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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 |
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OMC088 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc088/tasks/3082 | C | OMC088(C) | 200 | 181 | 210 | [
{
"content": "ãåç»è§£èª¬ãhttps:\\/\\/youtu.be\\/BYsTFR1AnuU \r\n\r\nãåæ°ã¯ $9x+10(49-x)$ ãšè¡šããïŒãã®ãšãïŒä»åãã®å ãæ¹ã«ã€ããŠ\r\n$${}\\_{49}\\mathrm{C}\\_{x}=6499270398159.$$\r\näžåŒãæºãã $x$ ã®å€ã® $1$ ã€ã $t$ ãšããã°ïŒ$x=49-t \\neq t$ ã解ãšãªãïŒãŸã\r\n$${}\\_{49}\\mathrm{C}\\_{0}\r\n\\lt {}\\_{49}\\mathrm{C}\\_{1}\r\n\\lt \\cdots \\lt {}\\_{49}\\... | ãã©ãããŠãç¡æã§æ¬ãèªã¿ããOMCåã¯ïŒèªåã®å®¶ã§OMCå³æžé€šãçµå¶ããã°ããããšã«æ°ãã€ããŸããïŒãŸãOMCåã¯ïŒç®ç«ã€å Žæã«ããããã®æ¬ããã¹ãŠæšªäžåã«äžŠã¹ãã®ã§ããïŒãã®ãŸãŸã§ã¯äžå®å®ãªã®ã§ä»¥äžã®æ¡ä»¶ã«åŸã£ãŠã¡ããã© $50$ åã®ä»åããå ããããšã«ããŸããïŒ
* $2$ åã®ä»åãã®éã«ã¯ïŒæ¬ã $9$ åãŸã㯠$10$ å䞊ã¹ã
* ãã ãïŒå·Šå³ã®ç«¯ã«ã¯å¿
ã $1$ ã€ãã€ä»åããå ãããã®ãšããïŒ
ãã®ãšãïŒä»åãã®å ãæ¹ã¯å
šéšã§ $6499270398159$ éãã§ãããšåãããŸããïŒOMCåã®ããããã®æ¬ã®åæ°ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ ãã ãïŒé©ããåæ°ã¯ååšããããšãä¿èšŒã... |
OMC088 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc088/tasks/3048 | D | OMC088(D) | 300 | 221 | 254 | [
{
"content": "ãåç»è§£èª¬ãhttps:\\/\\/youtu.be\\/te2E7al8Xho\r\n\r\n ãç¹ $A$ ããç¹ $P$ ãäžå³ã®ããã«å®ããïŒç¹ $E$ ããã³ $K$ ãæããããã«ã¯ïŒã«ã¡ã©ã¯çŽç· $EM$ ããäžåŽãšçŽç· $NK$ ããäžåŽã« $1$ ã€ãã€ãªããã°ãªããïŒããããç·å $HM,JN$ äžã«ãããšããŠãããïŒãã®ãšãã«ã¡ã©ã¯ãã®äœçœ®ã«ãããåè§åœ¢ $FPLO$ ã®å€éšãã¹ãŠãæããããïŒåè§åœ¢ $FPLO$ ã®ã¿ãèãããšïŒããããã®ã«ã¡ã©ãç¹ $M,N$ ã«è¿ã¥ãã»ã©æããé åã倧ãããªãïŒãã£ãŠïŒä»¥äžã«ã¡ã©ããã® $2$ ç¹ã«åºå®ããïŒ\\\r\nããã®ãš... | ãOMCåã¯OMCå³æžé€šãçµå¶ããããã«å°çšã®éšå±ãçšæããŠïŒæ¬æ£ãšç£èŠã«ã¡ã©ãèšçœ®ããããšã«ããŸããïŒããã§ïŒä»¥äžã®æ¡ä»¶ãã¿ããããã«ããŸãïŒ
- éšå±ã¯çäžããèŠãŠ $1$ 蟺ã®é·ãã $5 \ \mathrm{m}$ ã®æ£æ¹åœ¢ã§ããïŒ
- æ¬æ£ã¯çŽæ¹äœã§ãã®**奥è¡ã**㯠$1 \ \mathrm{m}$ ã§ããïŒäžå³ã®ããã«å·Šå³ããããã®å£ã«**åŽé¢ãæ¥è§ŠãããŠ**èš $2$ ã€èšçœ®ããïŒæ¥è§Šãããäœçœ®ã¯äžå³ã®éãïŒäžäžããããã« $1 \ \mathrm{m}$ ã®äœçœ®ïŒã§åºå®ã§ããïŒå¹
ã®ã¿èªç±ã«å€ããããïŒ
- ç£èŠã«ã¡ã©ã¯åšå² $360^\circ$ ã§ååé ããŸã§æãããããã®ã $2$ å°ïŒéšå±ã®ãã¹ãŠ... |
OMC088 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc088/tasks/3326 | E | OMC088(E) | 400 | 8 | 93 | [
{
"content": "ãåç»è§£èª¬ãhttps:\\/\\/youtu.be\\/Rha6JK_IwAc\r\n\r\n ãåã®æ¡ä»¶ãã¿ããé£æºã®æ§åã¯ïŒç¹å®ã® $2$ é ç¹ $X,Y$ ãå«ãäºãã«åºå¥å¯èœãª $30$ åã®é ç¹ã«ã€ããŠïŒ$X,Y$ ã«ã€ããŠã¯æ¬¡æ°ã $3$ ã«ïŒãã®ä»ã® $28$ é ç¹ã«ã€ããŠã¯æ¬¡æ°ã $2$ ã«ãªãããã«ïŒå
šäœãåçŽãã€é£çµãšãªãããç¡å蟺ã匵ã£ãã°ã©ããšã¿ãªããïŒ\r\nãã®ãšãïŒ$X$ ãåºçºããŠåäžã®é ç¹ã $2$ å以äžéãããšãªã $Y$ ãžç§»åããçµè·¯ (**è¯ãçµè·¯**ãšåŒã¶ããšã«ãã) 㯠$1$ éããŸã㯠$3$ éãããããšã確èªã§ããïŒ\\\r\nã次... | ãOMCå³æžé€šãããå°åã«ã¯ïŒOMCå³æžé€šãåãããŠå
šéšã§ $30$ è»ã®å³æžé€šããããŸãïŒ$2$ ã€ã®å³æžé€šã**é£æº**ããŠãããšãïŒãããã¯çžäºã«æ¬ãè²æž¡ããããšãã§ããŸãïŒ\
ãããŸïŒ$30$ è»ã®ãã¡ $X$ å³æžé€šãš $Y$ å³æžé€šã¯ããããèªèº«ä»¥å€ã® $3$ è»ã®å³æžé€šãšé£æºããŠããïŒæ®ãã® $28$ è»ã®å³æžé€šã¯ããããèªèº«ä»¥å€ã® $2$ è»ã®å³æžé€šãšé£æºããŠããŸãïŒããã§ïŒ$X$ å³æžé€šãš $Y$ å³æžé€šã¯ãšãã«OMCå³æžé€šãšã¯ç°ãªããŸãïŒ\
ãåå³æžé€šãé©åœãªè²æž¡ãç¹°ãè¿ãè¡ãããšã§OMCå³æžé€šãã©ã®å³æžé€šã«ããæ¬ãè²æž¡ããŠãããããšãå¯èœã§ãããšãïŒ$30$ è»ã®å³æžé€šã®é£æºã®ä»æ¹ãšããŠèãããããã®ã¯ $... |
OMC088 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc088/tasks/3323 | F | OMC088(F) | 400 | 21 | 90 | [
{
"content": "ãåç»è§£èª¬ãhttps:\\/\\/youtu.be\\/fhjXRI0DdmI\r\n\r\n ãç°ãªãããŒã¿ã«ã€ããŠã¯äºçŽçªå·ãç°ãªãå€ãšãªããã㪠$a,b$ ã®çµã**è¯ãçµ**ãšåŒã¶ããšã«ããïŒ\r\n\r\nã$2$ ã€ã®ããŒã¿ $(m,n)\\neq (m^\\prime,n^\\prime)$ ã®äºçŽçªå·ãäžèŽãããšãïŒ$a+b=2357$ ã¯çŽ æ°ãã $a,b$ ã¯äºãã«çŽ ã§ããããšã«æ³šæããã°ïŒ$a(m-m^\\prime)=b(n^\\prime-n)$ ããããæŽæ° $k$ ãååšã㊠$m-m^\\prime=bk,n^\\prime-n=ak$ ãã¿ããïŒ\r\... | ãOMCå³æžé€šã«ããå
š $M(\geq 1)$ åã®æ¬ã«ã¯ $1$ ãã $M$ ãŸã§ã®çªå·ãå²ãåœãŠãããŠããŸãïŒãŸãïŒå
š $N(\geq 1)$ 人ã®å©çšè
ã«ã¯ $1$ ãã $N$ ãŸã§ã®çªå·ãå²ãåœãŠãããŠããŸãïŒOMCå³æžé€šã§ã¯ãçªå· $m$ ã®æ¬ãçªå· $n$ ã®å©çšè
ãäºçŽããŠãããããšãçµ $(m,n)$ ã«ãã£ãŠè¡šãïŒããã**ããŒã¿**ãšåŒãã§ããŸãïŒ\
ã$a+b=2357$ ãã¿ããæ£æŽæ° $a,b$ ãçšããŠããŒã¿ $(m,n)$ ã®**äºçŽçªå·**ã $am+bn$ ãšå®ãããšãïŒç°ãªãããŒã¿ã«ã€ããŠã¯äºçŽçªå·ãç°ãªãå€ãšãªããããªæ£æŽæ°ã®çµ $(a,b)$ ã¯ã¡ããã© $333$ åãããŸããïŒãã®ãš... |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/4120 | A | OMC087(A) | 300 | 138 | 195 | [
{
"content": "ãç·å $BD$ ã®åçŽäºçåç·ã«é¢ã㊠$C$ ãšå¯Ÿç§°ãªç¹ã $C^\\prime$ ãšãã. $A,B,C^\\prime,D$ ãå
±åã§ããããšã«æ°ãã€ããã°\r\n$$\\triangle ABC^\\prime = AB\\times BC^\\prime \\times \\sin \\angle ABC^\\prime = C^\\prime D\\times DA\\times\\sin\\angle C^\\prime DA = \\triangle C^\\prime DA$$\r\nã§ãããã, ç·å $AC^\\prime$ ãšç·å $BD$ ã®äº€ç¹ã¯ç·å $BD$... | ãåã«å
æ¥ããåè§åœ¢ $ABCD$ 㯠$AB\times CD = AD\times BC$ ãæºãããŸã. ç·å $BD$ ã®äžç¹ã $M$ ãšããŸã.
