Beaver Math Olympiad - Revision history

ADucharme: /* 6. 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch) */ rename section to Space Needle like how BMO 2 is "Hydra and Antihydra"

2026-04-28T17:14:10Z

6. 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch): rename section to Space Needle like how BMO 2 is "Hydra and Antihydra"

← Older revision		Revision as of 17:14, 28 April 2026
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	Does there exist a positive integer <math>i</math> such that <math>b_i = f(a_i)-1</math>?		Does there exist a positive integer <math>i</math> such that <math>b_i = f(a_i)-1</math>?

	=== 6. ~~{{TM\|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD\|undecided}}~~ ===		=== 6. [[Space Needle]] ===
	Let <math>f(b) = b + k + 3a</math>, where <math>k</math> and <math>a</math> are non-negative integers satisfying <math>b = (2a+1)\cdot 2^k</math>.		Let <math>f(b) = b + k + 3a</math>, where <math>k</math> and <math>a</math> are non-negative integers satisfying <math>b = (2a+1)\cdot 2^k</math>.

Sligocki: /* 8. 1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch) */ Improve floor bracket sizes

2026-04-28T17:11:58Z

8. 1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch): Improve floor bracket sizes

← Older revision		Revision as of 17:11, 28 April 2026
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	<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}		<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
	(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n > \lfloor\frac{b_n}{2}\rfloor \\		(a_n-\left\lfloor\frac{b_n}{2}\right\rfloor-3, 3\left\lfloor\frac{b_n+1}{2}\right\rfloor+6) & \text{if }a_n > \left\lfloor\frac{b_n}{2}\right\rfloor \\
	(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor		(3a_n+5, b_n-2a_n) & \text{if }a_n \le \left\lfloor\frac{b_n}{2}\right\rfloor
	\end{cases}</math>		\end{cases}</math>

	for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_n = \lfloor\frac{b_n}{2}\rfloor + 1</math>?		for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_n = \left\lfloor \frac{b_n}{2} \right\rfloor + 1</math>?

	== Solved problems ==		== Solved problems ==

Coda: Add link to Lean formalizations

2026-04-11T13:57:09Z

Add link to Lean formalizations

← Older revision		Revision as of 13:57, 11 April 2026
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	The purpose of the BMO is twofold. First, statements where non-essential details (related to tape encoding, number of steps, etc.) are discarded are more suitable to be shared with mathematicians who perhaps are able to help. Second, it's a way to jokingly highlight how a hard question could appear deceptively simple.		The purpose of the BMO is twofold. First, statements where non-essential details (related to tape encoding, number of steps, etc.) are discarded are more suitable to be shared with mathematicians who perhaps are able to help. Second, it's a way to jokingly highlight how a hard question could appear deceptively simple.

			BMO problems have been formalized in Lean and added to the DeepMind formal-conjectures database ([https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Other/BeaverMathOlympiad.lean Github]).

	== Unsolved problems ==		== Unsolved problems ==

RobinCodes: Fix formatting

2026-04-10T09:30:16Z

Fix formatting

← Older revision		Revision as of 09:30, 10 April 2026
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	Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?		Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?

	=== 8. {{TM\|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA		=== 8. {{TM\|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA\|undecided}} ===
	\|undecided}} ===

	Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and		Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and

RobinCodes: /* 8. 1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch) */ Changed to canonical TM

2026-04-10T09:27:25Z

8. 1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch): Changed to canonical TM

← Older revision		Revision as of 09:27, 10 April 2026
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	Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?		Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?

