contest stringclasses 315
values | contest_url stringclasses 1
value | url stringlengths 53 65 | alphabet stringclasses 20
values | name stringlengths 9 17 | score stringclasses 10
values | correct int64 0 467 | total int64 0 485 | editorials listlengths 1 6 | task_content stringlengths 28 1.49k |
|---|---|---|---|---|---|---|---|---|---|
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/181 | F | OMCäžæ¬æ¯(F) | 200 | 46 | 50 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/181"
}
] | äœå: éŽæšã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãã©ã®è§ã $180^\circ$ æªæºã§ãããããªåè§åœ¢ $ABCD$ ã«ãããŠ, 察è§ç·ã®äº€ç¹ã $O$ ãšãããšã, 以äžã®æ¡ä»¶
$$AC=BD,\ \ AB=BC,\ \ AO:OD=1:2$$
ãæç«ããŸãã. $\angle CAD$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠãã ãã. æŽæ°ã«ãªããªãã£ãå Žåã¯, åæ¯ãååãæŽæ°ã®åæ°ã§è¡šããå€ã«ãªã£ãŠããŸã. ãã®å Žå, çãããã以äžçŽ... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/182 | G | OMCäžæ¬æ¯(G) | 200 | 50 | 54 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/182"
}
] | ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãã©ã®è§ã $180^\circ$ æªæºã§ãããããªåè§åœ¢ $ABCD$ ã«ãããŠ, 以äžã®æ¡ä»¶
$$\angle BAC=40^\circ,\ \ \angle CAD=80^\circ,\ \ AB+AD=AC=CD$$
ãæç«ããŠãããšã, $\angle CBD$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠãã ãã. æŽæ°ã«ãªããªãã£ãå Žåã¯, åæ¯ãååãæŽæ°ã®åæ°ã§è¡šããå€ã«ãªã£ãŠããŸã. ãã®å Žå, ç... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/183 | H | OMCäžæ¬æ¯(H) | 200 | 39 | 42 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/183"
}
] | ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãäžå³ã®ãããªäžèŸº $10\text{cm}$ ã®æ£æ¹åœ¢ $ABCD$ ããã, $EFGHIJKL$ ã¯æ£å
«è§åœ¢ã§ã. æ°Žè²éšå, æ©è²éšå, é»ç·è²éšå, æ¡è²éšåã®é¢ç©ã®åèšããããã $S,T,U,V$ ãšãããšã,
$$S\times 1+T\times 2+U \times 3+V\times 4$$
ã¯äœ $\text{cm}^2$ ã§ããïŒ
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/184 | I | OMCäžæ¬æ¯(I) | 200 | 37 | 50 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/184"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ã$3$ ã€ã®çžç°ãªãæ£æŽæ°ããããŸã. ãã®äžãã $2$ ã€ãéžã¶æ¹æ³ã¯ $3$ éããããŸãã, ããããã«ã€ã㊠$2$ ã€ã®æŽæ°ã®åãšç©ãé»æ¿ã«æžãæäœãè¡ããŸãã. ãã ã, éè€ããå Žå㯠$1$ ã€ã®ã¿æžããŸãã. çµæãšããŠ, é»æ¿ã«ã¯ $5$ ã€ã®æ°ãæžãã, ãã®ãã¡ $2$ çªç®ã«å°ãããã®ã¯ $28$ ã§ãã. $5$ ã€ã®ãã¡æã倧ããæ°ãšããŠããåŸããã®ããã¹ãŠæ±ã, ãããã®ç·åã解çããŠãã ãã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/185 | J | OMCäžæ¬æ¯(J) | 200 | 35 | 41 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/185"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãäžå³ã®ãããªäžè§åœ¢ããããŸãïŒäžèŸº $\displaystyle 1\text{cm}$ ã®æ£äºååè§åœ¢ãšäžèŸº $\displaystyle x$ ã®æ£äºååè§åœ¢ã®é¢ç©ã®å·®ã¯äœ $\text{cm}^{2}$ ã§ããïŒ
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠãã ãã. æŽæ°ã«ãªããªãã£ãå Žåã¯, åæ¯ãåå... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/186 | K | OMCäžæ¬æ¯(K) | 200 | 48 | 49 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/186"
}
] | äœå: 岩çšã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãäžå³ã®ãããªåè§åœ¢$\displaystyle ABCD$ããã,
$$\angle ABD=42^{\circ },\ \ \angle DBC=36^{\circ }\ \ \angle BCA=24^{\circ },\ \ \angle ACD=12^{\circ }$$
ãæãç«ã£ãŠããŸãïŒãã®ãšãïŒ$\displaystyle \angle DAC$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã.
... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/187 | L | OMCäžæ¬æ¯(L1) | 100 | 38 | 46 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/187"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãçœç³ãšé»ç³ãåèš $12$ åïŒæšªäžåã«äžŠã¹ãŸãïŒãã®ãšãïŒçœç³ã $3$ ã€é£ç¶ããŠäžŠã°ãªãããã«ããæ¹æ³ã¯ $M$ éããããŸã. $M$ ã解çããŠãã ãã.
ããã ã, å·Šå³å転ã§äžèŽãããã®ãåºå¥ãããã®ãšããŸã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠãã ãã. æŽæ°ã«ãªããªãã£ãå Žåã¯, åæ¯ãååãæŽæ°ã®åæ°ã§è¡šããå€ã«ãªã£ãŠããŸã. ãã®å Žå, çãããã以äžçŽåã§ããªãä»®åæ°ã§è¡šã, ååâåæ¯ããã®é ã§äžŠã¹ãŠ $... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/188 | M | OMCäžæ¬æ¯(L2) | 100 | 31 | 33 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/188"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãæ°åã®ã$\displaystyle 2$ããšã$\displaystyle 0$ããåèš $\displaystyle 12$ åïŒæšªäžåã«äžŠã¹ãŸãïŒé£ç¶ãã $\displaystyle 4$ ã€ã®æ°åãå·Šããé ã«èŠãŠãã£ããšãïŒã$\displaystyle 2020$ããšã$\displaystyle 0202$ããããããçŸããªãæ¹æ³ã¯ $M$ éããããŸã. $M$ ã解çããŠãã ãã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠã... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/189 | N | OMCäžæ¬æ¯(M) | 200 | 39 | 45 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/189"
}
] | äœå: é·è°·å·ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãã©ã®è§ã $180^\circ$ æªæºã§ãããããªåè§åœ¢ $ABCD$ ã $AD=BC=BD=CD=8\text{cm}$ ãã¿ãããŠãã, 蟺 $CD$ äžã« $AE=9\text{cm}$ ãªãç¹ $E$ ããšããš, $AE$ ãš $BC$ ã¯å¹³è¡ã§ãã. $AE$ ãš $BD$ ãç¹ $F$ ã§äº€ãã£ãŠãããšã, äžè§åœ¢ $ADF$ ã®é¢ç©ã¯äžè§åœ¢ $BCD$ ã®é¢ç©ã®äœåã§ããïŒ
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ããã®ãŸãŸè§£çããŠãã ãã... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/190 | O | OMCäžæ¬æ¯(N) | 200 | 9 | 21 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/190"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãäžèŸºãæŽæ° $\text{cm}$ ã®æ£å
è§åœ¢ãïŒäžèŸº $\displaystyle 1\text{cm}$ ã®æ£äžè§åœ¢ã®ãã¹ã«åå²ãããŠããç€ããããŸã. äžã€ç®ã®å³ã¯äžèŸºã $\displaystyle 3\text{cm}$ ã®å Žåã瀺ããŠããŸã. ãã $\displaystyle 2$ ã€ã®ãã¹ãïŒ$\displaystyle 1$ ã€ã®é ç¹ã ããå
±æããŠããŠïŒå
±æããŠããé ç¹ãäžå¿ã«äžæ¹ã®ãã¹ã $\displaystyle 180^{\circ }$ å転ãããããšã§ããäžæ¹ã®ãã¹ã«éãªããšãïŒããã $\displays... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/191 | P | OMCäžæ¬æ¯(O) | 200 | 15 | 23 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/191"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãäžå³ã®ãããªäžèŸº $\displaystyle 1\text{cm}$ ã®æ£åäºé¢äœ $\displaystyle S$ ããã. $\displaystyle M$ ãš $\displaystyle N$ ã¯ãããã蟺ã®äžç¹ã§ãïŒ$\displaystyle S$ ãçŽç· $\displaystyle MN$ ã軞㫠$90^\circ$ å転ããããã®ã $\displaystyle T$ ãšãïŒ$\displaystyle S$ ãš $\displaystyle T$ ã®å
±ééšåã®ç«äœã $\displaystyle U$ ãšããŸ... |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/58 | A | OMC009(A) | 100 | 227 | 231 | [
{
"content": "ãã$13$ ããã³ $19$ ãæ³ãšããŠãšãã« $-6$ ãšååã§ããããšè¡šçŸã§ãããã, æ±ããå€ã¯ $13\\times 19-6=\\textbf{241}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc009/editorial/58"
}
] | ã$13$ ã§å²ããš $7$ äœã, $19$ ã§å²ããš $13$ äœããããªæå°ã®æ£æŽæ°ãæ±ããŠãã ãã. |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/59 | B | OMC009(B) | 200 | 195 | 212 | [
{
"content": "ã$k=\\textbf{10403}=101\\times 103$ ã®ãšã $10403-4=10399$ ã§æ倧ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc009/editorial/59"
}
] | ã$k$ ã $1$ ä»¥äž $10404$ 以äžã®æŽæ°ãšãããšã, $k-(k\text{ã®æ£ã®çŽæ°ã®åæ°})$ ãæ倧ãšãªããã㪠$k$ ã®ç·åãæ±ããŠãã ãã. |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/60 | C | OMC009(C) | 300 | 159 | 186 | [
{
"content": "ãçŽç· $DH$ ã¯çŽç· $BF$ ãç¹ $A$ ãäžå¿ã«åæèšåãã« $90^{\\circ}$ å転ãããã®ã§ããããïŒç¹ $A$ ãšçŽç· $BF$ ã®è·é¢ã $d$ ãšããã° $AK^2=2d^2$ ãæç«ïŒèšç®ããã°çŽç· $BF$ ã®æ¹çšåŒã¯ $3x-7y=9$ ã§ãããã $$AK^2=2\\biggl(\\frac{|9|}{\\sqrt{3^2+7^2}}\\biggr)^2=\\frac{81}{29}$$\r\nãšãªãïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{110}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathco... | ã座æšå¹³é¢äžã«æ£æ¹åœ¢ $ABCD,EFGH$ ããã,é ç¹ã¯ã¢ã«ãã¡ãããé ã«åŸã£ãŠåæèšåãã«äžŠã³ãŸã.
$$A=E=(0,0),C=(3,3),G=(1,-2)$$
ã§ãããšã, çŽç· $DH$ ãšçŽç· $BF$ ã®äº€ç¹ã $K$ ãšãããš, $AK^2$ ã¯æ¢çŽåæ°ã«ãã£ãŠ $\dfrac{M}{N}$ ãšè¡šãããŸã. $M+N$ ãæ±ããŠãã ãã. |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/61 | D | OMC009(D) | 400 | 157 | 164 | [
{
"content": "ã$M$ ã $99$ åç®ã®æäœåŸã«ããã $X$ ã® $99!$ çªç®ãŸã§ã®ç·åãšããã° $M=99S+98,T=100M+99$ ã容æã«ç¢ºèªã§ããïŒåŸã£ãŠ $|T-9900S|=\\bf{9899}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc009/editorial/61"
}
] | ã次ã®ããã«ããŠæ°å $X$ ãäœããŸãïŒ
- ã¯ãã㯠$X=\\{1\\}$ ã§ã. ãã®åŸ, 以äžã®æäœã $99$ åç¹°ãè¿ããŸã.