$$\begin{aligned}
AB = 127,&& AD = 129,&& CM = 32
\end{aligned}$$
ã§ãããšã $AM$ ã®é·ããæ±ããŠäžãã. |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/2335 | B | OMC087(B) | 400 | 92 | 201 | [
{
"content": "ãããŸã¿ã€ã«åŒµããç¡éã«ç¶ããŠãããšããŠïŒããã¿ã€ã«ã®é ç¹ã«ãããç¹ã**è¯ãç¹**ãšåŒã¶ããšãšããïŒ\\\r\nã$2$ é ç¹ãè¯ãç¹ã§ããæ£äžè§åœ¢ã«ã€ããŠïŒãã $1$ é ç¹ãè¯ãç¹ã§ããããšããããããïŒããæ£å
è§åœ¢ã®é ç¹ãå
šãŠè¯ãç¹ã§ãããšãïŒãã®äžå¿ãè¯ãç¹ã§ããïŒ\\\r\nãéã«ïŒäžå¿ãš $1$ é ç¹ãè¯ãç¹ã«å®ãããšãïŒæ®ãã® $5$ é ç¹ãè¯ãç¹ã§ãããããªæ£å
è§åœ¢ãå®ãŸãïŒ\\\r\nãäžè¬ã« $300$ ã $3N$ ãšãïŒäžèŸº $3N$ ã®æ£äžè§åœ¢ã® $3$ é ç¹ã $A,B,C$ ãšããïŒè¯ãç¹ $P$ ã«ã€ããŠïŒ$P$ ãéã $CA$ ã«å¹³è¡ãªçŽç·ãšç·å $A... | ãäžèŸº $1$ ã®æ£äžè§åœ¢ã®ã¿ã€ã«ã $300^2$ åããïŒããããçµã¿åãããŠäžèŸº $300$ ã®å€§ããªæ£äžè§åœ¢ãäœããŸããïŒ\
ããã®å€§ããªæ£äžè§åœ¢ã«ãããŠïŒããã¿ã€ã«ã®é ç¹ã«ãããç¹ $6$ ã€ãé ç¹ã«ãã€æ£å
è§åœ¢ã¯ããã€ãããŸããïŒ |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/1458 | C | OMC087(C) | 500 | 61 | 108 | [
{
"content": "ãäžåŒãã $x_1x_2=x_2x_3-1=x_3x_4-2=\\cdots=x_8x_1-7=kx_1x_2-8$ ã§ãããã,\r\n$$(x_1x_2)(x_3x_4)(x_5x_6)(x_7x_8)=(x_2x_3)(x_4x_5)(x_6x_7)(x_8x_1)$$\r\nã $x_1x_2$ ã®ã¿ã®åŒã§è¡šãããšã§\r\n$$(x_1x_2)(x_1x_2+2)(x_1x_2+4)(x_1x_2+6)=(x_1x_2+1)(x_1x_2+3)(x_1x_2+5)(x_1x_2+7)$$\r\nããã解ã㊠$x_1x_2=-\\dfrac{7}{2},\\dfrac{-7\\p... | ã以äžã® $8$ ã€ã®åŒããã¹ãŠã¿ããå®æ°ã®çµ $(x_1,x_2,\ldots,x_8)$ ãååšãããããª, å®æ° $k$ ã®ç·åãæ±ããŠãã ããïŒ
$$\begin{aligned}
x_1+\frac{1}{x_2}=x_3,&&
x_2+\frac{1}{x_3}=x_4,
&&\cdots,
&&x_7+\frac{1}{x_8}=x_1,
&& x_8+\frac{1}{x_1}=kx_2
\end{aligned}$$
ãã ã,ãæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $-\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/2804 | D | OMC087(D) | 600 | 27 | 81 | [
{
"content": "ãã¯ããã®é
眮㧠$5$ ã€ã®ã°ã«ãŒãã $A,B,C,D,E$ ãšåœåããïŒãã®ãšãïŒä»¥äžã®ãããªè¡šãèããïŒ\r\n$$\\begin{array}{|c|c|c|c|c|}\r\n\\hline\r\n\\\\{D,E\\\\} & \\\\{C,E\\\\} & \\\\{A,B\\\\} & \\\\{A,B\\\\} & \\\\{C,D\\\\} \\\\\\\\\r\n\\hline \\hline\r\n\\\\{B,C\\\\} & \\\\{A,D\\\\} & \\\\{D,E\\\\} & \\\\{C,E\\\\} & \\\\{A,B\\\\} \\\\\\\... | ã$2$ ã€ã®å€§åŠãããïŒãããããã $10$ 人ã®åŠçãåå ããŠããŒãã£ãŒãå¬ããŸãïŒã¯ããïŒåå è
㯠$4$ åã〠$5$ ã€ã®ã°ã«ãŒãã«é
眮ãããŸããïŒããã§ïŒããããã®ã°ã«ãŒãã«ã¯å倧åŠã®åŠçã $2$ 人ãã€é
眮ãããŠããŸãïŒ\
ãããŸïŒåå è
ã®èŠªçŠãããæ·±ããããïŒã°ã«ãŒããçµã¿æ¿ããããšã«ããŸããïŒãã®ãšãïŒåæ§ã«åã°ã«ãŒãã«ããããã®å€§åŠã®åŠçã $2$ 人ãã€é
眮ãïŒãã€ä»¥äžã®æ¡ä»¶ãã¿ããããã«ããŸãïŒ
- å
¥ãæ¿ãã®ååŸã§ïŒãšãã«åãã°ã«ãŒãã«é
眮ããã $2$ 人çµãååšããªãïŒ$2$ 人ã®å€§åŠã®äžèŽã«ãããªãïŒïŒ
ã¯ããã®é
眮ãåºå®ãããšãïŒãããã®æ¡ä»¶ãã¿ããããã«ã°ã«ãŒããçµã¿æ¿ããæ¹æ³... |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/2472 | E | OMC087(E) | 800 | 3 | 20 | [
{
"content": "ãåé²è¡šèšãããšãã«ã©ã®æ¡ã«ã $9$ ãçŸããªã $n$ æ¡ä»¥äžã®éè² æŽæ°ã®ãã¡ïŒ$7$ ã®åæ°ã§ãããã®ã®åæ°ã $f(n)$ ãšããïŒ\r\nãã ãããã§ã¯ $0$ 㯠$0$ æ¡ã§ãããšããïŒãã®ãšãæ±ããç·å $S$ ã¯æ¬¡ã®ããã«è¡šããïŒ\r\n$$S= 5^{2022^{2022}} f(5^{2022^{2022}})-\\sum_{n=0}^{5^{2022^{2022}}-1}f(n)$$\r\n\r\nã$f(n)$ ãæ±ãããïŒåé²è¡šèšã§ã©ã®æ¡ã«ã $9$ ãçŸããªã $n$ æ¡ä»¥äžã®éè² æŽæ° $K$ 㯠$x_i\\in\\\\{0,1,\\dots,8\\\\}$... | ã次ã®æ¡ä»¶ãã¿ããæ£æŽæ°ãã¹ãŠã«ã€ããŠïŒåé²æ³è¡šèšã§ã®**æ¡æ°ã®ç·å**ã $9^7$ ã§å²ã£ãããŸããæ±ããŠãã ããïŒ
- åé²è¡šèšã§ã®æ¡æ°ã $5^{2022^{2022}}$ æ¡ä»¥äžã§ããïŒ
- $7$ ã®åæ°ã§ããïŒ
- åé²è¡šèšãããšãã«ïŒã©ã®æ¡ã«ã $9$ ãçŸããªãïŒ
äŸãã°ïŒæ¡ä»¶ãã¿ããæ£æŽæ°ã $3,14,15$ ã® $3$ ã€ã§ãã£ããšãïŒæ±ããå€ã¯ $1+2+2=5$ ã§ãïŒãŸãïŒ$5^{2022^{2022}}=5^{(2022^{2022})}$ ã§ãïŒ |
OMC087 (Mathpediaæ¯) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc087/tasks/3553 | F | OMC087(F) | 800 | 5 | 38 | [
{
"content": "ãäžè§åœ¢ $ABC$ ã®é¢ç©ã $S$ ãšãããš, \r\n\r\n$$AM = \\dfrac{2S}{BC} = \\dfrac{2S}{AB + BC - CA} = DI$$\r\n\r\nãåãã. ãŸã, äžè§åœ¢ $ABC$ ã® $A$ ã«å¯Ÿããåæ¥åãšèŸº $BC$ ã®æ¥ç¹ã $M$ ã§ããããšã«æ°ãã€ãããš, $AD = CM$ ãåŸã. åŸã£ãŠ, äžè§åœ¢ $ADI$ ãšäžè§åœ¢ $CMA$ ã¯ååã§ãããã, $AB = AC = AI$ ãåãã. ãŸã, \r\n\r\n$$\\angle PAM = \\angle PDI, \\quad \\angle PMA = \... | ã$AB = AC$ ãæºããäžè§åœ¢ $ABC$ ã«ã€ããŠ, 蟺 $BC$ ã®äžç¹ã $M$ ãšã, $B$ ã«å¯Ÿããåå¿ã $I$ ãšããŸã. $I$ ãéã蟺 $AC$ ã«åçŽãªçŽç·ãçŽç· $AC, AM$ ãšäº€ããç¹ããããã $D, E$ ãšããŸã. äžè§åœ¢ $ADE$ ã®å€æ¥åãšäžè§åœ¢ $EIM$ ã®å€æ¥åã® $E$ ã§ãªã亀ç¹ã $P$ ãšããŸã.
$$DP = 7, \quad MP = 9$$
ã§ãããšã, äžè§åœ¢ $ABC$ ã®é¢ç©ãæ±ããŠäžãã. \
ããã ã, æ±ããçãã¯äžã€ã®æ£ã®æŽæ° $a, b, c$ ïŒ $b$ ã¯å¹³æ¹å åãæããªãïŒãçšã㊠$a + \dfrac{c}{\sqrt{b... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3872 | A | 第25åçäžå
¥è©Šæš¡è©Š(A) | 100 | 114 | 134 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3872"
},
{
"content": "ã $(E,I,N,R,T,U,V)=(6,1,7,9,3,4,2)$ (ããªãã¡ïŒ $3497\\div13=269$ )ãšãã解ãè©Šè¡é¯èª€ããŠé 匵ã£ãŠã¿ã€ãããïŒ\\\r\n 解çãã¹ãæ°å€ã¯ $\\textbf{3497}$ ã§ããïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onli... | äœåïŒèç¢
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæŽæ°ãå
šãŠæ±ã, ãã®ç·åã解çããŠäžãã.
