User Indecisive - Mathematics Stack Exchange

8 votes

1 answer

253 views

Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$

asked Sep 25, 2023 at 17:13

7 votes

1 answer

197 views

Compute the largest natural number $n$ such that $2^n$ divides $S=\binom{2}{1}+\binom{4}{2}+\binom{8}{4}+\cdots+\binom{2^{100}}{2^{99}}$

asked Apr 14, 2023 at 13:22

5 votes

4 answers

264 views

Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square

asked Mar 9 at 15:51

5 votes

2 answers

136 views

Compute $\prod^{100}_{n=1} \left(5-4 \cos \left(\frac{\pi (2n-1)}{100} \right) \right)$.

asked Aug 26, 2023 at 2:28

5 votes

1 answer

101 views

Reciprocals of positive integers in arithmetic progression

asked Jul 13 at 14:35

4 votes

4 answers

399 views

Find ratio formed by the perpendicular bisector of an angle bisector

asked Jul 3, 2023 at 22:15

3 votes

2 answers

140 views

Solving $25^n + 16^n \equiv 1 \pmod{121}$

asked Mar 15 at 15:56

3 votes

2 answers

92 views

$I$ is the incenter of $ABC$. The extension of $AI$ meets the circumcircle of $ABC$ at $D$. If $AB=3$, $AC=4$, and $[IBC]=[DBC]$, find $BC$.

asked Feb 4 at 22:14

3 votes

1 answer

89 views

Find $k>0$ such that $(k+1)+\sum^{\infty}_{n=1} \left(\frac{\binom{4n}{n}}{3n+1} k^{2n}+\frac{2\binom{4n+1}{n}}{3n+2} k^{2n+1} \right)=2$

asked Apr 17, 2023 at 22:20

3 votes

1 answer

61 views

Find ratio formed by intersection of segments in a triangle

asked Jul 29, 2023 at 16:00

3 votes

1 answer

193 views

Finding area of hexagon inscribed in rectangle

asked Jul 31, 2023 at 1:41

2 votes

1 answer

157 views

In triangle $ABC$, $\angle A=50^{\circ}$. Point $D$ is constructed inside the triangle such that $\angle BDC=130^{\circ}$.

asked Aug 3, 2023 at 23:29

2 votes

2 answers

101 views

In $\Delta ABC$, $AB=\sqrt 5$, $BC=1$, $AC=2$. $I$ is the incenter of $\Delta ABC$ and the circumcircle of $\Delta IBC$ cuts $AB$ at $P$. Find $BP$.

asked Aug 14, 2023 at 1:07

2 votes

1 answer

186 views

Find the period of $f(n)=n^{n^n} \pmod{23}$

asked Apr 26, 2023 at 22:11

2 votes

1 answer

350 views

Let $p(x)=x^2-x+1$. Let $\alpha$ be a root of $p(p(p(p(x))))$. Compute $|(p(\alpha)-1)p(\alpha)p(p(\alpha))p(p(p(\alpha)))|$.

asked Jun 3, 2023 at 2:08

2 votes

1 answer

107 views

Let $N$ be the $44$-digit number $44 \cdots 44$. Compute the remainder when $\sum^{2021}_{n=0} 10^{2^n}$ is divided by $N$.

asked Jun 19, 2023 at 21:45

1 vote

0 answers

60 views

Checking Solution: For an integer $k \geq 2$, $f(k)$ is the number of positive integers $n$ such that $n|(n-1)!+k$. Compute $\sum^{100}_{r=2} f(r)$.

asked Sep 17, 2023 at 15:27

0 votes

1 answer

62 views

Comparing lengths in a triangle where the angle bisector makes an obtuse angle with the opposite side

asked Jul 15, 2023 at 20:14

0 votes

1 answer

47 views

Closed forms for $\sum^{n-1}_{r=1} (n-r) \sin \left(\frac{2\pi r}{n} \right)$ and $\sum^{n-1}_{r=1} (n-r) \omega^r$

asked Jun 29, 2023 at 17:10

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19 Questions

Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$

Compute the largest natural number $n$ such that $2^n$ divides $S=\binom{2}{1}+\binom{4}{2}+\binom{8}{4}+\cdots+\binom{2^{100}}{2^{99}}$

Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square

Compute $\prod^{100}_{n=1} \left(5-4 \cos \left(\frac{\pi (2n-1)}{100} \right) \right)$.

Reciprocals of positive integers in arithmetic progression

Find ratio formed by the perpendicular bisector of an angle bisector

Solving $25^n + 16^n \equiv 1 \pmod{121}$

$I$ is the incenter of $ABC$. The extension of $AI$ meets the circumcircle of $ABC$ at $D$. If $AB=3$, $AC=4$, and $[IBC]=[DBC]$, find $BC$.

Find $k>0$ such that $(k+1)+\sum^{\infty}_{n=1} \left(\frac{\binom{4n}{n}}{3n+1} k^{2n}+\frac{2\binom{4n+1}{n}}{3n+2} k^{2n+1} \right)=2$

Find ratio formed by intersection of segments in a triangle

Finding area of hexagon inscribed in rectangle

In triangle $ABC$, $\angle A=50^{\circ}$. Point $D$ is constructed inside the triangle such that $\angle BDC=130^{\circ}$.

In $\Delta ABC$, $AB=\sqrt 5$, $BC=1$, $AC=2$. $I$ is the incenter of $\Delta ABC$ and the circumcircle of $\Delta IBC$ cuts $AB$ at $P$. Find $BP$.

Find the period of $f(n)=n^{n^n} \pmod{23}$

Let $p(x)=x^2-x+1$. Let $\alpha$ be a root of $p(p(p(p(x))))$. Compute $|(p(\alpha)-1)p(\alpha)p(p(\alpha))p(p(p(\alpha)))|$.

Let $N$ be the $44$-digit number $44 \cdots 44$. Compute the remainder when $\sum^{2021}_{n=0} 10^{2^n}$ is divided by $N$.

Checking Solution: For an integer $k \geq 2$, $f(k)$ is the number of positive integers $n$ such that $n|(n-1)!+k$. Compute $\sum^{100}_{r=2} f(r)$.

Comparing lengths in a triangle where the angle bisector makes an obtuse angle with the opposite side

Closed forms for $\sum^{n-1}_{r=1} (n-r) \sin \left(\frac{2\pi r}{n} \right)$ and $\sum^{n-1}_{r=1} (n-r) \omega^r$