- $n\ (n=1, 2, \cdots, 99)$ åç®ã®æäœã§ã¯, $X$ ã®åŸã« $X$ ãšåããã®ã $n$ åä»ã足ã, ãã®åŸæ«å°Ÿã®èŠçŽ ã« $n$ ãå ããŸã.
åæäœã®æåŸã¯, $X$ ã®æ«å°Ÿã« $n$ ãè¿œå ããã®ã§ã¯ãªãããšã«æ³šæããŠãã ãã.\
ãäŸãã°, 以äžã®ããã«ãªããŸã.
- $1$ åç®ã®æäœåŸ, æ°å㯠$\\{1, 2\\}$ ã§ã.
- $2$ åç®ã®æäœåŸ, æ°å㯠$\\{1, 2, 1, 2, 1, 4\\}$ ã§ã.
ã$... |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/62 | E | OMC009(E) | 500 | 42 | 122 | [
{
"content": "ãæŽæ° $a$ ãçŽ æ° $p$ ã§å²ãåããåæ°ã $v_p(a)$ ã§è¡šãïŒ\r\nã«ãŒãã«æžãããæŽæ°ã $a_1,\\dots,a_n$ ãšãããšãïŒæ¡ä»¶ã¯å $k=1,\\dots,n$ ã«å¯Ÿã次ãæãç«ã€ããšãšåå€ïŒ\r\n\r\n- (a)ïŒä»»æã®çŽ æ° $p$ ã«å¯Ÿã $\\begin{cases}v_p(a_k)\\leq 50&(p\\leq 50)\\\\\\\\ v_p(a_k)=0&(p\\gt 50)\\end{cases}$ïŒ\r\n\r\n- (b)ïŒããçŽ æ° $p(\\leq 50)$ ãååšã㊠$v_p(a_k)\\gt\\bigl(\\sum_{i=1}... | ã$n$ ãæ£ã®æŽæ°ãšããŸã. 次ã®æ¡ä»¶ãæºããããã«, åºå¥ã§ãã $n$ æã®ã«ãŒãã« $1$ ã€ã〠$2$ 以äžã®æŽæ°ãæžã蟌ã¿ãŸã.
- ä»»æã®ã«ãŒãã«æžãããæŽæ° $a$ ãš, ãã®çŽ å æ° $p$ ã«ã€ããŠ, $p\leqq 50$ ã§ãã, $p^{51}$ 㯠$a$ ãå²ãåããªã.
- $n$ æã®ã«ãŒãããã©ã®ããã« $1$ æä»¥äž $n-1$ æ以äžã«ãŒããéžãã§ã, éžã°ããã«ãŒãã«æžãããæŽæ°ã®ç©ã¯, éžã°ããªãã£ãã«ãŒãã®ç©ã§å²ãåããªã.
ããã®ãããªæžã蟌ã¿æ¹ãååšãã $n$ ãšããŠèããããæ倧ã®å€ã $m$ ãšãããš, $n=m$ ã®ãšãæžã蟌ã¿æ¹ã¯ $K$ éããããŸã.... |
OMC009 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc009/tasks/63 | F | OMC009(F) | 600 | 12 | 55 | [
{
"content": "ã$i=1,\\dots,2020$ ã«å¯Ÿã $b_i=a_i-a_{i-1}$ ãšããã°ïŒ$S_0,S_1,\\dots,S_{2020}$ ã§èŠãããéšåã®é¢ç©( $S$ ãšãã)ã¯æ¬¡ã§æ±ããããïŒ\r\n$$S=4+\\sum_{i=1}^{2020}\\bigl(b_i(2i+2-b_i)+2i+3\\bigr)$$\r\n\r\nãŸãåã®æ§è³ªã¯æ¬¡ã®ããã«èšãæããããïŒ\r\n- å $i=1,2,\\dots,2020$ ã«å¯Ÿã㊠$0\\leq b_i\\leq 2$ ã§ããïŒ\r\n- å $i=1,2,\\dots,2019$ ã«å¯Ÿã㊠$b_i+b_{i+1}\\geq... | ã以äžã®æ§è³ªãã¿ããæŽæ°ã®çµ $(a_0, a_1, \dots, a_{2020})$ ãèããŸã.
- $a_0=0$ ã§ãã.
- å $i=1,2,\dots,2020$ ã«å¯ŸããŠ, $0\leqq a_{i}-a_{i-1}\leqq 2$ ã§ãã.
- å $i=2,3,\dots,2020$ ã«å¯ŸããŠ, $a_{i}-a_{i-2}\geqq 2$ ã§ãã.
ããã®ãããªæŽæ°åã«å¯ŸããŠ, 座æšå¹³é¢äžã®æ£æ¹åœ¢ $S_i$ ã,
$$(a_i,a_i),\quad (a_i+i+2,a_i),\quad (a_i+i+2,a_i+i+2),\quad (a_i,a_i+i+2)$$
ãé ç¹ã«æã€... |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/52 | A | OMC008(A) | 100 | 147 | 170 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc008/editorial/52"
}
] | ãäžèŸºã®é·ãã $12\sqrt 2$ ã®ç«æ¹äœ $ABCD-EFGH$ ããã, é ç¹ $A$ ããé·ã $24$ ã®çŽã§ç¹ãããç¹ $I$ ãç«æ¹äœã®é¢äžãåããŸã. ç¹ $I$ ãåããç¯å²ã®å€åšã®é·ãã $L$ ãšãããšã, $\cfrac{L}{\pi}$ ãæ±ããŠãã ãã.\
ããã ã, çŽã¯åŒãã§ãæ§ããŸããã, ç«æ¹äœã®å
éšã¯éããŸãã. |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/53 | B | OMC008(B) | 200 | 95 | 125 | [
{
"content": "ã$s$ ãå¥æ°ã®å Žå次ãæãç«ã€ããïŒä»»æã® $s,t$ ã§æ¡ä»¶ãã¿ããïŒ\r\n\r\n$$\\sum_{k=1}^{s}k^t=\\sum_{k=0}^{(s-1)\\/2}\\bigl(k^t+(s-k)^t\\bigr)\\equiv 0\\pmod{s}$$\r\n\r\nã$s$ ãå¶æ°ã®å Žå次ãæãç«ã¡ïŒ$\\bigl(\\frac{s}{2})^t\\equiv 0\\pmod{s}$ ãšãªã $s$ 㯠$t=1$ ã®ãšãååšããïŒ$t\\geq 3$ ã®ãšã $4$ ã®åæ°ã§ããïŒ\r\n\r\n$$\\sum_{k=1}^{s}k^t=\\Bigl(\\frac{s... | ããšãã« $1$ ä»¥äž $2020$ 以äžã®æŽæ° $s$ ãšå¥æ° $t$ ã®çµã§ãã£ãŠ, $\displaystyle \sum^{s}_{k=1}k^t$ ã $s$ ã®åæ°ãšãªããããªãã®ã®åæ°ãæ±ããŠãã ãã. |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/54 | C | OMC008(C) | 300 | 139 | 149 | [
{
"content": "$$y=\\begin{cases}\r\nx^2+4x+2=(x+2)^2-2&(x\\lt -1)\\\\\\\\\r\nx^2-4x-6=(x-2)^2-10&(x\\geq -1)\r\n\\end{cases}$$\r\n\r\nã«æ³šæããã° $M,m$ ã¯æ¬¡ã®ããã«æ±ããããïŒ\r\n\r\n$$M=\\begin{cases}\r\ns^2+4s+2&(s\\lt -3)\\\\\\\\\r\n-1&(-3\\leq s\\lt -1)\\\\\\\\\r\ns^2-4s-6&(-1\\leq s\\lt 1)\\\\\\\\\r\ns^2-10&(s\\geq 1)\\\\... | ã$y=x^2-|4x+4|-2$ ãšã, $x$ ã $s\leq x\leq s+2$ ã®ç¯å²ãåããšãã® $y$ ã®æ倧å€, æå°å€ããããã $M, m$ ãšããŸã. $M-m=4$ ãšãªããã㪠$s$ ã¯å°ããé ã« $-a, -\sqrt{b}, c, d$ ãšè¡šããã®ã§, $1000a+100b+10c+d$ ãæ±ããŠãã ãã.
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/55 | D | OMC008(D) | 400 | 84 | 122 | [
{
"content": "ã次ã®ããã«èããã°ïŒæ±ããå€ã¯ $2^{11}-2=\\bf{2046}$ïŒ\r\n- äžãã $1$ è¡ç®ã®ã©ã®é£æ¥ãã $2$ ãã¹ã«ãåãè²ãå¡ãããŠããªãå ŽåïŒã©ã®è¡ãçœé»äº€äºã«å¡ãããŠããïŒãã£ãŠåè¡ã®å¡ãæ¹ãèããã° $2^{10}$ éãïŒ\r\n- äžãã $1$ è¡ç®ã®ããé£æ¥ãã $2$ ãã¹ã«åãè²ãå¡ãããŠããå ŽåïŒ$2$ è¡ç®ä»¥éã®å¡ãæ¹ã¯äžæã«å®ãŸãïŒãã£ãŠ $1$ è¡ç®ã®å¡ãæ¹ãèããã° $2^{10}-2$ éãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contest... | ã$10\times 10$ ã®ãã¹ç®ããããŸã. 次ã®æ¡ä»¶ãæºããããã«, ãã¹ç®ãçœãšé»ã§å¡ãåããæ¹æ³ã¯äœéããããŸãã.
- æ¡ä»¶ïŒã©ã® $2\times 2$ ã®éšåãã¹ç®ã«ãããŠã, ãã®äžã«çœãã¹ãšé»ãã¹ã $2$ åãã€å«ãŸããŠãã.