***
ã$\text{E, I, N, R, T, U, V}$ ã® $7$ æåã«ã¯ $0$ ïœ $9$ ã®æŽæ°ã®ãããããåœãŠã¯ãŸã, åãæåã«ã¯åãæŽæ°ã, ç°ãªãæåã«ã¯ç°ãªãæŽæ°ãåœãŠã¯ãŸã. $\square$ ã«ã¯ $0$
ïœ $9$ ã®æŽæ°ã®ãããããåœãŠã¯ãŸã, å
šãŠåãæŽæ°ãåœãŠã¯ãŸããšã¯éããªã. 以äžã®çç®ãæãç«ã€ãšã, $\text{TURN}=\boxed{\phantom{nada}}$ ã§ãã. ãã ãæäžäœã® $\square$ ã«ã¯ $0$ ... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3873 | B | 第25åçäžå
¥è©Šæš¡è©Š(B) | 100 | 36 | 79 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3873"
},
{
"content": "ã(ããã§ã¯è§£çéå ±ãç®çãšããŠïŒè©³ããè«èšŒã¯ããã«ïŒçµæãåºãæ¹æ³ã®ã¿ãè¿°ã¹ãããšã«ããïŒ)\r\n\r\nã $987654321$ ã $14$ å䞊ã¹ãŠåºæ¥ã $126$ æ¡ã®æ°ãã $2$ æ¡åãé€ããŠåŸããã $124$ æ¡ã®æ°ãæ¡ä»¶ãæºãã $n$ ã§ããïŒ\\\r\nãã£ãŠïŒåãé€ã $2$ æ°ã®éžã³æ¹ãèããŠïŒæ±ããå... | äœåïŒå é
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæŽæ°ã解çããŠäžãã.
***
ã以äžã®æ¡ä»¶ãã¿ãã $1$ 以äžã®æŽæ° $n$ ã«ã€ããŠ, æãå°ããæ¡æ°ã®ãã®ã¯ $\boxed{\phantom{nada}}$ åãã.
- $9\times n$ ã®åäœã®å㯠$999$ .
- $n$ ã¯ã©ã®äœã $0$ ã§ãªã, ãã€é£ãåãäœã®æ°åã¯å
šãŠç°ãªã. |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3874 | C | 第25åçäžå
¥è©Šæš¡è©Š(C) | 100 | 115 | 142 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3874"
}
] | äœåïŒå®®æ
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæŽæ°ã解çããŠäžãã.
***
ã以äžã®æ¡ä»¶ãã¿ãã $1$ 以äžã®æŽæ° $n$ ã®ãã¡, $2022$ çªç®ã«å°ãããã®ã¯ $\boxed{\phantom{nada}}$ æ¡ã§ãã.
- $7\times n$ ã®åäœã®æ°ã¯ $1$ ã®äœããé ã« $3,9,2,1,1,2$ ã®äžŠã³ãç¹°ãè¿ã.
äŸãã°, $7\times13315899=93211293$ ã¯ãã®æ¡ä»¶ãã¿ãã. |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3875 | D | 第25åçäžå
¥è©Šæš¡è©Š(D) | 100 | 13 | 43 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3875"
},
{
"content": "ã $9$ æ°ã®æ倧å
¬çŽæ°ã $g$ ãšãããšïŒ $9$ æ°ã®åäœã®å㯠$30$ ãªã®ã§ïŒã©ãã $9$ ã§å²ã£ãŠ $3$ äœãæ°ã ããïŒ \\\r\n(i) $g$ 㯠$9$ ã§å²ã£ãŠ $3$ äœãæ°ã§ $9$ æ°ã¯ $g,4g,7g,10g,\\cdots$ ã®ãããã\\\r\n(ii) $g$ 㯠$9$ ã§å²ã£ãŠ $6$ ... | äœåïŒæ²
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæŽæ°ã解çããŠäžãã.
***
ãé»æ¿ã« $9$ ã€ã® $6$ æ¡ã®æŽæ°ãæžãããŠãã. é»æ¿ã«æžãããã©ã®æŽæ°ãåäœã®å㯠$30$ ã§, åæŽæ°ãäžäžã®äœã§åæšäºå
¥ããŠåäžã®äœãŸã§ã®æŠæ°ã«ãããš, ãããã $10$ äž, $20$ äž, $30$ äž, $40$ äž, $50$ äž, $60$ äž, $70$ äž, $80$ äž, $90$ äžã«ãªããšã, é»æ¿ã«æžãããæ°ã®æ倧å
¬çŽæ°ã¯æ倧㧠$\boxed{\phantom{nada}}$ ã§ãã. |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3876 | E | 第25åçäžå
¥è©Šæš¡è©Š(E) | 100 | 5 | 34 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3876"
}
] | äœåïŒèç¢ã»äžžå²¡
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ倧ã®æ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $m\times n\times n$ ã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ããã°ãåã¯, ãã $2$ ã€ã®äœã®åã $11$ ã§ãã, ãã€ãã $2$ ã€ã®äœã®åã $13$ ãšãªã $4$ æ¡ã®æŽæ°ãéžã³, ã¿ã€ã°ã¡åã¯ãã®ããšãç¥ã£ãŠãã. ã¿ã€ã°ã¡åã¯åãã« $100$ ç¹ãäžããããŠãã. $2$ 人ã¯æ¬¡ã®ãããªè¡åã $14$ åç¹°ãè¿ã.
- ã¿ã€ã°ã¡åã¯ã... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3877 | F | 第25åçäžå
¥è©Šæš¡è©Š(F) | 100 | 13 | 33 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3877"
}
] | äœåïŒäžç°æ
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæŽæ°ã解çããŠäžãã.
***
ã$1$ 以äžã®æŽæ° $m,n$ ã«ã€ããŠ, äžèŸº $1\text{cm}$ ã®æ£æ¹åœ¢ã瞊㫠$m$ å, 暪ã«
$n$ åããã€ããŠã§ããé·æ¹åœ¢ã® $1$ æ¬ã®å¯Ÿè§ç·ã§æ£æ¹åœ¢ãåå²ã, ã§ããäžè§åœ¢ã®é¢ç©ã®ç·åã $\lbrace m,n\rbrace\text{cm}^2$ ãšè¡šãããšã«ãã. äŸãã° $\lbrace 1,3\rbrace\text{cm}^2$ ã¯äžå³äžã®æç·éšã®äžè§åœ¢ã®é¢ç©ã®åã§ãããã, $\lbrace 1,3\rbrace=\dfrac{1}{3}$ ã§ãã. ãã®ãšã,... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3878 | G | 第25åçäžå
¥è©Šæš¡è©Š(G) | 100 | 42 | 71 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3878"
}
] | äœåïŒé£¯æ²¢
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæŽæ°ã解çããŠäžãã.
***
ã瞊ã®ãã¹ã®æ°ã $1$ ïœ $10$ , 暪ã®ãã¹ã®æ°ã $1$ ïœ $10$ ã®ãã¹ãŠã®çµã¿åããã§èš $100$ çš®é¡ã®ãã¹ç®ã $1$ ã€ãã€ãã, ãããã®èš $3025$ åã®ãã¹ã«å°æ£ã®é§ã§ãããè§ãã眮ã. 次ã®æ¡ä»¶ãã¿ãããªãã眮ãããšã®ã§ããé§ã®æ°ã¯æ倧㧠$\boxed{\phantom{nada}}$ åã§ãã.
- ã©ã®ãã¹ã $2$ ã€ä»¥äžã®é§ã眮ãããŠããªã.
- ã©ã®é§ãä»ã®é§ã®è¡ãå
ã«ãªã.
ãã ã, é§ã¯ååå€ãçšæãããŠãããã®ãšãã. ãªã, ãè§ãã¯äžå³ã®... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3879 | H | 第25åçäžå
¥è©Šæš¡è©Š(H) | 100 | 21 | 35 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3879"
}
] | äœåïŒå é
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãã㊠$\angle ADC$ 㯠$30^{\circ}$ ããå°ãã,
$$\begin{aligned}
AD:BD=11:13, && AB=BC, && FD=1\text... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3880 | I | 第25åçäžå
¥è©Šæš¡è©Š(I) | 100 | 4 | 18 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3880"
},
{
"content": "ã泚ã»ä»¥äžã®è§£çã§ã¯åäœã® $\\rm{cm}$ ã¯çç¥ããŠããŸãïŒ\r\n\r\nãæ£æ¹åœ¢ $ADEB$, $BFGC$, $CHIA$ ã®äžå¿ããããã $P$, $Q$, $R$ ãšãããŸãïŒãã®ãšãïŒäžè§åœ¢ $AHE$ ã¯äžè§åœ¢ $ARP$ ãç¹ $A$ ãäžå¿ã« $2$ åã«çžäŒŒæ¡å€§ãããã®ã«ãªã£ãŠããŠïŒãšãã«çŽç· $HE$ ... | äœåïŒæŸç°
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãããŠåè§åœ¢ $ADEB$ , $BFGC$, $CHIA$ ã¯ããããäžè§åœ¢ $ABC$ ã®å€åŽã«ããæ£æ¹åœ¢ã§ãã.
$$\begin{aligned}
DG=13\text{cm},... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3881 | J | 第25åçäžå
¥è©Šæš¡è©Š(J) | 100 | 64 | 98 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3881"
},
{
"content": "ã $ADGH$ ã¯çèå°åœ¢ãªã®ã§ $HD=AG=HB$ ãšãªãïŒ $H$ 㯠$EG$ ã®äžç¹ãšãªãïŒãã£ãŠïŒ $AD=GH=HE=\\frac{3}{2}$\\\r\n ããã§ïŒ$AF$ ãš $BD$ ã®äº€ç¹ã $P$ ãšãããšïŒ $AP=HF=\\frac{3}{2}$ ã§ããããïŒ $â³ABP=â³FHB$\\\r\n ãã£ãŠïŒ ... | äœåïŒå æ
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãã㊠$AD$ ãš $BC$ ã¯å¹³è¡ã§, äžè§åœ¢ $DBC$ ãšäžè§åœ¢ $HEF$ ã¯ãšãã«æ£äžè§åœ¢ã§ãã.
