^3</math>.
+. Suppose
+<math>x \equiv 2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6 \pmod{7!}</math>
+Find the remainder when <math>\min{x}</math> is divided by 1000.
+. Suppose <math>f(x)</math> is a <math>10000000010</math>-degrees polynomial. The Fundamental Theorem of Algebra tells us that there are <math>10000000010</math> roots, say <math>r_1, r_2, \dots, r_{10000000010}</math>. Suppose all integers <math>n</math> ranging from <math>-1</math> to <math>10000000008</math> satisfies <math>f(n)=n</math>. Also, suppose that
+<math>(2+r_1)(2+r_2) \dots (2+r_{10000000010})=m!</math>
+for an integer <math>m</math>. If <math>p</math> is the minimum possible value of
+<math>(1+r_1)(1+r_2) \dots (1+r_{10000000010})</math>.
+Find the number of factors of the prime <math>999999937</math> in <math>p</math>.

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User:Ddk001: Difference between revisions

Revision as of 17:47, 1 January 2024