ããã ã, $2\times 2$ ã®éšåãã¹ç®ãšã¯, é£æ¥ããè¡ãšé£æ¥ããåã®å
±ééšåãšãªã $4$ ãã¹ã®ããšãæããŸã. |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/56 | E | OMC008(E) | 500 | 38 | 57 | [
{
"content": "ã$\\triangle ABC$ ã®å€æ¥åãšçŽç· $AI$ ã®äº€ç¹ã®ãã¡ $A$ ã§ãªãæ¹ã $M$ ãšããã°ïŒæåäºå®ãšã㊠$M$ ã¯åŒ§ $BC$ ã®äžç¹ã§ãã $BM=CM=IM$ ãæç«ïŒä»®å®ãšããããã° $\\triangle ADO\\equiv\\triangle MIO$ ãåŸããïŒç¹ã« $AI=MD$ ã§ããïŒ\r\nãã㧠$AI=x,DI=y$ ãšããã°ïŒ$\\triangle ABM\\sim\\triangle BDM$ ãã $(x+y)^2=x(2x+y)$ ã§ãããã $x,y\\gt 0$ ãã $\\frac{x}{y}=\\frac{1+\\sqrt... | ãäžè§åœ¢ $ABC$ ã®å
å¿ã $I$, å€å¿ã $O$ ãšã, çŽç· $AI$ ãš $BC$ ã®äº€ç¹ã $D$ ãšãããš, $IO \parallel BC, AO\perp DO$ ãæãç«ã¡ãŸãã. äžè§åœ¢ $OBC$ ã®é¢ç©ã $39-13\sqrt 5$ ã§ãããšã, äžè§åœ¢ $ABC$ ã®é¢ç©ãæ±ããŠãã ãã. |
OMC008 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc008/tasks/57 | F | OMC008(F) | 600 | 28 | 68 | [
{
"content": "ã次ã®è£é¡ã瀺ããŠããïŒ\r\n\r\n----\r\n\r\n**è£é¡**ïŒéè² æŽæ° $N$ ã«å¯Ÿã次ãæãç«ã€ïŒ\r\n$$\\sum_{n=0}^{N+1}(-1)^{N+1-n}{}\\_{N+1}\\mathrm{C}\\_{n}n^{N+1}=(N+1)!$$\r\n**蚌æ**ïŒ$N+1$ åã®ããŒã«ãã¡ããã© $N+1$ è²ã§å¡ãåããæ¹æ³ã¯ $(N+1)!$ éãããïŒäžæ¹ã§ $N+1$ åã®ããŒã«ã $n$ è²ä»¥äžã§å¡ãåããæ¹æ³ã¯ $n^{N+1}$ éãããïŒãããã«å¯Ÿãå
é€åçãçšããã°è£é¡ã®åŒãåŸãããïŒ(蚌æçµ)\r\n\r\n----\r\n\r\nã$N=... | ã$\displaystyle \sum^{2020}\_{n=0}{}\_{2020}{\rm C}\_n(n+1)^{2020}$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ãã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/46 | A | OMC007(A) | 100 | 205 | 212 | [
{
"content": "ã$n$ ãå¶æ°ã®ãšã $P_n=\\dfrac{n}{2}$ïŒ$n$ ãå¥æ°ã®ãšã $P_n=\\dfrac{n-1}{2}$ ã§ããïŒ$P_1=0$ ã§ãããã $S_{2020}$ ã¯æ¬¡ã®ããã«èšç®ã§ããïŒ\r\n\r\n$$S_{2020}\r\n=\\sum_{k=1}^{1010}(P_{2k-1}+P_{2k})\r\n=\\sum_{k=1}^{1010}(2k-1)\r\n=\\bf{1020100}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc007/edit... | ãåã $n$ ã§ãããã㪠$2$ ã€ã®æ£ã®æŽæ° $A_1 \leq A_2$ ã®çµã¿åããã®æ°ã $P_n$ ãšã, $P_2$ ãã $P_n$ ã®ç·åã $S_n$ ãšããŸã. $S_{2020}$ ãæ±ããŠãã ãã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/47 | B | OMC007(B) | 200 | 156 | 179 | [
{
"content": "ã$a_n=\\tan\\theta_n~\\biggl(-\\dfrac{\\pi}{2}\\lt\\theta_n\\lt\\dfrac{\\pi}{2}\\biggr)$ ãšããã° $\\theta_1=\\dfrac{\\pi}{6}$ ã§ããïŒ$\\theta_n$ ã®ç¯å²ã«æ³šæããã°\r\n\r\n$$\\tan\\theta_{n+1}=\\frac{-1+\\sqrt{1+\\tan^2\\theta_n}}{\\tan\\theta_n}=\\frac{1-\\cos\\theta_n}{\\sin\\theta_n}=\\tan\\frac{\\theta_n}{2}$... | ã$\displaystyle a_1=\frac{1}{\sqrt{3}}, a_{n+1}=\frac{-1+\sqrt{1+a_n^2}}{a_n}$ ãæºããæ°å ${a_n}$ ã«ã€ããŠ, $a_{10}=\tan\dfrac{\pi}{x}$ ã§ã. $x$ ãæ±ããŠãã ãã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/48 | C | OMC007(C) | 300 | 155 | 180 | [
{
"content": "ã$\\\\\\{A_i\\\\\\}$ ã®é
æ° $(=n)$ ã $1$ ä»¥äž $5$ 以äžã§ããã®ã¯æããïŒ\r\n\r\n- $n=1$ ã®å ŽåïŒ$\\\\\\{5145\\\\\\}$ ã® $1$ åïŒ\r\n- $n=2$ ã®å ŽåïŒ$5145$ ã®æ£ã®çŽæ° $d$ 㧠$1\\lt d\\leq 5145\\/d$ ãã¿ãããã®ã«äžå¯Ÿäžã§å¯Ÿå¿ãïŒãã®ãã㪠$d$ 㯠$7$ åååšããïŒ\r\n- $n=3$ ã®å ŽåïŒæå°å€ãšããŠããåŸãã®ã¯ $3,5,7$ ã§ããïŒããããèããã°æ¡ä»¶ãæºããã®ã¯æ¬¡ã® $8$ åïŒ\r\n - æå°å€ã $3$ ã®å ŽåïŒ$\\\\\\{3,... | ã以äžã®æ¡ä»¶ãæºãã, æ£ã®æŽæ°ã®ã¿ãããªãæéå $\\{A_i\\}$ ã®æ°ãæ±ããŠãã ãã.
- $A$ ã®é
ã®ç·ç©ã¯ $5145 = 3\times5\times7\times7\times7$ ã§ãã.
- $A$ ã¯åºçŸ©å調å¢å ã§ãã.
- $A$ ã®ã©ã®é
ã $1$ ã§ãªã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/49 | D | OMC007(D) | 400 | 70 | 126 | [
{
"content": "ãå $k=1,\\dots,2019$ ã«å¯Ÿãäºæ¬¡æ¹çšåŒ $x^2-n_kx+n_{k+1}=0$ ã®æ£æŽæ°è§£ã $x=a,b~(a\\leq b)$ ãšãããšè§£ãšä¿æ°ã®é¢ä¿ããã³ $n_k\\geq n_{k+1}$ ãã $(a-1)(b-1)\\leq 1$ ãæãç«ã¡ïŒ$a,b$ ã¯æ£æŽæ°ã«æ³šæããã° $n_k=n_{k+1}+1$ ãŸã㯠$n_k=n_{k+1}=4$ ãåŸãïŒéã«ãããæç«ãããšã $x^2-n_kx+n_{k+1}=0$ ã®è§£ãæ£æŽæ°ã§ããããšã¯å®¹æã«ç¢ºèªã§ããïŒ\r\n\r\nããŸãæ¹çšåŒ $x^2-n_{2020}x+n_1=0$ ãæ£æŽæ°è§£ããã€ããš... | ãæ£ã®æŽæ° $n_1\geq n_2\geq \cdots\geq n_{2020}$ ã«ã€ããŠ, $x$ ã® $4040$ 次æ¹çšåŒ
$$(x^2-n_1x+n_2)\cdots(x^2-n_{2019}x+n_{2020})(x^2-n_{2020}x+n_1)=0$$
ã®è€çŽ æ°è§£ããã¹ãŠæ£ã®æŽæ°ã§ãããšã, $n_1 $ã®åãåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/50 | E | OMC007(E) | 500 | 45 | 99 | [
{
"content": "ãæäœãç¹°ãè¿ãããšã§å
šãŠã®ã³ã€ã³ãè¡šåãã«ã§ããããã®å¿
èŠååæ¡ä»¶ã¯ïŒå·Šããå¥æ°çªç®ã«ããã³ã€ã³ã®ãã¡è£åãã®ãã®ã®ææ°ãšå·Šããå¶æ°çªç®ã«ããã³ã€ã³ã®ãã¡è£åãã®ãã®ã®ææ°ãäžèŽããããšã§ããïŒãã®ãããªäžŠã¹æ¹ã¯ïŒå·Šããå¥æ°çªç®ã«ããã³ã€ã³ãå
šãŠè£è¿ãããšã§è£åãã®ã³ã€ã³ã $1010$ æã§ãã䞊ã¹æ¹ã«äžå¯Ÿäžã§å¯Ÿå¿ä»ãããããã $m={}\\_{2020}\\mathrm{C}\\_{1010}$ ã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{2017}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/co... | ãè¡šè£ã®åºå¥ãã€ã $2020$ æã®ã³ã€ã³ã暪äžåã«äžŠãã§ããŸã. é«æšåã¯ä»¥äžã®æäœãäœåã§ãè¡ãããšãã§ããŸã. $0$ åã§ãæ§ããŸãã.
- æäœïŒé£æ¥ãã $2$ æã®ã³ã€ã³ã, å
±ã«è¡šãåããŠãã, ãŸãã¯å
±ã«è£ãåããŠãããšã, ããããåæã«è£è¿ã.
ãé«æšåã以äžã®æäœãäœåãç¹°ãè¿ããããš, ã³ã€ã³ã¯å
šãŠè¡šãåããŸãã. æåã® $2020$ æã®ã³ã€ã³ã®åããšããŠããããçµã¿åãã㯠$m$ éãã§ã. $m$ ãå²ãåãæ倧ã®çŽ æ°ãçããŠãã ãã. |
OMC007 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc007/tasks/51 | F | OMC007(F) | 600 | 30 | 76 | [
{
"content": "ã$T_n$ ã®å
æ¥åã®ååŸã $r_n$ ãšããã°æ¡ä»¶ãã次ãæãç«ã€ïŒ\r\n\r\n- $\\pi r_1^2=1$ïŒ\r\n- $T_n$ ã®å€æ¥åã®ååŸã¯ $r_{n+1}$ïŒ\r\n- $T_n$ ã®å
å¿ãšå€å¿éã®è·é¢ã¯ $r_n$ïŒ\r\n\r\nãã®ãšãEulerã®å®çãã $r_n^2=r_{n+1}^2-2r_nr_{n+1}$ ã§ããããïŒ$r_n,r_{n+1}\\gt 0$ ãã $r_{n+1}=(1+\\sqrt{2})r_n$ ãåŸãïŒãããçšããã° $T_{20201017}$ ã®å€æ¥åã®é¢ç© $S$ 㯠$S=(3+2\\sqrt{2})^{202010... | ãåäžå¹³é¢äžã«ããäžè§åœ¢ $T_1, T_2, \cdots$ ã¯, 以äžã®æ¡ä»¶ãæºãããŸã.
- $T_1$ ã®å
æ¥åã¯é¢ç© $1$ ã§ãã, ã〠$T_1$ ã®å€å¿ãéã.
- ä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸããŠ, $T_n$ ã®å€æ¥å㯠$T_{n+1}$ ã«å
æ¥ã, ã〠$T_{n+1}$ ã®å€å¿ãéã.