$$\begin{aligned}
AG=BH, && AB=4\tex... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3882 | K | 第25åçäžå
¥è©Šæš¡è©Š(K) | 100 | 8 | 61 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3882"
}
] | äœåïŒæ²
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸã $0$ ãã倧ãã $180$ ããå°ããæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãããŠ
$$\begin{aligned}
BC+CE=AE=AD, && \angle BCE=\angle CBE=\angle CDE
... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3883 | L | 第25åçäžå
¥è©Šæš¡è©Š(L) | 100 | 9 | 31 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3883"
}
] | äœåïŒå é
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸã $0$ ãã倧ãã $180$ ããå°ããæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãã㊠$AP=DQ$, $CP=AQ$ ã®ãšã, $\angle ADC=\boxed{\phantom{nada}}^{\ \circ}$ ... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3884 | M | 第25åçäžå
¥è©Šæš¡è©Š(M) | 100 | 7 | 21 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3884"
}
] | äœåïŒæ²
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸã $0$ ãã倧ãã $180$ ããå°ããæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã«ãã㊠$AB=BD$, $AE=EF$, $BF=EC$ ã®ãšã, $\angle ADC=\boxed{\phantom{nada}}^{\ ... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3885 | N | 第25åçäžå
¥è©Šæš¡è©Š(N) | 100 | 39 | 45 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3885"
}
] | äœåïŒå±±å£
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã®æ£å
«é¢äœ $ABCDEF$ ã®äœç©ã¯ $36000\text{cm}^3$ ã§ãã. ãã®æ£å
«é¢äœã«ãããŠ, ç¹ $P,Q$ ã¯ãããã蟺 $AC, EF$ äžã®ç¹ã§, $CP:EQ:FQ... |
第25åçäžå
¥è©Šæš¡è©Š | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/nadachu2022/tasks/3886 | O | 第25åçäžå
¥è©Šæš¡è©Š(O) | 100 | 43 | 62 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/nadachu2022/editorial/3886"
}
] | äœåïŒäœè€
ã以äžã® $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãæ°ã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæŽæ°
$m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šããŸã. $\boxed{\phantom{nada}}$ ã«åœãŠã¯ãŸãåŸãæ°ãå
šãŠæ±ã, ããããã«ã€ã㊠$m\times n\times n$ ãèšç®ã, ãã®å€ã®ç·åã解çããŠäžãã. ãã ã, $n$ 㯠$1$ 以äžãšããŸã.
***
ãäžå³ã¯ããç«äœã®å±éå³ã§, äºç蟺äžè§åœ¢ $6$ ã€, ã²ã圢 $6$ ã€, æ£å
è§åœ¢ $1$ ã€ãããªã. åãèšå·ã¯åãé·ããè¡šã, $6$ ã€ã®ã²ã圢ã¯å
šãŠ $1\text{cm}$ ãš ... |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/2822 | A | OMC086(A) | 100 | 328 | 334 | [
{
"content": "ãæ£ããèšç®çµæã $57+a$ ã§ãããšãããšïŒèª€ã£ãèšç®çµæ㯠$57-a$ ã§ããïŒ\\\r\nããã£ãŠ $a=-34$ ã§ããïŒæ±ããå€ã¯ $57+a=\\textbf{23}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc086/editorial/2822"
}
] | ãæ£ã®æŽæ°ãšæŒç®å ($+$ ãŸã㯠$-$) ã亀äºã«äžŠãã§æ§æãããåŒããã, ãã®åŒã®å
é 㯠$57$ ã§ã. OMCåã¯ãã®åŒãèšç®ããããšããŠ, 誀ã£ãŠ $+$ ã $-$ ã«, $-$ ã $+$ ã«ãã¹ãŠèªã¿ééããŠããŸã£ããã, èšç®çµæã $91$ ã«ãªããŸãã. æ£ããèšç®çµæãæ±ããŠãã ãã. |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/3102 | B | OMC086(B) | 200 | 303 | 316 | [
{
"content": "ã$BD=30-DE=CE$ ãæç«ããïŒããã§ïŒåè§åœ¢ $DBCE$ ãåã«å€æ¥ããããšãã\r\n$$DE+30=DE+BC=BD+CE=2(30-DE)$$\r\nãæãç«ã€ããïŒä»¥äžãã $DE=\\bf{10}$ ãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc086/editorial/3102"
}
] | ãäžèŸºã $30$ ã§ããæ£äžè§åœ¢ $ABC$ ã«ãããŠïŒèŸº $AB,AC$ äžã«ããããç¹ $D,E$ ããšã£ããšããïŒ$AD=AE$ ã§ããïŒãã€åè§åœ¢ $DBCE$ ã¯åã«å€æ¥ããŸããïŒãã®ãšãïŒ$DE$ ã®é·ããæ±ããŠãã ãã. |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/2345 | C | OMC086(C) | 200 | 167 | 267 | [
{
"content": "ãäŸãã°, ã©ã®éåã $1$ ãå«ãã§ãããã¯, $2$ ã $A \\cup B \\cup C \\cup D$ ã«å«ãŸãããã $A \\cap B \\cap C \\cap D$ ã«å«ãŸããªããã«åœ±é¿ãåãŒããªã. ããªãã¡, å $1,2,3,4$ ã«ã€ããŠç¬ç«ã«èããããšãã§ãã.\\\r\nã$A \\cup B \\cup C \\cup D$ ã $1$ ãå«ãããšãã, $A,B,C,D$ ã®ãã¡å°ãªããšã $1$ ã€ã¯ $1$ ãå«ã¿, éã« $A \\cap B \\cap C \\cap D$ ã $1$ ãå«ãŸãªãããšãã, $A,B,C,D$ ã®ãã¡å°... | ã以äžã®æ¡ä»¶ãã¿ãã, $4$ ã€ã®éåã®é åºä»ããçµ $(A,B,C,D)$ ã¯ããã€ãããŸããïŒ
- åéå $A \cup B \cup C \cup D$ ã $\lbrace 1,2,3,4 \rbrace$ ã«äžèŽãã.
- ç©éå $A \cap B \cap C \cap D$ ã空éåã§ãã.
ãã ã, $A,B,C,D$ ã®ãã¡ã«ç©ºéåãå«ãããšãèªããŸã. |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/2068 | D | OMC086(D) | 300 | 63 | 194 | [
{
"content": "ã$N=2^p3^q5^r$ ãšè¡šã, ãã®ãšãåçŽ å æ°ã®åé
ãèããã° $\\textrm{lcm}(a,b)=N$ ãªãçµ $a\\lt b$ ã®åæ°ã¯\r\n$$\\dfrac{1}{2}\\left((2p+1)(2q+1)(2r+1)-1\\right)=2^9$$\r\nãã£ãŠç©ã $1025=5^2\\times 41$ ãšãªã $3$ ã€ã®å¥æ°ã®é åºä»ããçµãæ°ãäžããããšã«åž°çãã, ãã㯠$\\textbf{18}$ éãã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/conte... | ã$7$ 以äžã®çŽ å æ°ããããªãæ£æŽæ° $N$ ã§ãã£ãŠ, æå°å
¬åæ°ã $N$ ãšãªãããã«çžç°ãªã $2$ ã€ã®æ£æŽæ°ãéžã¶æ¹æ³ãã¡ããã© $2^9$ éãååšãããã®ã¯ããã€ãããŸããïŒããã§, $2$ ã€ã®æ£æŽæ°ã®é åºã¯èæ
®ããŸãã. |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/2748 | E | OMC086(E) | 300 | 138 | 186 | [
{
"content": "ã$DE$ äžã« $AF \\parallel BC$ ãªãç¹ $F$ ããšãã°, $\\triangle FAD$ ãš $\\triangle AED$ ã¯çžäŒŒã§ãããã,\r\n$$AE=FE=22-\\dfrac{20^2}{22}=\\dfrac{42}{11}$$\r\nãšæ±ããã, 解çãã¹ãå€ã¯ $\\bf{53}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc086/editorial/2748"
},
{
"content": "ãäžè§é¢æ°... | ã$\angle A = \angle B = \angle C\lt 90^\circ$ ãªãåžåè§åœ¢ $ABCD$ ã«ãããŠ, çŽç· $AB$ ãšçŽç· $CD$ ã®äº€ç¹ã $E$ ãšããŸã.