	=== 8. {{TM\|~~1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF~~---_0LA1LA\|undecided}} ===		=== 8. {{TM\|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA
			\|undecided}} ===

	Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and		Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and

Mxdys: /* 8. {{TM|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}} */

2026-04-10T05:25:17Z

8. {{TM|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}}

← Older revision		Revision as of 05:25, 10 April 2026
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	<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}		<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
	(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n ~~\ge \lfloor\frac{b_n}{2}\rfloor + 2 \\~~		(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n > \lfloor\frac{b_n}{2}\rfloor \\
	~~(0, 0) & \text{if }a_n =~~ \lfloor\frac{b_n}{2}\rfloor ~~+ 1~~ \\
	(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor		(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor
	\end{cases}</math>		\end{cases}</math>

	for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>~~a_i~~ = ~~b_i = 0~~</math>?		for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_n = \lfloor\frac{b_n}{2}\rfloor + 1</math>?

	== Solved problems ==		== Solved problems ==

Mxdys: /* 8. {{TM|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}} */

2026-04-10T05:21:07Z

8. {{TM|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}}

← Older revision		Revision as of 05:21, 10 April 2026
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	<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}		<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
	(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n \ge ~~\lfloor\frac{b_n}{2}\rfloor + 3 \\~~		(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n \ge \lfloor\frac{b_n}{2}\rfloor + 2 \\
	~~(2, 3\lfloor\frac{b_n+1}{2}\rfloor+8) & \text{if }a_n =~~ \lfloor\frac{b_n}{2}\rfloor + 2 \\
	(0, 0) & \text{if }a_n = \lfloor\frac{b_n}{2}\rfloor + 1 \\		(0, 0) & \text{if }a_n = \lfloor\frac{b_n}{2}\rfloor + 1 \\
	(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor		(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor

Mxdys: /* Unsolved problems */

2026-04-10T05:12:00Z

Unsolved problems

← Older revision		Revision as of 05:12, 10 April 2026
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	Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?		Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?

			=== 8. {{TM\|1RB0LD_0RC0RF_0RD0RA_1LE0RD_1LF---_0LA1LA\|undecided}} ===

			Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and

			<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
			(a_n-\lfloor\frac{b_n}{2}\rfloor-3, 3\lfloor\frac{b_n+1}{2}\rfloor+6) & \text{if }a_n \ge \lfloor\frac{b_n}{2}\rfloor + 3 \\
			(2, 3\lfloor\frac{b_n+1}{2}\rfloor+8) & \text{if }a_n = \lfloor\frac{b_n}{2}\rfloor + 2 \\
			(0, 0) & \text{if }a_n = \lfloor\frac{b_n}{2}\rfloor + 1 \\
			(3a_n+5, b_n-2a_n) & \text{if }a_n \le \lfloor\frac{b_n}{2}\rfloor
			\end{cases}</math>

			for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_i = b_i = 0</math>?

	== Solved problems ==		== Solved problems ==

RobinCodes: /* Solved problems */ Add links to Discord for the formalised solution

2026-04-01T14:49:19Z

Solved problems: Add links to Discord for the formalised solution

← Older revision		Revision as of 14:49, 1 April 2026
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	Link to Discord discussion: https://discord.com/channels/960643023006490684/1084047886494470185/1252634913220591728		Link to Discord discussion: https://discord.com/channels/960643023006490684/1084047886494470185/1252634913220591728

			Formalised solution: [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Initial announcement], [https://discord.com/channels/960643023006490684/1259770421046411285/1488743526882738276 Lean proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488781537699696821 LLM-translated Rocq proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488898995865784442 Proof of closure of existing mid-level rules].

	=== 4. {{TM\|1RB3RB---1LB0LA_2LA4RA3LA4RB1LB\|non-halt}} ===		=== 4. {{TM\|1RB3RB---1LB0LA_2LA4RA3LA4RB1LB\|non-halt}} ===

Polygon: Added Category:Individual machines

2026-02-20T14:02:06Z

Added Category:Individual machines

← Older revision		Revision as of 14:02, 20 February 2026
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	* {{TM\|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD}}		* {{TM\|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD}}
	* {{TM\|1RB0RD_0LC1RA_0RA1LB_1RE1LB_1LF1LB_---1LE}}		* {{TM\|1RB0RD_0LC1RA_0RA1LB_1RE1LB_1LF1LB_---1LE}}

			[[Category:Individual machines]]