ã$T_{20201017}$ ã®å€æ¥åã®é¢ç©ã® $1$ ã®äœããã³ $1\/10$ ã®äœãæ±ã, ããããç¶ããŠè§£çããŠãã ãã.\
ãäŸãã°, $1$ ã®äœã $1$ ã§, $1\/10$ ã®äœã $2$ ãªãã°, $12$ ãšè§£çããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/40 | A | OMC006(A) | 100 | 232 | 237 | [
{
"content": "ã$C_1,C_2$ ã®æ¹çšåŒã¯ãããã次ã®ããã«æ±ããããïŒ\r\n$$C_1:y^2=-x^2+4x,\\quad C_2:y=-2x+4$$\r\nãã£ãŠ $S,T$ éã®è·é¢ã¯ $|a^2-6a+4|$ ã§ããããïŒæ¹çšåŒ\r\n$$a^2-6a+4=\\pm 2020$$\r\nã®çžç°ãªãå®æ°è§£ã®ç·ç©ãæ±ããã°ããïŒ$a^2-6a+4=2020$ ã¯çžç°ãªãå®æ°è§£ãæã¡ïŒãããã®ç©ã¯è§£ãšä¿æ°ã®é¢ä¿ãã $-2016$ïŒãŸã $a^2-6a+4=-2020$ ã¯å®æ°è§£ããããªãïŒåŸã£ãŠæ±ããå€ã¯ $\\bm{2016}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"ur... | ã$3$ ç¹ $(0, 0), (2, 4), (4, 0)$ ãéã, äžã«åžãªæŸç©ç·ã $C_1$ ãšã, $2$ ç¹ $(0, 4), (2, 0)$ ãéãçŽç·ã $C_2$ ãšããŸã. ãŸã $a$ ãå®æ°ãšã, $C_1,C_2$ äžã§ $x$ 座æšã $a$ ã§ããç¹ããããã $S,T$ ãšããŸã.\
ã$2$ ç¹ $S, T$ ã®è·é¢ã $2020$ ã§ãããšã, $a$ ãšããŠããåŸãå€ã®ç·ç©ã®çµ¶å¯Ÿå€ãæ±ããŠãã ãã.\
ãäŸãã°, $a$ ãšã㊠$11, -4$ ãèãããããšãã¯, $44$ ãšè§£çããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/41 | B | OMC006(B) | 200 | 211 | 223 | [
{
"content": "ã$AO=BO(=\\frac{1}{\\sqrt{2}}),\\angle AOB=90^{\\circ}$ ãªãç¹ $O$ ãäžã€ãšãïŒ$O$ ãäžå¿ãšã $A,B$ ãéãååšã $\\Gamma$ ãšããïŒ$C,D$ 㯠$O$ ãšåãåŽã«ãããšããŠããïŒãã®ãšãæ¡ä»¶ãã $C,D$ 㯠$\\Gamma$ äžã«ãããã $CD\\leq (\\Gamma ã®çŽåŸ)=\\sqrt{2}$ïŒ\\\r\nãéã« $O$ ãéã $l$ ã«å¹³è¡ãªçŽç·ãš $\\Gamma$ ã®äº€ç¹ã $C,D$ ãšããã°æ¡ä»¶ãã¿ããïŒãªãã〠$CD=\\sqrt{2}$ ãšãªãïŒ\\\r\nããã£ãŠæ±ã... | ãçŽç· $l$ äžã« $AB=1$ ãªã $2$ ç¹ $A, B$ ããããŸã. ãŸã, 次ã®æ¡ä»¶ãæºããããã«ç°ãªã $2$ ç¹ $C, D$ ãåããŸã.
- $2$ ç¹ $C, D$ ãšçŽç· $l$ ã¯åäžå¹³é¢äžã«ãã.
- $2$ ç¹ $C, D$ ã¯çŽç· $l$ ã«é¢ããŠåãåŽã«ãã.
- $\angle ACB = \angle ADB = 45^{ \circ }. $
- äžè§åœ¢ $ACB$ ã®é¢ç©ãšäžè§åœ¢ $ADB$ ã®é¢ç©ãçãã.
ããã®ãšã, ç·å $CD$ ã®é·ããšããŠããåŸãæ倧å€ã¯, æ£ã®æŽæ° $a$ ãçšã㊠$\sqrt{a}$ ãšæžããŸã. $a$ ãæ±ããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/42 | C | OMC006(C) | 300 | 0 | 0 | [
{
"content": "$$x^2â4x+y^2â2y+z^2+6z=15\\iff (x-2)^2+(y-1)^2+(z+3)^2=29$$\r\nã§ããããïŒCauchy-Schwarzã®äžçåŒãã次ãæç«ïŒ\r\n$$\\begin{aligned}\r\n&\\quad(2x+3y+4z+5)^2\\\\\\\\\r\n&=\\left(2(x-2)+3(y-1)+4(z+3)\\right)^2\\\\\\\\\r\n&\\leq (2^2+3^2+4^2)\\left((x-2)^2+(y-1)^2+(z+3)^2\\right)\\\\\\\\\r\n&=29^2\r\n\\end{aligned}... | ãå®æ° $x,y,z$ ã«ã€ããŠ
$$x^2-4x+y^2-2y+z^2+6z=15$$
ã§ãããšã, $2x+3y+4z$ ã®ãšãåŸãæ倧å€ãæ±ããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/43 | D | OMC006(D) | 400 | 193 | 217 | [
{
"content": "ãæ£ã®é·ããçãé ã« $a_1, a_2, \\cdots a_9$ ãšãã. ãããã䜿ã£ãŠééå㪠$M$ è§åœ¢ãäœããªãããã®å¿
èŠæ¡ä»¶ã¯\r\n$$\\displaystyle \\sum_{i=1}^{M-1} a_i \\leq a_M$$\r\nã§ãããã, åž°çŽçã«èšç®ããããšã§ $a_9\\geq 2^{7}=128$ ãåŸã. éã«æ£ã®é·ãã $1,1,2,4,8,16,32,64,128$ ã§ãããšã, ééåãªå€è§åœ¢ãäœããªãããšã容æã«ç¢ºèªã§ãããã, æ±ããæå°å€ã¯ $\\textbf{128}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"u... | ã$9$ æ¬ã®æ£ããããŸã. äžçªçãæ£ã®é·ã㯠$1$ ã§ã. ãã®æ£ã®ãã¡, äœæ¬ãã®æ£ã䜿ã£ãŠ(ééåãª)å€è§åœ¢ãäœãããšããŸããã, ã©ããã£ãŠãäœãããšã¯ã§ããŸããã§ãã. ãã®ãšã, äžçªé·ãæ£ã®é·ããšããŠããåŸãæå°å€ãæ±ããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/44 | E | OMC006(E) | 500 | 54 | 95 | [
{
"content": "ã$S=a_1b_1+a_2b_2+\\cdots+a_{777}b_{777}$ ãšãããïŒæŒžååŒãã $a_{n+2}=10a_n,b_{n+2}=10b_n$ ããããããïŒ\r\n$$\\begin{aligned}\r\nS&=a_1b_1(1+10^2+\\cdots+10^{776})+a_2b_2(1+10^2+\\cdots+10^{774})\\\\\\\\\r\n&=8(1+10^2+\\cdots+10^{776})+a_2b_2(1+10^2+\\cdots+10^{774})\r\n\\end{aligned}$$\r\nãåŸãããïŒãã£ãŠ $a_2b_2$ ã... | ãæ°å $\\{a_{n}\\}$ ãš $\\{b_{n}\\}$ ã¯, ä»»æã®æ£ã®æŽæ° $n$ ã«ã€ããŠæ¬¡ã®åŒãã¿ãããŸãïŒ
$$a_{n+1}=3\sqrt{2}a_{n}-2b_{n},\ \ b_{n+1}=4a_{n}-3\sqrt{2}b_{n}$$
ã$a_{1},b_{1}$ ã $a_{1}b_{1}=8$ ãæºãããªããåããšã, $a_{1}b_{1}+a_{2}b_{2}+\dots +a_{777}b_{777}$ ã®ãšãåŸãæå°å€ã $m$ ãšããŸã. $m$ ã®åæ¡ã®ç·åãæ±ããŠãã ãã. |
OMC006 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc006/tasks/45 | F | OMC006(F) | 600 | 72 | 196 | [
{
"content": "ã$f$ ã®ã¿ããã¹ãæ¡ä»¶ã¯æ¬¡ãšåå€ã§ããããšã容æã«ç¢ºèªã§ããïŒ\r\n- (i)ïŒ$f(1)f(2)\\cdots f(10)=10!=2^8\\times 3^4\\times 5^2\\times 7$ ïŒ\r\n- (ii)ïŒ$n=1,\\dots,10$ ã«å¯Ÿã $\\dfrac{n+10}{n}\\cdot f(n)$ ã¯æŽæ°ïŒ\r\n\r\nã$n=1,\\dots,10$ ã«å¯Ÿã $\\dfrac{n+10}{n}$ ã®å€ã¯é ã« $11,6,\\dfrac{13}{3},\\dfrac{7}{2},3,\\dfrac{8}{3},\\dfrac{17}{7},\\dfra... | ãæ£ã®æŽæ°ããæ£ã®æŽæ°ãžã®é¢æ° $f$ ã§, ä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸããŠ, 以äžã®çåŒãæãç«ã€ãã®ã¯ããã€ãããŸããïŒ
$$f(n)f(n+1)f(n+2) \cdots f(n+9) = n(n+1)(n+2) \cdots (n+9)$$ |
OMCB003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb003/tasks/168 | A | OMCB003(A) | 100 | 0 | 0 | [
{
"content": "ãçã®åŒ·ãæ° $n$ ãæ£ã®çŽæ°ã $d$ åãã€ãšããïŒ$n$ ãçŽ å æ° $p$ ã $a(\\gt 0)$ åãã€ãšãïŒ$n$ ã®æ£ã®çŽæ°ã®ç·ç©ãš $n^3$ ã¯çŽ å æ° $p$ ããããã $ad\\/2,3a$ åãã€ïŒããããçã®åŒ·ãæ°ã¯æ£ã®çŽæ°ã $6$ åãã€æ°ã§ããããšããããïŒ\\\r\nããã®ãããªæ°ã¯ $p,q$ ãçŽ æ°ãšã㊠$p^5,p^2q$ ã®ããããã®åœ¢ã«çŽ å æ°å解ããïŒããããã®åœ¢ã«ã€ã㊠$200$ 以äžã®ãã®ãæ°ããã°ïŒæ±ããå€ã¯ $\\bm{28}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinema... | ãæ£ã®æŽæ° $n$ ã«ã€ããŠ, $n$ ã®ãã¹ãŠã®æ£ã®çŽæ°ã®ç©ã $n^3$ ã«çãããšã, **çã®åŒ·ã**æ°ãšåŒã³ãŸã.\
ã$200$ 以äžã®çã®åŒ·ãæ°ã¯ããã€ãããŸããïŒ
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMCB003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb003/tasks/169 | B | OMCB003(B) | 200 | 0 | 0 | [
{
"content": "ã$\\triangle XYZ$ ã¯æ£äžè§åœ¢ã§ãããã $XY=XZ$ ã§ããïŒãŸãç·å $CP,BQ$ äžã«ããããç¹ $R,S$ ã\r\n$$PR\\parallel AB,\\quad QS\\parallel AC$$\r\nãã¿ããããã«ãšãã°ïŒæ¬¡ãæãç«ã€ïŒ\r\n$$PR=\\frac{4}{5},\\quad\r\nQS=\\frac{9}{5},\\quad\r\nXY=\\frac{QX\\cdot PR}{PQ}=\\frac{9QX}{5PQ},\\quad\r\nXZ=\\frac{PX\\cdot QS}{PQ}=\\frac{4PX}{5PQ}$$\r\n... | ãäžèŸºã®é·ãã $5$ ã§ããæ£äžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $AB, AC$ äžã«ãããã $AP=3, AQ=2$ ãã¿ããç¹ $P, Q$ ããã, $BQ$ ãš $CP$ ã®äº€ç¹ã $R$ ãšããŸã. ããã§, ç·å $PQ, QR, RP$ äžã«ããããç¹ $X, Y, Z$ ã,
$$XY\parallel CA,\ \ YZ\parallel BC\ \ ZX\parallel AB$$
ãšãªãããã«ãšããŸãã.\
ã$XY$ ã®é·ãã¯äºãã«çŽ ãªæ£ã®æŽæ° $m, n$ ãçšã㊠$\displaystyle \\frac{m}{n}$ ãšè¡šããã®ã§, $m+n$ ãæ±ããŠãã ãã. |
OMCB003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb003/tasks/170 | C | OMCB003(C) | 300 | 0 | 0 | [
{
"content": "ãäžåŒã¯æ¬¡ã®ããã«å€åœ¢ã§ããïŒ$$(x+1)(x-2)(x-4)=12y(y-3)$$\r\n$y=2$ ã®ãšããããã¿ãã $x$ ã¯ååšããïŒ$y=3$ ã®ãšã $x=2,4$ ã§ããïŒ\\\r\nã$y\\geq 5$ ã®ãšãïŒå³èŸºã«çŽ å æ° $3$ ã¯ã¡ããã© $1$ åå«ãŸããããšãã $x\\equiv 1\\pmod{3}$ ã§ããïŒ$x\\neq 1,4$ ã¯å®¹æã«ç¢ºèªã§ããããïŒæ£æŽæ° $k$ ãçšã㊠$x=6k+1$ ãŸã㯠$x=6k+4$ ãšãããïŒ\r\n\r\n- $x=6k+1$ ã®ãšãäžåŒãã $$(3k+1)(6k-1)(2k-1)=2y(y-3).$$ ... | ã以äžã®çåŒãã¿ããæ£ã®æŽæ° $x$ ãšçŽ æ° $y$ ã®çµãã¹ãŠã«ã€ããŠ, $x+y$ ã®ç·åãæ±ããŠãã ãã.