$$AD=20, \quad DE=22$$
ã®ãšã, $AE$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle\frac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC086 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc086/tasks/1449 | F | OMC086(F) | 400 | 45 | 105 | [
{
"content": "ãä»»æã®å®æ° $a$ ã«å¯Ÿã㊠$ a-1\\lt\\\\left\\lfloor a\\right\\rfloor \\leq a $ ãæãç«ã€ããšãã,\r\n$$x^3+7x^2+4x-12\\lt\\left\\lfloor x^3\\right\\rfloor +7\\left\\lfloor x^2\\right\\rfloor +4\\left\\lfloor x\\right\\rfloor\\leq x^3+7x^2+4x$$\r\nãæç«ãã. ãããã, $x$ ã¯ä»¥äžã®ç¯å²ã«å«ãŸããããšãå¿
èŠã§ããïŒ\r\n$$\\Big[-6,-2-2\\sqrt{3}\\Bi... | ãå®æ° $x$ ã«ã€ããŠã®æ¹çšåŒ
$$ \left\lfloor x^3\right\rfloor +7\left\lfloor x^2\right\rfloor +4\left\lfloor x\right\rfloor =12$$
ã® $x\geq -5$ ã«ããã解ã¯, å®æ° $a\lt b\lt c\lt d\lt e$ ã«ãã£ãŠ
$$\begin{aligned}
a \leq x\lt b,&& x=c,&& d\leq x\lt e
\end{aligned}$$
ãšè¡šãããŸã. $a^6+b^6+c^6+d^6+e^6$ ã解çããŠãã ãã.\
ããã ã, å®æ° $x$ ã«å¯Ÿã, $\left\lfl... |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2384 | A | OMC085(A) | 100 | 290 | 299 | [
{
"content": "ãåºãç®ã®ãã¡, ã¡ããã©äžã€ãçŽ æ°ã§æ®ãã $1$ ã§ããã°ãã. ãµã€ã³ãã®ç®ã§çŽ æ°ã§ãããã®ã¯ $2,3,5$ ã§ãããã,\r\n$$\\frac{3}{6}Ã\\Bigl(\\frac{1}{6}\\Bigr)^2Ã3=\\frac{1}{24}$$\r\nãæ±ãã確çã§ãã, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{25}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc085/editorial/2384"
}
] | ãã©ã®ç®ãç確çã§åºããããª, äžè¬çãªå
é¢äœã®ãµã€ã³ãã $3$ ã€åæã«æ¯ããŸã. ãã®ãšã, åºãç®ã®ç©ãçŽ æ°ãšãªã確çã¯, äºãã«çŽ ãªæ£æŽæ° $x,y$ ãçšã㊠$\dfrac{x}{y}$ ãšè¡šããã®ã§, $x+y$ ã解çããŠãã ãã. |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2371 | B | OMC085(B) | 200 | 207 | 263 | [
{
"content": "ãäžè¬ã«åž $n$ è§åœ¢ã«å¯ŸããŠç·å $A_n$ ãèãã. ãã㯠$2\\leq b-a\\leq n-2$ ã〠$b\\leq n$ ãªãæ£æŽæ°ã®çµ $(a,b)$ ã«ã€ã㊠$ab$ ã足ãåããããã®ã§ãã. ãããã£ãŠ, 以äžã®æç«ã確ããããã.\r\n$$A_n=\\frac{1}{2}\\Bigl((1+\\cdots+n)^2-(1^2+\\cdots+n^2)\\Bigr) -(1\\times 2+\\cdots+(n-1)\\times n+n\\times 1)$$\r\nããã§\r\n$$1\\times 2+\\cdots+(n-1)\\times n+n... | ãåž $100$ è§åœ¢ã®åé ç¹ã«, æèšåãã« $1$ ãã $100$ ãŸã§ã®æŽæ°ãæ¯ãããŠããŸã. ãã¹ãŠã®å¯Ÿè§ç·ã«å¯Ÿã, ãã®äž¡ç«¯ã® $2$ æ°ã®ç©ãèšç®ã, ãããã®ç·åãæ±ããŠãã ãã. ãã ã, 察è§ç·ã«ã¯èŸºãå«ãŸãªãããšãšããŸã. |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2388 | C | OMC085(C) | 200 | 214 | 249 | [
{
"content": "ã$S_n=\\dfrac{n(n+1)}{100}$ ãšãã. $n\\leq 50$ ã®ãšã $S_{n+1}-S_{n}\\leq 1$ ãã, $S_n$ ã®æŽæ°éšå㯠$S_{50}=25.5$ 以äžã®éè² æŽæ°å€ããã¹ãŠãšã. éã« $n\\geq 51$ ã®ãšã $S_{n+1}-S_{n}\\gt 1$ ãã, $S_{51},S_{52},\\cdots,S_{10000}$ ã®æŽæ°éšåã¯çžç°ãªã.\\\r\nã以äžãã, æ±ããåæ°ã¯ $26+9950=\\textbf{9976}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https... | ã$10000$ 以äžã®æ£æŽæ° $n$ ã«å¯Ÿã, $\dfrac{n(n+1)}{100}$ ã®æŽæ°éšåãšããŠããåŸããã®ã¯ããã€ãããŸããïŒ |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2051 | D | OMC085(D) | 300 | 163 | 225 | [
{
"content": "ãçžå ã»çžä¹å¹³åã®é¢ä¿ãã,\r\n$$x+\\left(2y^2+\\frac{1}{8}\\right)+\\left(3z^3+\\frac{1}{9}+\\frac{1}{9}\\right)\\geq x+2\\sqrt{\\frac{y^2}{4}}+3\\sqrt[3]{\\frac{z^3}{27}}\\geq x+y+z=1$$\r\néã« $(x,y,z)=\\left(\\dfrac{5}{12},\\dfrac{1}{4},\\dfrac{1}{3}\\right)$ ã§çå·ãæç«ãããã, 解çãã¹ãå€ã¯ $47+72=\\textbf{119}$ ã§ãã.",
... | ãæ£ã®å®æ° $x,y,z$ ã $x+y+z=1$ ãã¿ãããšã, $x+2y^2+3z^3$ ã®ãšãåŸãæå°å€ãæ±ããŠãã ãã. ãã ã, æ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2383 | E | OMC085(E) | 400 | 86 | 160 | [
{
"content": "ãããããç·å $BC,BE$ äžã«ç¹ $F,G$ ã, $DBF$ ããã³ $GBC$ ãæ£äžè§åœ¢ãšãªãããã«ãšããš, $DFC$ ãš $EGC$ ã¯çžäŒŒã§ãã, ç¹ã«ä»¥äžãæãç«ã€ïŒ\r\n$$BC:FC=GC:FC=EC:DC=10:DC$$\r\nãŸã, $D$ ãã $BF$ ã«ããããåç·ã®è¶³ã $H$ ãšãããš, \r\n$$(7-DC):7=AD:AC=BH:BC=(BC-FC):2BC$$\r\nããããé£ç«ãããããšã§ $DC=70\\/13$ ãåŸããããã, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{83}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
... | ãè§ $B$ ãçŽè§ã§ããäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $AC$ äžã®ç¹ $D$ ããã³çŽç· $BD$ ã® $D$ åŽã®å»¶é·ç·äžã®ç¹ $E$ ã以äžã®æ¡ä»¶ãã¿ãããŠããŸãïŒ
$$AC=7,\quad CE=10,\quad \angle DBC=\angle ECD=60^\circ$$
ãã®ãšã, $CD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $x,y$ ã«ãã£ãŠ $\dfrac{x}{y}$ ãšè¡šãããã®ã§, $x+y$ ã解çããŠãã ãã. |
OMC085 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc085/tasks/2593 | F | OMC085(F) | 500 | 17 | 86 | [
{
"content": "ãäžæ¹çšåŒã® $3$ 解ã $0\\lt x\\lt y\\lt z$ ãšããã°, 解ãšä¿æ°ã®é¢ä¿ãã\r\n$$x+y+z=a,\\quad xy+yz+zx=\\dfrac{a^2}{4}$$\r\nãããã $a$ ãæ¶å»ããã°, $x,y,z$ ã®å€§å°é¢ä¿ãã\r\n$$x^2+y^2+z^2-2xy-2yz-2zx=0 \\implies z=(\\sqrt{x}+\\sqrt{y})^2=x+y+2\\sqrt{xy}$$\r\nãããŸ, $\\gcd(x,y)=g$ ãšããã°, $x=gs^2,y=gt^2$ ãšãã, $z=g(s+t)^2$ ã§ãã. ãããã£ãŠ, $x,... | ã以äžã®æ¡ä»¶ãã¿ããæ£æŽæ°ã®çµ $(a,b)$ ã®ãã¡, $a\leq 300$ ãªããã®ã¯ããã€ãããŸããïŒ
- $t$ ã«ã€ããŠã®æ¹çšåŒ $t(2t-a)^2=b$ ãçžç°ãªã $3$ ã€ã®æ£æŽæ°è§£ããã¡, ãããã®æ倧å
¬çŽæ°ã¯å¹³æ¹æ°ã§ãã. |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/2812 | A | OMC084(A) | 100 | 325 | 327 | [
{
"content": "ãåè§åœ¢ $ABCD$ ãšååãªåè§åœ¢ãå³ã®ããã« $4$ ã€çµã¿åããããšæ£æ¹åœ¢ãã§ãïŒç¹ã« $C$ ã¯ãã®æ£æ¹åœ¢ã®äžå¿ãšãªãïŒãããã $\\angle CAD=45^\\circ$ ã§ããããïŒæ±ããè§ã®å€§ãã㯠$90^\\circ-45^\\circ=\\bf{45}^\\circ$ ãšãããïŒ\\\r\nããªã $A,B,C,D$ ã¯å
±åã§ãã $BC=CD$ ã§ããããšããã $\\angle CAD=45^\\circ$ ã¯åŸãïŒ\r\n",
... | ãåžåè§åœ¢ $ABCD$ ã¯æ¬¡ãã¿ãããŸãïŒ
$$BC=CD,\quad \angle A=\angle C=90^\circ$$
ç·å $AC$ ã®åçŽäºçåç·ãšçŽç· $AD$ ãæãè§ã®å€§ããã床æ°æ³ã§æ±ããŠãã ããïŒ |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/2528 | B | OMC084(B) | 200 | 327 | 331 | [
{
