User ETS1331

11 votes

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Olympiad Algebra Practice Question

answered Aug 31, 2019 at 22:34

11 votes

Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$

answered Mar 12, 2023 at 20:43

6 votes

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Evaluate $\sum\limits_{k=1}^n (k^{3} +k^{2} +1) / (k^{2} +k)$

answered Feb 10, 2019 at 2:04

5 votes

How do you compute the sum of k * a^k

answered Feb 10, 2019 at 1:50

5 votes

Functions satisfying: $f(f(x)^2+f(y))=xf(x)+y$

answered Jun 17, 2019 at 0:26

5 votes

A Straightforward optimization problem, but without calculus

answered Jan 16, 2021 at 18:34

4 votes

Prove that the number of "good" subsets in $A$ and the number of "good" subsets in $B$ have the same parity.

answered Nov 26, 2022 at 2:18

4 votes

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How many ways to place plus and minus signs in front of numbers from 0 to 12 so that the sum is divisible by 5?

answered Nov 26, 2022 at 1:10

4 votes

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For all natural numbers $a_i$ of the set $\{1,2,\dots\,2017\}$ show that $\sum_{i=1}^{2017}\frac{{a_i}^2-i^2}{i} \geq 0$

answered Nov 28, 2022 at 21:51

4 votes

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Combinatorial proof of $\sum_{k=0}^{n} \binom{3n-k}{2n} = \binom{3n+1}{n}$

answered Jul 17, 2019 at 19:32

3 votes

Inequality related with $abcd=(1-a)(1-b)(1-c)(1-d)$

answered Apr 10, 2019 at 1:52

3 votes

Accepted

Find $21^{1234}\pmod{100}\equiv \ ?$

answered Feb 5, 2019 at 21:32

3 votes

Solve the equation $\sqrt{3x+2}+\dfrac{x^2}{\sqrt{3x+2}}=2x$

answered Nov 18, 2021 at 19:43

2 votes

Accepted

Simple cyclic inequality, similar to Shapiro's

answered May 5, 2021 at 18:34

2 votes

Accepted

How many anagrams are there for the word "MERCABLE" with given conditions?

answered Aug 25, 2019 at 1:26

2 votes

Accepted

Let $G$ be a disconnected graph. Then $\overline{G}$ is connected. Prove that if $u,v \in V(G)$, then $d_{\overline{G}}(u,v)=1$ or $2$,$diam(G)\leq 2$

answered Jan 14, 2023 at 19:09

2 votes

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Isosceles triangle generated by 3 touching circles

answered Mar 12, 2023 at 20:36

2 votes

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A multiple of $(1111+912)^{(1111+912)^{(1111+912)}}$ without using the digits "$0$", "$13$" and "$666$"?

answered Dec 25, 2023 at 18:09

2 votes

Stuck on this - In ABC right triangle AC= $2+\sqrt{3}$ and BC = $3+2\sqrt{3}$. circle touches point C and D, Find the Area of $AMD$

answered Jun 24, 2019 at 16:50

1 vote

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What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?

answered Feb 6, 2019 at 1:28

1 vote

Accepted

3 unknowns and 1 equation

answered Nov 9, 2018 at 22:02

1 vote

Toss a fair coin four times independently. Let X be the number of heads minus the number of tails. Find the distribution of X.

answered Feb 23, 2017 at 23:35

1 vote

How to show that $\sum_{i=1}^n i^2$ is a polynomial in $n$?

answered Nov 27, 2022 at 20:13

1 vote

In $\triangle ABC$, if angle bisectors $AE$ and $CD$ meet at incenter $F$, and $|FE|=|FD|$, then the triangle is isosceles or $\angle B=60^\circ$

answered Aug 23, 2019 at 19:05

1 vote

Prove that $TK=TO$

answered Jul 5, 2020 at 16:44

1 vote

Problem $5$ of $2011$ USAJMO

answered Aug 5, 2022 at 23:04

0 votes

I got 72 as an answer for this question but it says 12 is the right answer on the worksheet?

answered Feb 26, 2017 at 1:17

0 votes

Converse of Equal Area Theorem for Medians

answered Oct 21, 2018 at 17:05

0 votes

Prove that $f$ is additive if $f(x)f(x-y)+f(y)f(x+y)= f(x)^2+f(y)^2$

answered Nov 3, 2018 at 3:43

0 votes

Finding the $100$-th term of $1,3,4,9,10,12,13\dots$ (powers of $3$, or sums of distinct powers of $3$)

answered Nov 7, 2018 at 5:40

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34 Answers

Olympiad Algebra Practice Question

Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$

Evaluate $\sum\limits_{k=1}^n (k^{3} +k^{2} +1) / (k^{2} +k)$

How do you compute the sum of k * a^k

Functions satisfying: $f(f(x)^2+f(y))=xf(x)+y$

A Straightforward optimization problem, but without calculus

Prove that the number of "good" subsets in $A$ and the number of "good" subsets in $B$ have the same parity.

How many ways to place plus and minus signs in front of numbers from 0 to 12 so that the sum is divisible by 5?

For all natural numbers $a_i$ of the set $\{1,2,\dots\,2017\}$ show that $\sum_{i=1}^{2017}\frac{{a_i}^2-i^2}{i} \geq 0$

Combinatorial proof of $\sum_{k=0}^{n} \binom{3n-k}{2n} = \binom{3n+1}{n}$

Inequality related with $abcd=(1-a)(1-b)(1-c)(1-d)$

Find $21^{1234}\pmod{100}\equiv \ ?$

Solve the equation $\sqrt{3x+2}+\dfrac{x^2}{\sqrt{3x+2}}=2x$

Simple cyclic inequality, similar to Shapiro's

How many anagrams are there for the word "MERCABLE" with given conditions?

Let $G$ be a disconnected graph. Then $\overline{G}$ is connected. Prove that if $u,v \in V(G)$, then $d_{\overline{G}}(u,v)=1$ or $2$,$diam(G)\leq 2$

Isosceles triangle generated by 3 touching circles

A multiple of $(1111+912)^{(1111+912)^{(1111+912)}}$ without using the digits "$0$", "$13$" and "$666$"?

Stuck on this - In ABC right triangle AC= $2+\sqrt{3}$ and BC = $3+2\sqrt{3}$. circle touches point C and D, Find the Area of $AMD$

What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?

3 unknowns and 1 equation

Toss a fair coin four times independently. Let X be the number of heads minus the number of tails. Find the distribution of X.

How to show that $\sum_{i=1}^n i^2$ is a polynomial in $n$?

In $\triangle ABC$, if angle bisectors $AE$ and $CD$ meet at incenter $F$, and $|FE|=|FD|$, then the triangle is isosceles or $\angle B=60^\circ$

Prove that $TK=TO$

Problem $5$ of $2011$ USAJMO

I got 72 as an answer for this question but it says 12 is the right answer on the worksheet?

Converse of Equal Area Theorem for Medians

Prove that $f$ is additive if $f(x)f(x-y)+f(y)f(x+y)= f(x)^2+f(y)^2$

Finding the $100$-th term of $1,3,4,9,10,12,13\dots$ (powers of $3$, or sums of distinct powers of $3$)