$$x^3-5x^2+2x-12y^2+36y+8=0$$
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMCB003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb003/tasks/171 | D | OMCB003(D) | 400 | 0 | 0 | [
{
"content": "ã$N=9999$ ãšããïŒ$5^0,5^1,\\dots,5^{N-1},5^N$ ã®ä»£ããã« $10^N,5\\cdot 10^{N-1},\\dots,5^{N-1}\\cdot 10,5^N$ ïŒããªãã¡\r\n$$5^N,2\\cdot 5^N,\\dots,2^{N-1}\\cdot 5^N,2^N\\cdot 5^N$$\r\nã«ã€ããŠèããŠãæ§ããªãïŒ$a_n$ ã $2^n\\cdot 5^N$ ã®æ¡æ°ãšããïŒãã®ãšã $n=1,\\dots,N$ ã«å¯ŸãïŒ$2^n\\cdot 5^N$ ã®æäžäœã $1$ ãªãã° $a_n=a_{n-1}+1$ïŒããã§ãªããªãã° $a_... | ã$5^0, 5^1, \\cdots, 5^{9999}$ ã®ãã¡, åé²æ³ã§æé«äœã $1$ ã§ããæ°ã¯ããã€ãããŸããïŒ\
ããã ã, $5^{9999}$ ã¯åé²æ³ã§ $6990$ æ¡ã§, æé«äœã $1$ ã§ããããšãä¿èšŒãããŸã. |
OMCB002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb002/tasks/164 | A | OMCB002(A) | 100 | 0 | 0 | [
{
"content": "ã$24=2^3\\times 3$ ã§ããããïŒæ£ã®çŽæ°ã $24$ åæã€æ£æŽæ°ã¯æ¬¡ã®ããããã®åœ¢ã«çŽ å æ°å解ãããïŒ\r\n$$p^{23},\\quad p^7q^2,\\quad p^5q^3,\\quad p^5qr,\\quad p^3q^2r,\\quad p^2qrs$$\r\nããããã®åœ¢ã«ã€ããŠæå°å€ã¯ $2^{23},1152,864,480,360,420$ ã§ããããïŒæ±ããå€ã¯ $\\bm{360}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb00... | ãæ£ã®çŽæ°ã®åæ°ã $24$ åãã€æ£ã®æŽæ°ã®ãã¡, æå°ã®ãã®ãæ±ããŠãã ãã.
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMCB002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb002/tasks/165 | B | OMCB002(B) | 200 | 0 | 0 | [
{
"content": "ããã¹ãŠã®ããŒã«ãåãåºããŸã§ã²ãŒã ãçµäºããªããšããŠæ§ããªãïŒãã®ãšãæ±ãã確çã¯ïŒæåŸã«åãåºãããããŒã«ãèµ€è²ã§ãã確çã«äžèŽãïŒããã¯æããã« $1001\\/2000$ ã§ããïŒãã£ãŠè§£çãã¹ãå€ã¯ $\\bm{3001}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb002/editorial/165"
}
] | ãç®±ã®äžã« $999$ åã®çœãããŒã«ãš $1001$ åã®èµ€ãããŒã«ããããŸã. ããããã©ã³ãã ã«ç確çã§ããŒã«ã $1$ ã€ãã€åãåºããŠãã, çœãããŒã«ãèµ€ãããŒã«ã®ã©ã¡ãããå
šãŠåãåºããããã²ãŒã ãçµäºããŸã. ãã®ãšã, çœãããŒã«ãå
šãŠåãåºãããŠçµãã確çã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæ£ã®æŽæ° $p, q$ ãçšã㊠$\displaystyle \\frac{p}{q}$ ãšè¡šãããŸã. $p+q$ ã解çããŠãã ãã. |
OMCB002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb002/tasks/166 | C | OMCB002(C) | 300 | 0 | 0 | [
{
"content": "ã$x^3+ax^2+bx-b=0$ ã®è§£ãæ£æŽæ° $p,q,r\\\\,(p\\leq q\\leq r)$ ã§ãããšããã°ïŒè§£ãšä¿æ°ã®é¢ä¿ãã次ãæãç«ã€ïŒ\r\n$$p+q+r=-a,\\quad pq+qr+rp=b,\\quad pqr=b.$$\r\nãã®ãšã $\\dfrac{1}{p}+\\dfrac{1}{q}+\\dfrac{1}{r}=1$ ã§ããïŒ$p\\leq q\\leq r$ ã«æ³šæããã°ãããã¿ããçµã¯\r\n$$(p,q,r)=(2,3,6),(2,4,4),(3,3,3).$$\r\nããããã«å¯Ÿã $(a,b)=(-11,36),(-10,32),(-... | ã$x$ ã«ã€ããŠã®æ¹çšåŒ $x^3+ax^2+bx-b=0$ ã®è€çŽ æ°è§£ããã¹ãŠæ£æŽæ°ã§ãããããªå®æ°ã®çµ $a, b$ ãã¹ãŠã«å¯ŸããŠ, $a+b$ ã®ç·åãæ±ããŠãã ãã. |
OMCB002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb002/tasks/167 | D | OMCB002(D) | 400 | 0 | 0 | [
{
"content": "ãæ¡ä»¶ãã $AB=x-d,BC=x,CA=x+d~(0\\leq d\\lt x)$ ãšãããïŒãã®ãšãçŽç· $AI$ ãšèŸº $BC$ ã®äº€ç¹ã $D$ ãšãããšïŒè§ã®äºçåç·å®çãã $BD=(x-d)\\/2$ ãåŸããïŒåã³è§ã®äºçåç·å®çãã $AI:ID=2:1$ ããããïŒãŸã蟺 $BC$ ã®äžç¹ã $M$ ãšãããšïŒæåäºå®ãšã㊠$A,G,M$ ã¯ãã®é ã«å
±ç·ã§ïŒ$AG:GM=2:1$ ã§ããïŒ\\\r\nã以äžãã $IG:DM=2:3$ ããããïŒ$IG=1,DM=d\\/2$ ãã $d=3$ ãåŸãïŒãã®ãšãäžè§åœ¢ã®æç«æ¡ä»¶ãã $x\\gt 6$ ã§ããããïŒç¹... | ãå¹³é¢äžã®äžè§åœ¢ $ABC$ ã«ã€ããŠ, $AB, BC, CA$ ã®é·ãã¯ãã®é ã§çå·®æ°åããªããŠããŸã. ãŸã, å
å¿ã $I$, éå¿ã $G$ ãšãããš, $GI=1$ ãšãªããŸã. ãã®ãšã, $BC$ ã®é·ããšããŠããåŸãæå°ã®æŽæ°å€ãæ±ããŠãã ãã.\
ããã ã, $XY$ ã§ç·å $XY$ ã®é·ããè¡šããŸã. |
OMCE001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce001/tasks/172 | A | OMCE001(A) | 100 | 0 | 0 | [
{
"content": "ãçŽç· $y=px+2q$ ãæŸç©ç· $y=x^2$ ã«ãã£ãŠåãåãããç·åã®é·ãã¯\r\n$$\\sqrt{p^2+1}\\times \\sqrt{p^2+8q}$$\r\nãšå®¹æã«èšç®ã§ããïŒãã£ãŠæ¬¡ã®åŒã«ã€ããŠèããã°ããïŒ\r\n$$(p^2+1)(p^2+8q)=4r^2$$\r\nã$p=2$ ã®ãšã $5(2q+1)=r^2$ ãšãªãïŒãã®ãšã $r=5,q=2$ïŒãŸã $p\\geq 3$ ãªãã°\r\n$$(p^2+1)(p^2+8q)\\equiv 2\\pmod{4}$$\r\nããæ¡ä»¶ãã¿ãã $p,q,r$ ã¯ååšããªãïŒåŸã£ãŠæ±ããçµã¯ $(p,q,r)=(... | ã$p, q, r$ ãçŽ æ°ãšããŸã. $xy$ å¹³é¢äžã«ãããŠ, çŽç· $y=px+2q$ ãæŸç©ç· $y=x^2$ ã«ãã£ãŠåãåãããç·åã®é·ãã $2r$ ã«ãªããšã, ãã®ãããªçµãã¹ãŠã«ã€ã㊠$p+q+r$ ã®ç·åãæ±ããŠãã ãã.
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMCE001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce001/tasks/173 | B | OMCE001(B) | 200 | 0 | 0 | [
{
"content": "ãå¹³é¢äžã«ç¹ $F$ ããšãïŒæ¬¡ãã¿ããç¹ $A,B,C$ ããšãïŒ\r\n$$AF=x,\\quad BF=y,\\quad CF=z,\\quad \\angle AFB=\\angle BFC=\\angle CFA=120^{\\circ}$$\r\nãã®ãšã $\\triangle ABC$ ã®é¢ç© $S$ ã¯\r\n$$S=\\frac{\\sqrt{3}}{4}(xy+yz+zx).$$\r\näžæ¹ã§äœåŒŠå®çãã $AB=5,BC=7,CA=8$ ã§ããããïŒHeronã®å
¬åŒãã $S=10\\sqrt{3}$ ã§ããïŒ\\\r\nãåŸã£ãŠ $xy+yz+zx=40$ ããã... | ãæ£ã®å®æ° $x, y, z$ ã以äžã®çåŒãã¿ãããšã, $x^2+y^2+z^2$ ãæ±ããŠãã ãã.
$$\begin{cases}x^2+xy+y^2=25 \\\\ y^2+yz+z^2=49 \\\\ z^2+zx+x^2=64\end{cases}$$
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMCE001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce001/tasks/174 | C | OMCE001(C) | 300 | 0 | 0 | [
{
"content": "ãäžè¬ã« $N=100$ ãšããïŒåºãç®å
šãŠã®ç©ã¯ $2^a 3^b 5^c$ ã®åœ¢ã«çŽ å æ°å解ã§ãïŒç°¡åãªè°è«ã«ãã次ã確èªã§ããïŒ\r\n\r\n- $c$ ã®ãšãåŸãå€ã®ç¯å²ã¯ $0\\leq c\\leq N$ïŒ\r\n- $c$ ãåºå®ãããšã $b$ ã®ãšãåŸãå€ã®ç¯å²ã¯ $0\\leq b\\leq N-c$ïŒ\r\n- $c,b$ ãåºå®ãããšã $a$ ã®ãšãåŸãå€ã®ç¯å²ã¯ $0\\leq a\\leq 2N-2c-b$ïŒ\r\n\r\nããã£ãŠæ±ããå€ã¯æ¬¡ã®ããã«èšç®ã§ããïŒ\r\n$$\\sum_{c=0}^{N}\\sum_{b=0}^{N-c}(2N-2c-b+1... | ãäžè¬ç㪠$6$ é¢ã®ãµã€ã³ãã $100$ åæ¯ã£ããšã, åºãç®ãã¹ãŠã®ç©ãšããŠèããããæ°ã¯ããã€ãããŸããïŒ |
OMCE001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce001/tasks/175 | D | OMCE001(D) | 400 | 0 | 0 | [
{
"content": "ãäžåŒã§ $(x,y)=(a,b)$ ãšããåŒã $P(a,b)$ ã§è¡šãããšãšããïŒ$f$ ãå®æ°é¢æ°ãªãã° $f(x)\\equiv 0,1$ ã§ããããšã¯å®¹æã«ãããããïŒä»¥äž $f$ ãå®æ°é¢æ°ã§ãªãå ŽåãèããïŒãã®ãšã $P(x,0)$ ãèããã° $f(0)=1$ ããããïŒ\r\n\r\n----\r\n**è£é¡1**ïŒä»»æã® $x\\geq 0$ ã«å¯Ÿã $\\dfrac{1}{x+1}\\leq f(x)\\leq 1$\\\r\n**蚌æ**ïŒå³åŽã®äžçåŒã¯ $f(0)=1$ ããæããïŒ\\\r\nãŸãå·ŠåŽã®äžçåŒã¯èçæ³ã«ãã瀺ãããïŒå
·äœçã«ã¯ïŒ$P(x,-x)... | ã$-1$ ãã倧ããå®æ°ã«å¯ŸããŠå®çŸ©ãã, å®æ°å€ããšãé¢æ° $f$ ã¯, 次ã®æ¡ä»¶ãã¿ãããŸã.