"content": "ãäž $4$ æ¡ã®æ°ãå®ããããšã§, äžæã«æ¡ä»¶ãã¿ããåææ°ãæ§æã§ããããšã«æ³šæãã. ããããã®äœãšãªãåŸãæ°ã¯, äžã®äœã§ã¯ $1, 2$ ã® $2$ éã, ãã以å€ã®äœã§ã¯ $0, 1, 2$ ã® $3$ éãããããšãã, 解çãã¹ãå€ã¯ $2\\times 3^3=\\mathbf{54}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc084/editorial/2528"
}
] | ã$20211202$ ã®ããã«, åäœã®æ°ã $0, 1, 2$ ã®ããããã§ãã $8$ æ¡ã®åææ°ã¯ããã€ãããŸããïŒ\
ããã ã, æ£æŽæ°ã**åææ°**ã§ãããšã¯, äžã®äœã $0$ ã§ãªã, äžã®äœããéé ã«èªãã å Žåã§ãå
ã®æ°ãšäžèŽããããšãæããŸã. ãŸã, æ¡æ°ãèããå Žåã«, å
é ã®äœã¯ $0$ ã§ãªããã®ãšããŸã. |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/1955 | C | OMC084(C) | 200 | 280 | 317 | [
{
"content": "ãç¹ã« $20301+n$ ã $n$ ã§å²ãåããããšãã, $n$ 㯠$20301=3\\times 67\\times101$ ã®çŽæ°ã§ãã. ãŸãæ¡ä»¶ãã\r\n$$20301+n\\geq n^2 \\implies n\\leq 142$$\r\nããããããåŸã $n$ ã調ã¹ãã°, $n=1,3,101$ ãé©ãããã, æ±ããå€ã¯ $1+3+101=\\bold{105}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc084/editorial/1955... | ã$20301+n$ ã $n^2$ ã®åæ°ã§ãããããªæ£æŽæ° $n$ ã®ç·åãæ±ããŠãã ãã. |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/294 | D | OMC084(D) | 300 | 142 | 168 | [
{
"content": "ãå $C_1,C_2$ ã®æ¹çšåŒã¯ãããã\r\n$$(x-a^2)^2+(y-b^2)^2=a^3b,\\quad (x-b^2)^2+(y-a^2)^2=ab^3$$\r\nãããã, çŽç· $AB$ (æ ¹è»ž) ã®æ¹çšåŒã¯\r\n$$((x-a^2)^2+(y-b^2)^2)-((x-b^2)^2+(y-a^2)^2)=a^3b-ab^3$$\r\nãããã $a^2-b^2$ ãæ¬ãã ãããšã§ $y-x=ab\\/2$ ãåŸã.\\\r\nãäžæ¹ã§çŽç· $PQ$ ã®æ¹çšåŒã¯ $x+y=a^2+b^2$ ã§ããããšãã,\r\n$$ab=\\dfrac{21}{8},\\quad a... | ã$a\gt b$ ãªãæ£ã®å®æ°ã«ã€ããŠ, çŽäº€åº§æšå¹³é¢å
ã«ç¹ $P:(a^2,b^2)$ ããã³ $Q:(b^2,a^2)$ ããããŸã. ãŸã $P$ ãäžå¿ãšããååŸ $a\sqrt{ab}$ ã®åã $C_1$, $Q$ ãäžå¿ãšããååŸ $b\sqrt{ab}$ ã®åã $C_2$ ãšããŸã.\
ãããŸ, $C_1$ ãš $C_2$ ã¯çžç°ãªã $2$ ç¹ $A,B$ ã§äº€ãã£ãŠãã, çŽç· $AB$ ãšçŽç· $PQ$ ã®äº€ç¹ã¯ $\left(2,\dfrac{53}{16}\right)$ ã§ãã. ãã®ãšã, $a,b$ ã¯ããããæ¢çŽåæ°ãšã㊠$\dfrac{p}{q},\dfrac{r}{s}$ ãšè¡šãããã®ã§... |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/1932 | E | OMC084(E) | 300 | 149 | 203 | [
{
"content": "ãååšè§ã®å®çãã $ABCD$ ã¯åã«å
æ¥ãããã, $DP=x$ ãšããã°æ¹ã¹ãã®å®çãã $BP=24\\/x$ ã§ãã. ããã§è§ã®äºçåç·ãã $AB=BC$ ããã³ $AD:CD=AP:PC=3:8$ ã§ãããã, Ptolemyã®å®çãã\r\n$$80\\times\\left(1+\\dfrac{3}{8}\\right)=AB\\times CD+AD\\times BC=AC\\times BD=11\\times\\left(x+\\dfrac{24}{x}\\right)$$\r\nããã解ã㊠$x=4,6$ ãåŸã. äŸãã° $x=4$ ã®ãšã, $AB:CD... | ãåžåè§åœ¢ $ABCD$ ã«ãããŠ, 察è§ç·ã®äº€ç¹ã $P$ ãšããã°, 以äžã®æ¡ä»¶ãæç«ããŸããïŒ
$$\angle ADB=\angle BAC=\angle ACB,\quad AB\times CD=80,\quad AP=3,\quad PC=8$$
ãã®ãšã, $AB^2$ ãšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC084 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc084/tasks/2169 | F | OMC084(F) | 300 | 106 | 158 | [
{
"content": "ã$X$ ã®éšåéåã§ãã£ãŠ, ããèŠçŽ $a$ ãå«ã¿èŠçŽ ã $n$ åã§ãããã®ã¯ ${}\\_{15}\\mathrm{C}\\_{n-1}$ åãã. ãã£ãŠ, $a$ ã¯ç·åã«å¯ŸããŠ\r\n$$\\sum^{16}\\_{n=1} \\frac{a}{n} {}\\_{15}\\mathrm{C}\\_{n-1} = \\frac{a}{16} \\sum^{16}\\_{n=1} {}\\_{16}\\mathrm{C}\\_{n} = \\frac{a}{16}(2^{16}-1)$$\r\nã ãå¯äžãã. ãããã£ãŠ, æ±ããç·åã¯\r\n$$\\sum^{16}\\_{a... | ãæééå $A$ ã«å¯ŸããŠ, $S(A)$ ã $A$ ã®èŠçŽ ã®ç·å, $|A|$ ã $A$ ã®èŠçŽ ã®åæ°ãšããŸã. ãã®ãšã, éå
$$X=\\{1, 2, \cdots, 16 \\}$$
ã®ç©ºã§ãªãéšåéå $A$ ãã¹ãŠã«ã€ããŠ, $\dfrac{S(A)}{|A|}$ ã®ç·åãæ±ããŠãã ãã.\
ããã ã, æ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\displaystyle \frac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/2514 | A | OMC083(A) | 200 | 254 | 261 | [
{
"content": "ã$(x+1)^2+2(x+1)+6=x^2+4x+9$ ãã,\r\n$$\\prod_{x=1}^{500} \\frac{x^2+4x+9}{x^2+2x+6} = \\frac{500^2+4\\times 500 + 9}{1^2 + 2 \\times 1 + 6} = \\bf{28001}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc083/editorial/2514"
}
] | ã$\displaystyle\prod_{x=1}^{500} \displaystyle\frac{x^2+4x+9}{x^2+2x+6}$ ãèšç®ããŠãã ãã. |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/2459 | B | OMC083(B) | 300 | 215 | 245 | [
{
"content": "ãæ¡ä»¶ã¯ä»¥äžã®ããã«èªã¿æ¿ããããããšããããïŒ\r\n- $f_2(X)$ 㯠$2$ ã§å²ã£ãŠ $1$ äœã, $3$ ãš $5$ ã§å²ã£ãŠ $2$ äœã.\r\n\r\nãããã£ãŠ, æ±ããæå°å€ã¯ $\\bf{17}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc083/editorial/2459"
}
] | ãæ£æŽæ° $m$, $2$ 以äžã®æŽæ° $n$ ã«ã€ããŠ, $f_n(m)$ ã $m$ ã $n$ ã§å²ãåããæ倧ã®åæ°ãšããŠå®çŸ©ããŸã.\
ã以äžãã¿ããæ£æŽæ° $X$ ã«ã€ããŠ, $f_2(X)$ ãšããŠããåŸãæå°å€ã解çããŠãã ãã.
$$f_2\left(\frac{X}{4^{f_4(X)}}\right)=1,\quad f_2\left(\frac{X}{8^{f_8(X)}}\right)=f_2\left(\frac{X}{32^{f_{32}(X)}}\right)=2$$ |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/2692 | C | OMC083(C) | 400 | 160 | 208 | [
{
"content": "**解æ³1.** $F$ ã $BC$ ã®äžç¹ãšã, $G$ ã $F$ ãã $AB$ ã«äžãããåç·ã®è¶³ãšãã. ãŸã $FG$ ãš $AD$ ã®äº€ç¹ã $H$ ãšãããš,\r\n$$\\angle CAE =\\angle FAH,\\quad\\angle EAF = \\angle HAG,\\quad\\angle AFC = \\angle AGF = 90^\\circ$$\r\nãã $GH:HF = FE:EC = 7:32$ ã§ãã, Menelausã®å®çãã $AG:GB = 7:9$ ãåãã. \\\r\nã$\\triangle ABF\\sim \\tr... | ã$AB=AC$ ãªãäžè§åœ¢ $ABC$ ã®èŸº $BC$ äžã« $B$ ã«è¿ãé ã«ç¹ $D, E$ ããšããš, 以äžãæç«ããŸãã.
$$BD=13,ãDE=33,ãEC=32,ã\angle DAE = \frac12 \angle BAC$$
ãã®ãšã, $AB$ ã®é·ããæ±ããŠãã ãã. |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/2770 | D | OMC083(D) | 500 | 33 | 78 | [
{
"content": "ã$c=1,2,\\ldots,P-1$ ãåºå®ãããšã, **[OMC033(B)](https:\\/\\/onlinemathcontest.com\\/contests\\/omc033\\/editorial\\/240)** ã®èŠé 㧠$p_1=p_2$ ã§ãã. ãã£ãŠ, ä»¥äž $c=P$ ã®å Žåãèãã. $p_2$ ã«ã€ããŠãåæ§ã«, $1\\/P$ ã§ãã. $p_1$ ã«ã€ããŠ, $(P-1)! \\equiv -1 \\pmod P$ ãã $(P-1)!$ 以äžã®æ£æŽæ°ã«\r\n- $x \\equiv 1,2,\\cdots,P-1 \\pmod P$ ãªã ... | ãçŽ æ° $P=2^{82589933}-1$ ã«ã€ããŠ, 以äžã®åè©Šè¡ã§ $a+b$ ã $c$ ã®åæ°ãšãªã確çããããã $p_1,p_2$ ãšããŸã.
- $(P-1)!$ 以äžã®æ£æŽæ°ããç確çã« $a,b$ ã, $P$ 以äžã®æ£æŽæ°ããç確çã« $c$ ãéžãã ãšã.
- $(P-1)!$ 以äžã®æ£æŽæ°ããç確çã« $a$ ã, $P$ 以äžã®æ£æŽæ°ããç確çã« $b,c$ ãéžãã ãšã.
