2024 Year Challenge · Integrity

	Core	Further Core	Expansion
$-1$	$(2+0!)/(2-5)$
$0$	$(2+0!+2)!-5!$
$1$	$(2+0!+2)!/5!$
$2$	$20/(2 \times 5) \\ (2+0!)!/(-2+5)$
$3$		$(2+0!) \times 2 \times .5$
$4$	$(2+0!)^2-5$	$2+0!+2 \times .5$
$5$	$-2+0! \times (2+5)$	$20 \times .25$
$6$	$(2-0!) \times (-2+5)$	$(2+0!)! \times 2 \times .5$
$7$	$(2+0!)! \times 2-5$	$(2+0!)!+2 \times .5$
$8$	$20 \times 2/5 \\ -2 \times 0!+2 \times 5$	$20/2.5$
$9$			$-2+\arcsin(0!)/(2 \times 5) \\ 2 \times \arctan(0!)/2/5$
$10$		$(2+0!+2)/.5$
$11$		$(.2+0+2) \times 5$	$2+\arcsin(0!)/(2 \times 5) \\ -2+\arcsin(0!) \times .2-5$
$12$	$2 \times 0!+2 \times 5 \\ -20+2^5$	$(2+0!)/.25$
$13$	$((2+0!)!+2+5$		$-2+\arctan(0!)/(-2+5)$
$14$	$2 \times 0! \times (2+5) \\ (2+0!)^2+5$
$15$	$20/2+5$	$(2+0!)! \times 2.5$
$16$		$((2+0!)!+2)/.5 \\ (.2+0!+2) \times 5$	$-2+\arctan(0!) \times 2/5$
$17$	$(2+0!)! \times 2+5$		$2+\arctan(0!)/(-2+5)$
$18$	$(2+0!)! \times (-2+5)$	$(2+0!)!^2 \times .5$	$(2+\arcsin(0!)-2)/5$
$19$		$20-2 \times .5$
$20$		$2/.02/5 \\ (20^2)^{.5}$	$2+\arctan(0!) \times 2/5$
$21$		$20+.2 \times 5$
$22$	$-(2+0!)+25$
$23$	$20-2+5$
$24$	$(2+0!+2)!/5$	$(2+0!)!/.25$
$25$	$(2+0!+2) \times 5$	$(2+0!)!/.2-5$
$26$	$-(2+0!)!+2^5$		$2 \times (\arctan(0!)-2^5) \\ 2 \times (\arcsin(0!) \times .2-5)$
$27$	$20+2+5 \\ (2+0!)^{(-2+5)}$
$28$	$(2+0!)+25$		$-2+\arcsin(0!)/(-2+5)$
$29$	$-2-0!+2^5$
$30$	$(2+0!) \times 2 \times 5$	$(.2+0!) \times 25$	$\arcsin(2-0!)/(-2+5)$
$31$	$(2+0!)!^2-5$	$(.2+(0!+2)!) \times 5$
$32$	$2 \times 0+2^5$		$2+\arcsin(0!)/(-2+5)$
$33$	$(2-0!)+2^5$
$34$	$2 \times 0!+2^5$		$.2 \times \arctan(0!)+25$
$35$	$20 \times 2-5$	$(2+0!)!/.2+5$	$(.2 \times \arctan(0!)-2) \times 5$
$36$	$(2+0!)! \times (-2+5)!$	$((2+0!)!^2)^{.5}$	$2 \times \arctan(0!) \times 2/5$
$37$
$38$	$(2+0!)!+2^5$
$39$
$40$	$((2+0!)!+2) \times 5$		$2 \times (\arctan(0!)-25)$
$41$	$(2+0!)!^2+5$
$42$			$-2+(\arcsin(0!)-2) \times .5$
$43$			$-2+\arctan(0!)/.2/5$
$44$			$2+\arctan(0!)+2-5 \\ -2+(\arcsin(0!)+2) \times .5$
$45$	$20 \times 2+5$	$20 \times 2.5$
$46$		$2 \times (-0!+(2/.5)!)$	$-2+\arctan(0!)-2+5 \\ 2 \times (\arcsin(0!) \times .2+5)$
$47$			$2+\arctan(0!)/.2/5$
$48$			$2+(\arcsin(0!)+2) \times .5$
$49$			$\arctan(2-0!)+2/.5$
$50$		$2/.02 \times .5 \\ 2 \times (0! \times (2/.5)!)$
$51$			$2+\arctan(0!)+2/.5$
$52$	$20+2^5$		$2+\arcsin(0!)/2+5$
$53$			$2+\arctan(0!)+(-2+5)!$
$54$			$.2 \times \arcsin(0!) \times (-2+5)$
$55$			$(.2 \times \arctan(0!)+2) \times 5$
$56$			$-2+\arcsin(0!)-2^5$
$57$
$58$
$59$	$2^{(0!+2)!}-5$
$60$	$(2+0!) \times 2 \times 5 \\ 20 \times (-2+5)$	$(2+0!)!/.2/.5$	$2+\arcsin(0!)-2^5 \\ 2 \times \arcsin(0!)/(-2+5)$
$61$
$62$
$63$
$64$	$2^{(-0!+2+5)}$		$-2+\arcsin(0!)-(2/.5)!$
$65$
$66$
$67$			$-2+\arctan(0!)+(2/.5)!$
$68$			$2+\arcsin(0!)-(2/.5)! \\ -2+\arctan(0!)+25$
$69$	$2^{(0!+2)!}+5$		$\arctan(2-0!)+(2/.5)!$
$70$			$2 \times (\arctan(0!)-2 \times 5)$
$71$			$2+\arctan(0!)+(2/.5)!$
$72$			$2+\arctan(0!)+25$
$73$
$74$
$75$	$(2+0!) \times 25$		$-2+\arctan(0!)+2^5$
$76$			$2 \times (\arctan(0!)-2-5) \\ (.2 \times \arctan(0!))^2-5$
$77$
$78$			$2 \times (\arctan(0!)-(-2+5)!)$
$79$			$2+\arctan(0!)+2^5$
$80$	$20^2/5$	$2/.025 \\ 20/.25$
$81$
$82$			$2 \times (\arctan(0!)-2/.5)$
$83$
$84$	$-((2+0!)!)^2+5!$		$2 \times (\arctan(0!)+2-5)$
$85$
$86$			$(.2 \times \arctan(0!))^2+5$
$87$			$\arcsin(2-0!)+2-5$
$88$		$-2^{(0!/.2)}+5!$	$2+\arcsin(0!)-2/.5 \\ 2 \times (\arctan(0!)-2 \times .5)$
$89$			$2+\arcsin(0!)+2-5$
$90$			$\arctan(2+0!-2)/.5$
$91$			$-2+\arcsin(0!)-2+5$
$92$			$-2+\arcsin(0!)+2/.5$
$93$			$\arcsin(2-0!)-2+5$
$94$			$-2+\arcsin(0!)-2/.5$
$95$		$2/.02-5$
$96$	$(2+0!) \times 2^5$		$2+\arcsin(0!)+2/.5$
$97$			$\arcsin(2-0!)+2-5$
$98$			$-2+\arcsin(0!)+2 \times 5 \\ (2+\arctan(0!)+2)/.5$
$99$			$2+\arcsin(0!)+2+5$
$100$			$\arcsin(2-0!)+2 \times 5 \\ 2 \times \arctan(0!)+2 \times 5$