- $x+y, x, yf(x)$ ããã¹ãŠ $-1$ ãã倧ãããããªä»»æã® $x, y$ ã«ã€ããŠ, 以äžã®åŒãæãç«ã€.$$f(x+y)=f(x)f(yf(x))$$
- $-1$ ãã倧ããä»»æã®å®æ° $a, b$ ã«ã€ããŠ, $a \lt b$ ãªãã° $f(a) \\geq f(b)$ ãæãç«ã€.
ã$\displaystyle f(100) \gt \frac{1}{2020}$ ã®ãšã, $\displaystyle f\left(f\left(100\right)-\frac{2021}{2020}... |
OMCB001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb001/tasks/160 | A | OMCB001(A) | 100 | 0 | 0 | [
{
"content": "ãè§åºŠèšç®ã«ãã $\\angle BAD=65^{\\circ}$ ã§ãããã $AB=BD$ ãåŸãããïŒãã®ãšã $BD=CD$ ãã $\\angle CBD=\\angle BCD=60^{\\circ}$ ãããããã $\\triangle BCD$ ã¯æ£äžè§åœ¢ïŒãã®ãšã $AB=BC$ ãã $\\angle BCA=\\bm{35}^{\\circ}$ ãšèšç®ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb001/editorial/160"
}
] | ãåžåè§åœ¢ $ABCD$ ã以äžã®æ¡ä»¶ãããããã¿ãããŸã.
$$\angle ABD=50^\circ,\ \ \angle ADB=65^\circ,\ \ \angle BDC=60^\circ,\ \ AB=CD$$
ãã®ãšã, $\angle BCA$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã. |
OMCB001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb001/tasks/161 | B | OMCB001(B) | 200 | 0 | 0 | [
{
"content": "ãæ£æ¹åœ¢ $ABCD$ ã®äžèŸºã®é·ãã $1$ ã§ãããšããŠäžè¬æ§ã倱ããªãïŒ\\\r\nã$\\triangle ABM\\equiv \\triangle A^\\prime BM$ ã«æ³šæããã° $\\triangle MDP\\equiv\\triangle MA^\\prime P,\\triangle BCQ\\equiv\\triangle BA^\\prime Q$ ã¯å®¹æã«ãããïŒãã®ãšãè§åºŠèšç®ããã° $\\triangle ABM\\sim\\triangle DMP$ ããããïŒçžäŒŒæ¯ãèããã° $DP=1\\/4$ ãåŸãïŒãããã $CP=3\\/4$ïŒãŸãäžå¹³... | ãæ£æ¹åœ¢ $ABCD$ ã«ãããŠ, 蟺 $AD$ ã®äžç¹ã $M$ ãšã, $BM$ ã«é¢ã㊠$A$ ãšå¯Ÿç§°ãªç¹ã $A^\prime$ ãšããŸã. 蟺 $CD$ ãšçŽç· $A^\prime B,A^\prime M$ ã®äº€ç¹ããããã $P,Q$ ãšãããšã, $DP:PQ:QC$ ã¯äºãã«çŽ ãªæ£æŽæ° $p,q,r$ ã«ãã£ãŠ $p:q:r$ ãšè¡šãããŸã. $p+q+r$ ã解çããŠãã ãã. |
OMCB001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb001/tasks/162 | C | OMCB001(C) | 300 | 0 | 0 | [
{
"content": "ãæäœãéé ã«èããã°ïŒãã®åé¡ã¯æ¬¡ã®ããã«èšãæããããïŒ\r\n\r\n- äºãã«åºå¥ã§ãã $16$ åã®ç³ã暪äžåã«äžŠãã§ããïŒèªèº«ããå³åŽã«ç³ãå¶æ°å ($0$ åãå«ã) 䞊ãã§ããç³ãäžã€éžã³ãããåãå»ãïŒãšããæäœã $16$ åç¹°ãè¿ããŠãã¹ãŠã®ç³ãåãå»ãïŒãã®ãšãïŒæäœã®æé ãšããŠããåŸããã®ã¯äœéããããïŒ\r\n\r\nããã®çãã¯å®¹æã« $(8!)^2=\\bm{1625702400}$ éããšèšç®ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb... | ã$16$ åã®çœç³ã暪äžåã«äžŠãã§ããŸã. 以äžã®æäœã $16$ åç¹°ãè¿ã, ãã¹ãŠã®ç³ãé»ç³ã«ããããšãèããŸã,
- èªèº«ããå³åŽã«é»ç³ãå¶æ°å ($0$ åãå«ã) 䞊ãã§ãããããªçœç³ãäžã€éžã³, é»ç³ã«çœ®ãæãã.
ãæäœã®æé ãšããŠããåŸããã®ã¯äœéããããŸããïŒ |
OMCB001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb001/tasks/163 | D | OMCB001(D) | 400 | 0 | 0 | [
{
"content": "ãäžåŒã®äž $2$ åŒã®å·®ããšãã° $$2q=r^2-s^4=(r-s^2)(r+s^2)=(r-s^2)q$$ ã§ãããã $q\\geq 2$ ãã $r-s^2=2$ ãåŸãïŒãããäžåŒã«ä»£å
¥ãèšç®ããã°æ¬¡ãåŸãããïŒ\r\n$$p=s^4+2s^2+2,\\quad q=2s^2+2,\\quad r=s^2+2$$\r\nã$p,q,r,s$ ãäºãã«çŽ ã«ãªããã㪠$s$ ã®æå°å€ã¯ $s=3$ ã§ïŒãã®ãšã $pq=\\bm{2020}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/... | ã$2$ 以äžã®äºãã«çŽ ãªæ£æŽæ°ã®çµ $(p,q,r,s)$ ã, 以äžã®çåŒãã¿ãããŸã.
$$\begin{cases}p+q=r^2 \\\\ p-q=s^4 \\\\ r+s^2=q \end{cases}$$
ãã®ãããªçµã®ãã¡ $s$ ãæå°ã®ãã®ãã¹ãŠã«ã€ããŠ, $pq$ ã®ç·åãæ±ããŠãã ãã. |
OMC005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc005/tasks/36 | A | OMC005(A) | 100 | 0 | 0 | [
{
"content": "ãç¹ã« $n^3$ ãš $n$ ã®äžã®äœã¯äžèŽããããïŒ$n$ ã®äžã®äœã¯ $0,1,4,5,6,9$ ã®ããããïŒ\\\r\nã$n$ ã®åã®äœã $k$ ãšããïŒäžã®äœã§å ŽååãããïŒ\r\n- $0$ ã®ãšã $n^3$ ã®äžäºæ¡ã¯ $00$ ã«ãªã $n$ ãšäžèŽããªãïŒ\r\n- $1$ ã®ãšã $n^3\\equiv 30k+1\\pmod{100}$ ãã $k=5\\\\,(n=51)$ ãåŸããïŒããã¯æ¡ä»¶ãã¿ããïŒ\r\n- $4$ ã®ãšã $n^3\\equiv 80k+64\\pmod{100}$ ãã $k=2\\\\,(n=24)$ ãåŸããïŒããã¯æ¡ä»¶ãã¿ãã... | ã$2$ æ¡ã®æŽæ° $n$ ã«ã€ããŠ, $n^2$ ã®äžäºæ¡ã¯ $n$ ãšäžèŽããŸããã, $n^3$ ã®äžäºæ¡ã¯ $n$ ãšäžèŽããŸã. ãã®ãã㪠$n$ ã®ç·åãæ±ããŠãã ãã. |
OMC005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc005/tasks/37 | B | OMC005(B) | 200 | 0 | 0 | [
{
"content": "ã$\\theta=\\dfrac{\\pi}{7}$ ãšããïŒååšã®äžå¿ã $O$ ãšããïŒæ£åŒŠå®çãã $\\sin\\angle ACB=\\sin\\theta$ ã§ããïŒ$a\\gt 1$ ãã $\\angle ACB\\lt\\dfrac{\\pi}{2}$ ãããããã $\\angle ACB=\\theta$ ãåŸãïŒãŸãäžå¿è§ã®å®çããã³ $BC=CD=DA$ ãã $$\\angle BOC=\\angle COD=\\angle DOA=\\frac{1}{3}(2\\pi-\\angle AOB)=4\\theta$$ ãšèšç®ã§ããããïŒç°¡åãªèšç®ã«ãã£ãŠæ¬¡ãã... | ã$a, b$ ã $1$ ãã倧ããæ£ã®å®æ°ãšããŸã. ååŸ $\displaystyle \\frac{1}{2\\sin\\frac{\\pi}{7}}$ ã®ååšäžã«, åæèšåãã§é ã«ç¹ $A, B, C, D$ ããããŸã.
$$AB=1,\quad BC=CD=DA=a,\quad AC=BD =b$$
ã®ãšã, $a+b-ab$ ã®å€ãæ±ããŠãã ãã. |
OMC005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc005/tasks/38 | C | OMC005(C) | 300 | 0 | 0 | [
{
"content": "ã$\\alpha=1010+\\sqrt{1020101}$ ãšããïŒçµè«ããè¿°ã¹ãã°ïŒ$\\{a_n\\}$ ã¯æ¬¡ã®æŒžååŒãã¿ããïŒ\r\n$$a_0=1,\\quad a_1=2020,\\quad a\\_{n+2}=2020a\\_{n+1}+a\\_n-1\\quad (n\\geq 0)\\tag{\\\\#}$$\r\nãããèªããã°ïŒ$a_{n+2}\\equiv a_n-1\\pmod{2020}$ ã§ãããã容æã« $a_{2020}\\equiv\\bm{1011}\\pmod{2020}$ ããããïŒ\\\r\nããªã $(\\\\#)$ ã蚌æããã«ã¯ $n\\... | ãæ°å $\\{a_n\\}$ ã以äžã§å®ãããšã, $a_{2020}$ ã $2020$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ
$$a_0=1,\quad a_{n+1}= \\lfloor (1010+\\sqrt{1020101})a_n \\rfloor \quad (n=0,1,2,\cdots)$$
ããã ã, $\\lfloor x \\rfloor$ 㯠$x$ ãè¶
ããªãæ倧ã®æŽæ°ãè¡šããŸã. |
OMC005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc005/tasks/39 | D | OMC005(D) | 400 | 0 | 0 | [
{
"content": "ãå¶å¥ãèããã° $p,q,r$ ã®ãã¡ã¡ããã©äžã€ã $2$ ã§ããïŒ$2$ ã§ãªãäºã€ã¯å¥çŽ æ°ã§ããããšã«æ³šæããïŒ\\\r\nã$p=2$ ã®å ŽåïŒ$q=3$ ã§ãªãããšã¯ä»£å
¥ããã°ç¢ºèªã§ããïŒãã£ãŠ $4+5q^6\\equiv 0\\equiv r^n\\pmod{3}$ ãã $r=3$ ãåŸãïŒããã« $4+5q^6\\equiv 1\\equiv 3^n \\pmod{4}$ ãã $n$ ã¯å¶æ°ã§ããïŒãã®ãšã$$(3^{n\\/2}-2)(3^{n\\/2}+2)=5q^6$$ãšãªããïŒ$q$ ã¯å¥çŽ æ°ãããããã¿ãã $q,n$ ã¯ååšããªãïŒ\\\r\nã$q=2$... | ãçŽ æ° $p, q, r$ ãšæ£ã®æŽæ° $n$ ã¯
$$p^2+5q^6=r^n$$
ãæºãããŸã. ãã®ãããªçµãã¹ãŠã«ã€ããŠ, $pqrn$ ã®ç·åãæ±ããŠãã ãã.