ãã ã, åãç¯å²ããäºã€ã®æ£æŽæ°ãéžã¶ãšãã¯, äžã€ãã€é ã«ç¬ç«ã«éžã¶ããšãšããŸã. \
ããã®ãšã, $|p_1-p_2|$ ã®éæ°ã¯æ£æŽæ° $n$ ã«ãªããŸã. $n$ ã $2$ ã§å²ãåããæ倧åæ°ã $M$ ... |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/3123 | E | OMC083(E) | 500 | 77 | 144 | [
{
"content": "ãäžè¬ã« $a_0, a_1, \\ldots, a_n$ ã«ã€ããŠïŒä»¥äžã® $2^n$ çš®é¡ã®å€ã®å¹³åãèããïŒ\r\n$$(a_0 \\pm a_1 \\pm \\cdots \\pm a_n)^2$$\r\nçžç°ãªã $i,j$ ã«ã€ããŠïŒå±éããæã® $a_ia_j$ ã®ä¿æ°ã $2$ ã§ãããããªç¬Šå·ã®éžã³æ¹ãšïŒ$-2$ ã§ãããããªç¬Šå·ã®éžã³æ¹ãåæ°ãã€ååšããããšããïŒãã®å¹³å㯠$a_0,a_1, \\ldots, a_n$ ã® $2$ ä¹ã®ç·åã«çããïŒ\\\r\nãæ¬å㯠$n=3123$ ãã€\r\n$$a_0=\\overbrace{66\\cdots6}^{næ¡}... | ã$3123$ æ¡ã®æ£æŽæ°ã§ãã£ãŠïŒã©ã®æ¡ã®æ°åã $4$ ãŸã㯠$8$ ã§ãããããªãã®ãã¹ãŠã«ã€ããŠïŒããããã® $2$ ä¹ã®ïŒçžå ïŒå¹³åã¯æ£æŽæ° $M$ ã«ãªãããšã蚌æã§ããŸãïŒ$M$ ã®åæ¡ã®åã解çããŠãã ããïŒ |
OMC083 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc083/tasks/2992 | F | OMC083(F) | 600 | 21 | 70 | [
{
"content": "ã以äžã $4$ é ç¹ãšããæ£æ¹åœ¢ã«å¹³è¡ç§»åã®ã¿ã§äžèŽãããããæ Œåæ£æ¹åœ¢ã $S(x,y)$ ã§è¡šãïŒ\r\n$$(x,0),ã(0,y),ã(x+y,x),ã(y,x+y)$$\r\nãã ãïŒ$x\\geq0,y\\gt 0$ ãšããïŒãŸãïŒæ Œåæ£æ¹åœ¢ã®**ã¬ãã«**ã以äžã§å®çŸ©ããïŒ\r\n\r\n- $x=0$ ã®ãšãïŒ$S(x,y)$ ã®ã¬ãã«ã $y$ ã§å®ããïŒ\r\n- $x\\neq 0$ ã®ãšãïŒ$S(x,y)$ ã®ã¬ãã«ã $x+y-1$ ã§å®ããïŒ\r\n\r\nããã®ãšãïŒã¬ãã« $L$ ã®æ Œåæ£æ¹åœ¢ã«å«ãŸããæ Œåæ£æ¹åœ¢ã®ã¬ãã«ã¯ $L$ æªæºã§ããããšã容æã«ç¢º... | ã$N$ ãæ£æŽæ°ãšããŸã. $xy$ 座æšå¹³é¢äžã«æ£æ¹åœ¢ $S_1, S_2, \ldots, S_n$ ããã, 以äžã®æ¡ä»¶ãæºãããŠããŸã.
- å
šãŠã®æ£æ¹åœ¢ã®é ç¹ã¯æ Œåç¹ã§ãã.
- $S_1$ 㯠$(0,0), (0,N), (N,0), (N,N)$ ãé ç¹ãšãã.
- ä»»æã® $1\leq k \leq n-1$ ã«ã€ããŠ, $S_{k+1}$ 㯠$S_k$ ã®èŸºäžãå«ãå
éšã«ãã, ãã€äžèŽããªã.
ãã®ãã㪠$(S_1, S_2, \ldots, S_n)$ ã®ãã¡, $n$ ãæ倧å€ããšããã®ã®åæ°ã $M_N$ ã§è¡šããŸã.\
ã$M_N$ ã $2$ ã§å²ãåããåæ°ãã¡ããã© $1234... |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/2112 | A | OMC082(A) | 100 | 253 | 255 | [
{
"content": "ãååãéããŠ, çéã $1$ åšããéã«é·éã $12$ åšãããã, ããã㯠$11$ åéãªã. ãã®ãã¡ $1$ å㯠$0$ æã¡ããã©ã§ããããšã«çæããã°, æ±ããåæ°ã¯ $\\textbf{10}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc082/editorial/2112"
}
] | ãäžè¬çãªã¢ããã°æèšã«ãããŠ, åå $0$ æ $1$ åããåå $11$ æ $59$ åãŸã§ã«, é·éãšçéãéãªãã®ã¯äœåã§ããïŒ |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/1981 | B | OMC082(B) | 200 | 206 | 251 | [
{
"content": "ãäžè¬ã«, æ£æŽæ° $n$ ã $n=p_1^{e_1}\\times p_2^{e_2}\\times \\ldots p_m^{e_m}$ ãšçŽ å æ°å解ããããšã, $n$ ãšäºãã«çŽ 㪠$1$ ä»¥äž $n$ 以äžã®æŽæ°ã®åæ° $\\phi(n)$ ã«ã€ããŠ\r\n$$\\phi(n)=n\\times \\left(1-\\frac{1}{p_1}\\right)\\times \\left(1-\\frac{1}{p_2}\\right)\\times \\ldots \\times \\left(1-\\frac{1}{p_m}\\right)=n\\prod _{k=1}^m ... | ã$420$ ãšäºãã«çŽ 㪠$420$ 以äžã®æ£æŽæ°ã¯ããã€ãããŸããïŒ |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/286 | C | OMC082(C) | 200 | 167 | 195 | [
{
"content": "$$2S=AB\\times BC\\times \\sin\\angle B+CD\\times DA\\times \\sin\\angle D\\leq AB\\times BC+CD\\times DA$$\r\nã§ãã. éã« $AB^2+BC^2=CD^2+DA^2$ ããçå·ãæç«ããããããã, æ±ããæ倧å€ã¯ \r\n$$\\dfrac{AB\\times BC+CD\\times DA}{2}=\\dfrac{5\\sqrt{19}+2\\sqrt{589}}{2}$$\r\nããªãã¡è§£çãã¹ãå€ã¯ $475+2356=\\textbf{2831}$ ã§ãã.",
... | ã以äžã®æ¡ä»¶ãã¿ããåžåè§åœ¢ $ABCD$ ã«ã€ããŠ, ãã®é¢ç©ã $S$ ãšããŸã.
$$AB=\sqrt{5},\ \ BC=\sqrt{95},\ \ CD=\sqrt{38},\ \ DA=\sqrt{62}$$
ãã®ãšã, $S$ ãšããŠããåŸãæ倧å€ãæ±ããŠãã ãã. ãã ã, çãã¯æ£æŽæ° $p\lt q$ ã«ãã£ãŠ $\dfrac{\sqrt{p}+\sqrt{q}}{2}$ ãšäžæã«è¡šããã®ã§, $p+q$ ã解çããŠãã ãã. |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/2582 | D | OMC082(D) | 300 | 152 | 188 | [
{
"content": "ãæããã« $S_2=2$ ã§ãããã, ä»¥äž $n\\geq 3$ ãšãã.\\\r\nããŸã $1$ ãæžã蟌ãå Žæã®æ±ºãæ¹ã¯ $n$ éããã. $1$ ã®äž¡é£ã㯠$2,3$ ãšãªããã, ãããã®æ±ºãæ¹ã $2$ éããã. ãã®ãšã, $k=2,3,\\ldots,n-2$ ã«ã€ããŠé çªã«, $k$ ã®é£ã«ã¯ $k+2$ ãæžã蟌ãã»ããªããªã, å
šäœãäžæã«å®ãŸã.\\\r\nã以äžãã, æ±ããå€ã¯ $2+2\\times3+2\\times4+\\cdots+2\\times2022=\\textbf{4090502}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬"... | ãååšäžã« $n$ åã®ç¹ããã, ããããã« $1$ ãã $n$ ãŸã§ã®æ°åãã¡ããã©äžåºŠãã€, 以äžãã¿ããããã«æžã蟌ã¿ãŸã.
- é£ãåã $2$ ç¹ã«ãããŠ, æžã蟌ãŸãã $2$ æ°ã®å·®ïŒã®çµ¶å¯Ÿå€ïŒã¯åžžã« $2$ 以äžã§ãã.
ãã®ãããªæ¹æ³ã $S_n$ éãã§ãããšãããšã, $S_{2}+S_{3}+\cdots+S_{2022}$ ãæ±ããŠãã ãã.\
ããã ã, å転ãå転ã«ãã£ãŠäžèŽãããã®ãåºå¥ããŠèãããã®ãšããŸã. |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/2214 | E | OMC082(E) | 300 | 108 | 155 | [
{
"content": "ã$1$ ãã $2021$ ã®æå°å
¬åæ°ã $L$ ãšããã°, ãã¹ãŠã®ç¹ãåããŠåæã« $A$ ãžæ»ã£ãŠããã®ã¯ $2L$ ç§ã§ãã, ãããŸã§ã«åé¡ã®ç¶æ³ãèµ·ãããšããã° $L$ ç§åŸã§ãã. ãã®ãšã, æ¡ä»¶ã¯ $L\\/x$ ãå¥æ°ã§ãããšèšãæããã, ãããã $2021$ 以äžã§ $2$ ã§å²ãåããåæ°ãæãå€ã $x=\\textbf{1024}$ ãé©ããå¯äžã®ãã®ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc082/editorial/2214"
... | ã$2021$ åã®ç¹ $P_1,P_2,\cdots,P_{2021}$ ã, åæã«ç¹ $A$ ããåºçºã, ç¹ $B$ ãšã®éãåŸåŸ©ãç¶ããŸã. ç¹ $P_n$ ãçéãé²ãã®ã«ãããæé㯠$n$ ç§ã§ã. ãã®ãšã, ããäžã€ã®ç¹ $P_x$ ã®ã¿ãç¹ $B$ ã«ãã, æ®ãã®ãã¹ãŠã®ç¹ãç¹ $A$ ã«ãããããªç¬éãååšããŸãã. $x$ ãšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC082 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc082/tasks/1475 | F | OMC082(F) | 400 | 71 | 123 | [
{