$101$		$202 \times .5$
$102$	$-20+2+5!$		$2 \times (\arctan(0!)+(-2+5)!) \\ 2+\arcsin(0!)+2 \times 5$
$103$
$104$			$2 \times (\arctan(0!)+2+5)$
$105$
$106$
$107$
$108$	$-2 \times (0!+2)!+5!$
$109$
$110$	$-20/2+5!$
$111$	$-(2+0!)^2+5!$
$112$			$-2+\arcsin(0!)+(2/.5)!$
$113$		$-2-0!/.2+5!$	$-2+\arcsin(0!)+25$
$114$	$-(2-0!+2)!+5!$
$115$			$\arcsin(2-0!)+25$
$116$			$2+\arcsin(0!)+(2/.5)!$
$117$		$2-0!/.2+5!$	$2+\arcsin(0!)+25$
$118$	$2 \times 0-2+5!$
$119$	$2-0!-2+5!$
$120$	$(2 \times 0! \times 2)! \times 5 \\ 2+0-2+5! \\ 20 \times (-2+5)!$		$-2+\arcsin(0!)+2^5$
$121$	$2+0!-2+5!$
$122$			$\arcsin(2-0!)+2^5$
$123$	$2-0!+2+5!$
$124$	$(2+0!)!-2+5!$		$2+\arcsin(0!)+2^5$
$125$	$2+0!+2+5!$
$126$	$(2-0!+2)!+5!$
$127$		$2+0!/.2+5!$
$128$	$2^{(0+2+5)} \\ (2+0!)!+2+5!$
$129$	$(2+0!)^2+5!$
$130$	$(4!+2) \times 5 \\ 20/2+5!$
$132$	$2 \times (0!+2)!+5!$
$133$			$-2+\arctan(0!) \times (-2+5)$
$134$
$135$			$\arctan(2-0!) \times (-2+5)$
$136$
$137$			$2+\arctan(0!) \times (-2+5)$
$138$	$20-2+5!$
$139$
$140$	$20 \times (2+5)$
$141$
$142$	$20+2+5! \\ -2+((0!+2)!)!/5$
$144$	$((2-0!+2)!)!/5$	$(2+0!)! \times (2/.5)!$
$146$	$2+((0!+2)!)!/5$
$148$	$(2+0+2)!+5!$		$2 \times \arcsin(0!)-2^5$
$150$	$(2+0!)! \times 25$	$(2+0!)!/.2 \times 5$
$160$	$20 \times 2+5!$
$180$	$(2+0!)!^2 \times 5$	$((2+0!)!)! \times .25$	$2 \times \arcsin(0!) \times 2 \times .5$
$200$	$20 \times 2 \times 5$	$2/.02/.5$
$300$
$400$
$500$	$20 \times 25$	$2/.02 \times 5$
$600$	$((2+0!) \times 2)!-5!$
$700$	$-20+((-2+5)!)!$
$800$		$20^2/.5$
$900$			$\arcsin(2-0!) \times 2 \times 5$
$1000$
$2000$	$20^2 \times 5$

Comments

Damn, this year was super fruitful!
- Honestly, the $5$ is just too nice to work with. It’s great for getting a large number with $5! = 120$ , and works nicely with $.5 = 1/2$ .
As per usual, I’m saving the small numbers for the most ridiculous solutions I can find, which is why some perhaps ‘obvious’ ones are still empty.
$\arcsin(0!)$ is identical to $\arccos(0)$ , I just think it’s more fun.