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc004/tasks/32 | A | OMC004(A) | 100 | 0 | 0 | [
{
"content": "ã$C$ ã®æ¹çšåŒã¯ $a,b,c\\geq 0$ ãçšããŠ\r\n$$(x-a)^2+(y-b)^2+(z-c)^2=a^2+b^2+c^2$$\r\nãšãããïŒãã®ãšãæ¡ä»¶ãã $$a^2+b^2=13,\\quad b^2+c^2=20,\\quad c^2+a^2=25$$ ãšãªãããïŒæ±ããå€ã¯ $a^2+b^2+c^2=\\dfrac{13+20+25}{2}=\\bm{29}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc004/editorial/32"
... | ã座æšç©ºéäžã«åç¹ãéãçé¢ $C$ ããããŸã. $C$ ã $xy$ å¹³é¢, $yz$ å¹³é¢, $zx$ å¹³é¢ãšäº€ãã£ãŠã§ããåã®é¢ç©ããããã $13 \\pi, 20 \\pi, 25 \\pi$ ã§ãããšã, ç $C$ ã®ååŸã¯ $\\sqrt A$ ã§ã. $A$ ãæ±ããŠãã ãã. |
OMC004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc004/tasks/33 | B | OMC004(B) | 200 | 0 | 0 | [
{
"content": "ã$n=\\sqrt{2^a\\times 3^b+1}$ ãšããã° $(n-1)(n+1)=2^a\\times 3^b$ ãšãªãïŒ$a=0$ ã®ãšã $(b,n)=(1,2)$ïŒ\\\r\nã$a\\geq 1$ ã®ãšãïŒèããããã®ã¯æ¬¡ã®2éãïŒãã ã $p,q$ ã¯æ£æŽæ°ã§å°ãªããšãäžæ¹ã¯ $1$ïŒ\r\n$$\\mathrm{(i)}:\\begin{cases}n-1=2^p\\times 3^b\\\\\\\\ n+1=2^q\\end{cases}\\quad\\mathrm{(ii)}:\\begin{cases}n-1=2^p\\\\\\\\ n+1=2^q\\time... | ã$0$ 以äžã®æŽæ° $a, b$ ãçšã㊠$\\sqrt {2^a \\cdot 3^b + 1}$ ãšè¡šãããšãã§ãããããªæ£ã®æŽæ°ã«ã€ããŠ, ãã®ç·åãæ±ããŠãã ãã. |
OMC004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc004/tasks/34 | C | OMC004(C) | 300 | 0 | 0 | [
{
"content": "ã$Q$ ãéã $AQ$ ã«åçŽãªçŽç·ã蟺 $BC$ ã®äžç¹ $M$ ãéãããšã¯å®¹æã«ç¢ºèªã§ããããïŒ$Q$ ã¯ç·å $AM$ ãçŽåŸãšããååš $\\Gamma$ äžã«ããããšããããïŒ$\\Gamma$ ãšçŽç· $AB,AC$ ã®äº€ç¹ã®ãã¡ $A$ ã§ãªããã®ã $D,E$ ãšããã° $Q$ ã®è»è·¡ã¯ $A$ ãå«ãŸãªã匧 $DE$ (以äžåã«åŒ§ $DE$ ãšåŒã¶) ã§ããïŒ\\\r\nãäžç·å®çãã $AM=7\\/2$ ïŒãŸãäœåŒŠå®çãçšããã° $\\cos \\angle BAC=1\\/2$ ãã $\\angle BAC=60^{\\circ}$ ã§ããããïŒåŒ§ $DE$... | ã$AB=3, BC=\\sqrt {19}, CA=5$ ã§ããäžè§åœ¢ $ABC$ ã«ãããŠ, ç·å $BC$ äžã«ç¹ $P$ ããã, ç¹ $B,C$ ããçŽç· $AP$ ã«äžãããåç·ã®è¶³ããããã $X, Y$ ãšããŸã. ãŸã, ç·å $XY$ ã®äžç¹ã $Q$ ãšããŸã. ç¹ $P$ ãç·å $BC$ äžãç¹ $B$ ããç¹ $C$ ãŸã§åããšã, ç¹ $Q$ ã®åããé·ãã¯äºãã«çŽ ãªæ£ã®æŽæ° $m, n$ ãçšã㊠$\displaystyle \\frac{m \\pi}{n}$ ãšè¡šããŸã. $m+n$ãæ±ããŠãã ãã. |
OMC004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc004/tasks/35 | D | OMC004(D) | 400 | 0 | 0 | [
{
"content": "ãäžè¬ã« $n$ åã®ç¹ã®å Žåã®æžãæ¹ã $N_n$ ãšãïŒ$N_n=n!$ ã§ããããšã瀺ããïŒ\\\r\nãåž°çŽçã«æ¡ä»¶ã¯æ¬¡ã®ããã«èšãæããããïŒ\r\n- ã©ã®ããã« $k(\\geq 3)$ ç¹ $A_1,\\dots,A_k$ ããšã£ãŠãïŒ $A_1\\to A_2\\to\\cdots\\to A_k\\to A_1$ ãšç¢å°ãæžãããŠããããšã¯ãªãïŒ\r\n\r\nãç¹ã«é©åœãªé 㧠$P_1,P_2,\\dots,P_n$ ãšååãã€ããŠããïŒãã®ãšãïŒæ¡ä»¶ãæºããç¢å°ã®æžãæ¹ã«ãããŠïŒæ¬¡ãã¿ãã $1\\leq m\\leq n$ ããã äžã€ååšããïŒãªãååšæ§ã¯èçæ³ã«... | ãå¹³é¢äžã«åºå¥ããã $100$ åã®ç¹ãåã, çžç°ãªãäºç¹ã®çµãã¹ãŠã«ã€ããŠ, ãã®éã«ä»¥äžã®æ¡ä»¶ãæºããããã«ç¢å°ãã¡ããã© $1$ æ¬æžããŸã. ãã®ãšã, ç¢å°ã®æžãæ¹ã¯ $N$ éããããŸã. $N$ ã«å«ãŸããæ倧ã®çŽ å æ°ãçããŠãã ãã.
- æ¡ä»¶ïŒã©ã®ããã«äžç¹ $A, B, C$ ããšã£ãŠã, $A \\to B \\to C \\to A$ ã®ãããªã«ãŒããååšããªã. |
OMC003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc003/tasks/28 | A | OMC003(A) | 100 | 0 | 0 | [
{
"content": "ã$2$ çŽåæ±ã®äžå¿è»žããšãã«å«ãå¹³é¢ã«å¹³è¡ã§è·é¢ã $t(\\leq 3)$ ã®å¹³é¢ã«ãã£ãŠïŒå
±ééšåãåæãããšãã®æé¢ç©ã $S(t)$ ãšãããïŒåãå£ã¯å
è§ã®äžã€ã $30^{\\circ}$ïŒå¯ŸèŸºéã®è·é¢ã $2\\sqrt{9-t^2}$ ã®è±åœ¢ã«ãªãããïŒç°¡åãªèšç®ã«ãã£ãŠ $S(t)=8(9-t^2)$ ãšæ±ããããïŒåŸã£ãŠå
±ééšåã®äœç©ã¯æ¬¡ã®ããã«èšç®ã§ããïŒ\r\n$$2\\int_{0}^{3}S(t)dt=2\\left[8\\left(9t-\\frac{t^3}{3}\\right)\\right]_{t=0}^{3}=\\bm{288}$$",
"... | ãåãå£ãååŸ $3$ ã®åã§ãããããªç¡éã«é·ãçŽåæ±ã $2$ ã€ããïŒãããã®äžå¿è»žã $30^{\\circ}$ ã§äº€ãã£ãŠããŸãïŒãã®ãšãïŒå
±ééšåã®äœç©ãæ±ããŠãã ããïŒ
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc003/tasks/29 | B | OMC003(B) | 200 | 0 | 0 | [
{
"content": "ã座æšå¹³é¢ã§ $O(0,0),A(8,0),B(0,6)$ ãšããã°ïŒæ±ããå€ã¯æ¬¡ã®ããã«èšç®ã§ããããšã容æã«ãããïŒ\r\n$$\\begin{aligned}\r\n&\\quad OA^2+OB^2+OC^2+OD^2+OE^2+OF^2\\\\\\\\\r\n&=\\sum_{k=0}^{5}\\left(\\left(\\frac{8k}{5}\\right)^2+\\left(\\frac{6(5-k)}{5}\\right)^2\\right)\\\\\\\\\r\n&=\\frac{6^2+8^2}{5^2}\\sum_{k=0}^{5}k^2\\\\\\\\\r\n&=... | ã$\\angle AOB=90^{ \\circ }$ ãªãçŽè§äžè§åœ¢ $AOB$ ã«ã€ããŠ, ç·å $AB$ ã® $5$ çåç¹ã $A$ ã«è¿ãé ã« $C, D, E, F$ ãšããŸã. $OA=8, OB=6$ ã®ãšã,
$$OA^2+OB^2+OC^2+OD^2+OE^2+OF^2$$
ãæ±ããŠãã ãã.
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc003/tasks/30 | C | OMC003(C) | 300 | 0 | 0 | [
{
"content": "ã$m=n^a,m^2-m+6=2(n+1)^b$ ãã $$n^{2a}-n^a+6=2(n+1)^b$$ ã§ããïŒäž¡èŸºã®å°äœãèããã°æ¬¡ããããïŒ\r\n$$\\begin{aligned}6&\\equiv 2\\pmod{n},\\\\\\\\ 7-(-1)^a&\\equiv 0\\pmod{n+1}\\end{aligned}$$\r\näžåŒãã $n$ 㯠$2,4$ ã®ã©ã¡ãããšãããïŒããããã«ã€ããŠäžåŒãèããã°\r\n- $n=2$ ã®ãšã $1\\equiv (-1)^a\\pmod{3}$ ãã $(a,b,m)=(2,2,4)$ïŒ\r\n- $n=4$ ã®ãšã ... | ã$2$ 以äžã®æ£ã®æŽæ° $m, n$ ã«å¯ŸãïŒä»¥äžã®ããã«å®ããŸãïŒ
$$a=\log\_{n}m,\quad b=\log\_{(n+1)}\left(\dfrac{m^2-m+6}{2}\right).$$
$a, b$ ããšãã«çŽ æ°ãšãªããããªãã¹ãŠã®çµ $(m,n)$ ã«å¯ŸããŠïŒ$m+n$ ã®ç·åãæ±ããŠãã ããïŒ
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc003/tasks/31 | D | OMC003(D) | 400 | 0 | 0 | [
{
"content": "ã$D_1(\\theta)$ ã¯3ç¹ $\\vec{b},3\\vec{a}-2\\vec{b},5\\vec{a}+4\\vec{b}$ ãé ç¹ãšããäžè§åœ¢ã®åšããã³å
éšã§ããïŒ$\\vec{a},\\vec{b}$ ã¯ãããã $\\begin{pmatrix}3\\\\\\\\ 2\\end{pmatrix},\\begin{pmatrix}1\\\\\\\\ 3\\end{pmatrix}$ ãåç¹äžå¿ã« $\\theta$ ã ãå転ãããã®ã§ããããïŒ$D_2$ ã¯\r\n$$X(1,3),\\quad Y(7,0),\\quad Z(19,22)$$\r\nãšã㊠$\\tr... | $$\\vec{a}=\\left( \\begin{array}{c} 3\\cos \\theta -2\\sin \\theta\\\\ 2\\cos \\theta +3\\sin \\theta \\\\ \\end{array} \\right),\quad\\vec{b}=\\left( \\begin{array}{c} \\cos \\theta -3\\sin \\theta \\\\ 3\\cos \\theta +\\sin \\theta \\\\ \\end{array} \\right)$$
ãšããŸãïŒãŸãïŒç¹ $P(\\ve... |
OMC002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc002/tasks/24 | A | OMC002(A) | 100 | 0 | 0 | [
{
"content": "ã$X=x^2-10x-29$ ãšããã°ïŒ\r\n$$0=\\frac{1}{X}+\\frac{1}{X-16}-\\frac{2}{X-40}=\\frac{-64(X-10)}{X(X-16)(X-40)}$$\r\nãã $X=10$ ãåŸãïŒ$x^2-10x-29=10$ ã®è§£ã¯ $x=-3,13$ ã§ããããïŒè§£çãã¹ãå€ã¯ $\\bm{13}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc002/editorial/24"
}
] | ã$x$ ã®æ¹çšåŒ
$$\displaystyle \frac{1}{x^2-10x-29}+\frac{1}{x^2-10x-45}-\frac{2}{x^2-10x-69}=0$$
ã®æ£ã®å®æ°è§£ã®ç·åãæ±ããŠãã ããïŒ |
OMC002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc002/tasks/25 | B | OMC002(B) | 200 | 0 | 0 | [
{
"content": "$$\\begin{aligned}\r\nz&=x^6+y^6+3^3-9x^2y^2\\\\\\\\\r\n&=(x^2+y^2+3)(x^4+y^4+3^2-x^2y^2-3x^2-3y^2)\\\\\\\\\r\n&=(x^2+y^2+3)\\times\\frac{(x^2-3)^2+(y^2-3)^3+(x^2-y^2)^2}{2}\\\\\\\\\r\n\\end{aligned}$$\r\nãšå€åœ¢ã§ããïŒ$x,y$ ã¯æ£æŽæ°ãã $x^2+y^2+3\\geq 5$ ãæãç«ã€ããïŒ$z$ ãçŽ æ°ã§ããããšãã次ãåŸãïŒ\r\n$$\\begin{cases}x^2+y^2... | ãæ£ã®æŽæ° $x, y$ ãšçŽ æ° $z$ ã¯ïŒä»¥äžã®çåŒãã¿ãããŸãïŒ
$$x^6+y^6-9x^2y^2=z-27.$$
ãã®ãããªçµ $(x,y,z)$ ãã¹ãŠã«ã€ããŠïŒ$xyz$ ã®ç·åãæ±ããŠãã ããïŒ
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc002/tasks/26 | C | OMC002(C) | 300 | 0 | 0 | [
{
"content": "ã$5\\times 2^m=(n-1)(n+1)$ ããïŒæ¡ä»¶ãã¿ãã $n$ ã¯æ£æŽæ° $a,b$ ãçšã次ã®ããããã®åœ¢ã«è¡šããããšããããïŒãã ã $a,b$ ã®ãã¡å°ãªããšãäžã€ã¯ $1$ ã§ããïŒ\r\n$$\\mathrm{(i)}:\\begin{cases}n-1=5\\times 2^a\\cr n+1=2^b\\end{cases}\\quad\\mathrm{(ii)}:\\begin{cases}n-1=2^a\\cr n+1=5\\times 2^b\\end{cases}$$\r\nã$\\mathrm{(i)}$ ã®å Žå $2^b-5\\times 2^a=... | ãæ£ã®æŽæ°ã®çµ $(m, n)$ ã¯
$$1+5\\cdot2^m=n^2 $$
ãæºãããŸãïŒãã®ãããªçµãã¹ãŠã«ã€ããŠïŒ$m+n$ ã®ç·åãæ±ããŠãã ããïŒ
ã**ãã¡ãã®åé¡ã®æäŸè
ãæ¢ããŠããŸãïŒå¿åœããã®ããæ¹ã¯ãé£çµ¡ãã ããïŒ** |
OMC002 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc002/tasks/27 | D | OMC002(D) | 400 | 0 | 0 | [
{
"content": "ãåé¢äœã®äœç©ã $V$ ïŒè¡šé¢ç©ã $S$ ãšããã° $3V=rS$ ãæç«ïŒåãåºãããå°åé¢äœã®ãã¡å
æ¥çã®ååŸã $a$ ã§ãããã®ãåé¢äœã®é ç¹ $A$ ãå«ããšããŠïŒ$A$ ã®å¯Ÿé¢ã®é¢ç©ã $S_A$ ãšãããïŒçžäŒŒãèããã° $$\\frac{3V}{S_A}:r=\\left(\\frac{3V}{S_A}-2r\\right):a$$ ãã $S_A=\\dfrac{S(r-a)}{2r}$ ããããïŒä»ã®é ç¹ã«ã€ããŠãåæ§ã®åŒãæãç«ã€ããïŒãããã足ãåãããŠ\r\n$$S=\\frac{S(4r-a-b-c-d)}{2r}=\\frac{S(4r-811114)}{... | ãååŸ $r$ ã®å
æ¥çãæã€åé¢äœããããŸãïŒåé¢ã«å¹³è¡ã§ïŒãã€å
æ¥çã«æ¥ãããããªé¢ã§åé¢äœãåæãããšïŒåãåºããã $4$ ã€ã®å°ããªåé¢äœã¯ããããå
æ¥çãæã¡ïŒããããã®ååŸã $a, b, c, d$ ã«ãªããŸããïŒ$a+b+c+d=811114$ ã§ãããšãïŒ$r$ ãæ±ããŠãã ãã. |
OMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc001/tasks/20 | A | OMC001(A) | 100 | 0 | 0 | [
{
"content": "ã$x=5\\pm4\\sqrt{2}$ ã®æå°å€é
åŒã¯ $x^2-10x-7$ ã§ããïŒ$x^3-mx-n$ ã¯ãããå æ°ã«ãã€ïŒäºæ¬¡ã®ä¿æ°ã«çç®ãããšãã㯠$(x^2-10x-7)(x+10)$ ã§ããã»ããªãããïŒæ±ããå€ã¯ $107+70=\\textbf{177}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc001/editorial/20"
}
] | ã$x=5\pm4\sqrt{2}$ ã $x^3-mx-n=0$ ã®è§£ã§ãããšãïŒãã®ãããªæŽæ°ã®çµ $(m, n)$ ãã¹ãŠã«å¯ŸããŠïŒ$m+n$ ã®ç·åãæ±ããŠãã ããïŒ |
OMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc001/tasks/21 | B | OMC001(B) | 200 | 0 | 0 | [
{
"content": "$$\\begin{aligned}\r\nN+321&=10+100+1000+\\cdots+\\underbrace{100\\cdots 00}\\_{322æ¡}\\\\\\\\\r\n&=\\underbrace{11\\cdots 110}\\_{322æ¡}\r\n\\end{aligned}$$\r\nãã $N=\\underbrace{11\\cdots 10789}_{322æ¡}$ ãšèšç®ã§ããïŒåŸã£ãŠ $N$ ã®åæ¡ã®ç·å㯠$\\bm{342}$ ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontes... | $$N=9+99+999+9999+\\cdots+\\underbrace{99\\ldots99}_{321æ¡}$$
ãšããŸãïŒãã®ãšãïŒ$N$ ã®åæ¡ã®ç·åãæ±ããŠãã ããïŒ |
OMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc001/tasks/22 | C | OMC001(C) | 300 | 0 | 0 | [
{
"content": "ã$X=x^2+18x+30$ ãšãããšäžæ¹çšåŒã¯ $X=2\\sqrt{X+15}$ ãšãªãïŒäž¡èŸºãäºä¹ããã°äºæ¬¡æ¹çšåŒ\r\n$$X^2=4(X+15)$$\r\nãåŸããïŒãã®è§£ã¯ $X=-6,10$ ã§ããïŒãã㧠$X=2\\sqrt{X+15}\\geq 0$ ã§ãããã $X=10$ ã®ã¿ãé©ããããïŒçµå±äºæ¬¡æ¹çšåŒ $x^2+18x+20=0$ ã®å®æ°è§£ã®ç·ç©ãèããã°ããïŒå€å¥åŒãèããã°ããã¯çžç°ãªãå®æ°è§£ããã€ããšããããããïŒè§£ãšä¿æ°ã®é¢ä¿ããããã®å®æ°è§£ã®ç·ç©ã¯ $\\bm{20}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "... | ã$x$ ã®æ¹çšåŒ
$$x^2+18x+30=2\sqrt{x^2+18x+45}$$
ã®ãã¹ãŠã®å®æ°è§£ã®ç©ãæ±ããŠãã ããïŒ |
OMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc001/tasks/23 | D | OMC001(D) | 400 | 0 | 0 | [
{
"content": "ãäžåã«äžŠãã $2020$ åã®çœäžžã®ãã¡, $1009$ åãé»ãå¡ãã€ã¶ãæ¹æ³ãèããïŒãã®ãã¡ïŒå·Šãã $2,4,\\ldots,1008$ çªç®ã®é»äžžãå·Šãã $a_{1}+1,a_{1}+a_{2}+2,\\ldots,a_{1}+\\cdots+a_{504}+504$ çªç®ã«ãããããªãã®ã¯ $a_{1}a_{2}\\cdots a_{505}$ éãã§ããããšããããïŒããªãã¡ïŒæ±ããç·å㯠$\\_{2020}\\mathrm{C}\\_{1009}$ ã«çããïŒããã®ãã€æ倧ã®çŽ å æ°ã¯ $\\textbf{2017}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬... | ã$a_1+a_2+\cdots+a_{505}=1516$ ãªãæ£æŽæ°ã®çµ $(a_1, a_2, \ldots, a_{505})$ ãã¹ãŠã«ã€ããŠïŒç© $a_1a_2\cdots a_{505}$ ã®ç·åã $M$ ãšããŸãïŒ$M$ ããã€çŽ å æ°ã®ãã¡ïŒæ倧ã®ãã®ã解çããŠãã ãã. |
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