"content": "ãäžè¬ã« $OG:GH=1:2$ ã§ããããšãã $OG=GI$ ãæç«ãã. $BC$ ã®äžç¹ $M$ ã«ã€ã㊠$BM=MC=1$ ãšããŠãã, $AB=AC=a(\\neq 2),AM=h$ ãšãããšä»¥äžã®æç«ããããïŒ\r\n$$AG=\\frac{2}{3}h,\\quad AI=h - \\frac{2|\\triangle ABC|}{AB + BC + CA} = h - \\frac{2h}{2a+2} = \\frac{ah}{a+1} = \\frac{a(a-1)}{h}$$\r\nãŸã, $AO = BO$ ã«æ°ãã€ããã°\r\n$$AO^2 = BO^2 = (... | ã$AB=AC\neq BC$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, ãã®å€å¿ $O$, éå¿ $G$, å
å¿ $I$, åå¿ $H$ ããã®é ã«çééã«äžŠãã§ãããšã, ãã®å€æ¥åã®ååŸãšå
æ¥åã®ååŸã®æ¯ã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ãçšã㊠$p:q$ ãšè¡šãããŸã. $p+q$ ã解çããŠãã ãã. |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/2499 | A | OMC081(A) | 300 | 181 | 198 | [
{
"content": "ãå€é¢äœã®é ç¹ã®åæ°ã $v$ïŒèŸºã®æ¬æ°ã $e$ïŒé¢ã®åæ°ã $f$ ãšè¡šãããšã«ããïŒä»»æã®å€é¢äœã«å¯ŸããŠïŒé ç¹å
šäœãåäžã®å¹³é¢äžã«ãªãããšãã $v \\geq 4$ïŒå€é¢äœã®å
éšãéãã空éã§ããããšãã $f \\geq 4$ ãåŸãïŒ \r\nã$f = 4$ ãŸã㯠$v = 4$ ã®ãšãïŒã©ã¡ããåé¢äœã«ãªãïŒ$e = 6$ïŒãŸã $f \\geq 5$ ã〠$v \\geq 5$ ã®ãšãïŒEuler ã®å€é¢äœå®çãã $e \\geq 8$ ã§ããïŒããã§åè§éïŒäºè§éã¯ãããã $e = 8, 10$ ã§ããããšã«çæããïŒ \r\nãããã§ïŒåžå€é¢äœã«ãã㊠$... | ãåžå€é¢äœã®èŸºã®æ¬æ°ã«**ãªãåŸãªã**æ£æŽæ°ã®**ç·ç©**ãæ±ããŠãã ããïŒãã ãïŒãã¹ãŠã®é ç¹ãåäžã®å¹³é¢ã«ãããã®ã¯å€é¢äœãšã¯ã¿ãªããïŒãŸãå€é¢äœãåžã§ãããšã¯ä»¥äžã®æ¡ä»¶ãã¿ããããšãæããŸãïŒ
- ãã¹ãŠã®èŸºã«ãããŠ, ããã蟺ã«ã〠$2$ é¢ã®ãªãè§ã $180^\circ$ æªæºã§ãã
- èªå·±äº€å·®ããããªã |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/1265 | B | OMC081(B) | 500 | 136 | 170 | [
{
"content": "ã$n=1$ ã¯é©ããäžæ¹ã§, $n=2,3$ ã¯é©ããªãããšã容æã«ããã. ä»¥äž $n\\geq 4$ ã§èãã.\\\r\nã$n$ ãçŽ æ°ã§ãããšã, Wilsonã®å®çãã $(n-2)!-1$ 㯠$n$ ã§å²ãåãããã, ç¹ã«æ¡ä»¶ãã¿ãã. \\\r\nããã㧠$n\\leq 120$ ãçŽ æ°ã§ãªããšã, ãã㯠$2,3,5,7$ ã®å°ãªããšãäžã€ã§å²ãåãã. äžæ¹ã§ $m\\geq 6$ ã«ã€ã㊠$m!-1$ 㯠$2,3,5,7$ ã®ãããã§ãå²ãåããªãããšãã, 以äžã® $4$ æ°ã«ã€ããŠèŠãã°ååã§ãã.\r\n$$2!-1=1,\\quad3!-1=5,... | ã以äžã®æ¡ä»¶ãã¿ããæ£æŽæ° $n$ ã®ãã¡, $10,20,30$ çªç®ã«å°ãããã®ã®**ç©**ãæ±ããŠäžããïŒ
- ããæŽæ° $m\ge2$ ãååšããŠ, $m!-1$ ã $n$ ã®åæ°ã§ãã. |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/2530 | C | OMC081(C) | 500 | 56 | 124 | [
{
"content": "ãçœãç¢ç³ã $1$ ã«ïŒé»ãç¢ç³ã $0$ ã«å¯Ÿå¿ããïŒååŽã $1$ ã®äœãšãªãããã« $2$ é²æ³è¡šç€ºããéè² æŽæ°ã«å¯Ÿå¿ãããããšãèããïŒç¢ç³ã®è²ãšéè² æŽæ°ã¯äžå¯Ÿäžã«å¯Ÿå¿ããïŒãã®ãšãïŒ**æäœ**ã¯æäœåã®æ£æŽæ°ãè¶
ããªããã㪠$2$ ã¹ããæžããããšã«å¯Ÿå¿ããïŒããã $0$ ã«ããæ¹ãè² ãã§ããïŒå¯Ÿå¿ããéè² æŽæ°ã¯å¿
ãæžå°ããããïŒç¹ã«åæãå¿
ã決ããïŒ\\\r\nã$n=1,\\ldots,32767$ ã«å¯Ÿãé¢æ° $f(n)$ ãïŒæåã®æ£æŽæ°ã $n$ ã§å
æ»ãåã€ãªãã° $1$ïŒåŸæ»ãåã€ãªãã° $0$ ãšããŠå®ãããšïŒ$f(1) = 0$ ããã³ $f(2)=f(... | ãååäžåã« $15$ åã®ç¢ç³ã䞊ãã§ããŸãïŒããããã®ç¢ç³ã¯çœè²ãŸãã¯é»è²ã§ããïŒãã¹ãŠé»è²ã§ã¯ãããŸããïŒ\
ã$2$ 人ã®ãã¬ã€ã€ãŒãããããçšããŠã²ãŒã ãè¡ããŸãïŒå
æ»ããå§ããŠïŒä»¥äžã®**æäœ**ã亀äºã«è¡ããŸãïŒ
- **æäœ**ïŒ$0$ å以äžã®é»ãç¢ç³ãš $1$ åã®çœã®ç¢ç³ãïŒ**åããé ã«**é£ç¶ããŠãé»é»$\ldots$é»çœãã®ããã«çœ®ãããŠããç®æãäžã€éžã³ïŒããã«å«ãŸããç¢ç³ã®è²ããã¹ãŠå
¥ãæ¿ããïŒ
ãäŸãã°ïŒ$4$ åã®ç¢ç³ãåããé ã«ãçœé»é»çœããšäžŠãã§ãããšãïŒäžåºŠã®æäœã«ãã£ãŠãé»é»é»çœããçœé»é»é»ããçœé»çœé»ããçœçœçœé»ãã®ããããã«å€åããŸãïŒå
ã«ãã¹ãŠã®ç¢ç³ãé»è²ã«ãã人ã**... |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/2500 | D | OMC081(D) | 600 | 24 | 83 | [
{
"content": "ãäžè¬ã« $10000$ ã $N$ ãšããïŒå $r = 0, \\ldots, N$ ã«å¯ŸãïŒ$M \\ge r$ ãšãªã $A,B$ ã®äžŠã¹æ¿ãã®ç·æ°ã¯ïŒã«ã¿ã©ã³æ°ãšåæ§ã®é¡åæ³ã«ãã£ãŠ ${}\\_{2N}\\mathrm C\\_{N-r}$ ãšåããïŒãã£ãŠæ±ããç·å $S$ ã¯\r\n$$ S = \\sum\\_{r=0}^{N-1} r^2 \\left( {}\\_{2N}\\mathrm C\\_{N-r} - {}\\_{2N}\\mathrm C\\_{N-(r+1)} \\right) + N^2 {}\\_{2N}\\mathrm C\\_0 = \\sum\... | ãé»æ¿ã« $0$ ãäžã€ã ãæžãããŠããŸãïŒãããžä»¥äžã®æäœ $A,B$ ããããã $10000$ åãã€é©åœãªé åºã§æœããŸãïŒ
- æäœ $A$ïŒé»æ¿ã«æžãããŠããæ°ã $n$ ã®ãšãïŒããã $n+1$ ã«æžãæããïŒ
- æäœ $B$ïŒé»æ¿ã«æžãããŠããæ°ã $n$ ã®ãšãïŒããã $n-1$ ã«æžãæããïŒ
ãã®éçšã§é»æ¿ã«æžããããã¹ãŠã®æ°ã«ã€ããŠïŒãã®æ倧å€ã $M$ ãšããŸãïŒ\
ãæäœ $A,B$ ãè¡ãé åºã¯å
šéšã§ ${}\_{20000}\mathrm C\_{10000}$ éããããŸããïŒãã¹ãŠã«ã€ã㊠$M^2$ ã®ç·åãæ±ãïŒããã $10000$ ã§å²ã£ãäœããæ±ããŠãã ããïŒãã ãïŒ... |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/2524 | E | OMC081(E) | 700 | 8 | 45 | [
{
"content": "ã$Q, T, U$ ãéãåã $\\Gamma^\\prime$ ãšãïŒçŽç· $IT, IU$ ã«ã€ããŠïŒ$\\Gamma^\\prime$ ãšã® $T,U$ ã§ã¯ãªã亀ç¹ããããã $T^\\prime, U^\\prime$ ãšãããšïŒ\r\n$$ \\angle PTU = \\angle DTU = \\angle DEU = \\angle QUU^\\prime = \\angle QT^\\prime U^\\prime. $$\r\nåæ§ã« $\\angle PUT = \\angle QU^\\prime T^\\prime$ ããïŒ$\\triangle PTU$ ... | ã$I$ ãå
å¿ãšããäžè§åœ¢ $ABC$ ã«ãããŠïŒå $\Gamma$ ã¯ç¹ $A$ ãéãïŒçŽç· $BI, CI$ ã«ããããç¹ $T, U$ ã§æ¥ããŸãïŒãã®ãšãç·å $AB, AC$ ã«ã€ããŠïŒãããã $\Gamma$ ãšã® $A$ ã§ãªãäº€ç¹ $D, E$ ãååšããŸããïŒããã«çŽç· $DT, EU$ ãç¹ $P$ ã§ïŒçŽç· $DU, ET$ ãç¹ $Q$ ã§äº€ãããŸããïŒããå®æ° $x$ ã«ã€ããŠ
$$ AD = 20,\qquad AE = 22,\qquad IP = IQ = x $$
ãšãªããšãïŒ$x^2$ ã¯å¹³æ¹å åãæããªãæ£æŽæ° $c$ ããã³æ£æŽæ° $a$, $b$ ãçšã㊠$a - b \sqr... |
OMC081 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc081/tasks/2518 | F | OMC081(F) | 700 | 6 | 24 | [
{
"content": "**è£é¡.**ã$k$ ãæ£æŽæ°ãšããïŒä»»æã® $0$ ã§ãªãè€çŽ æ° $e\\_1,\\ldots,e\\_k$ ããã³ $0$ ã§ãªãçžç°ãªãè€çŽ æ° $z\\_1,\\ldots, z\\_k$ ã«å¯ŸãïŒ$\\displaystyle S(x) := \\sum\\_{i=1}^k e\\_i{z\\_i}^x$ ã¯æ£æŽæ° $x$ ã«å¯ŸããŠæççã« $0$ ã«ãªãããšã¯ãªãïŒ\\\r\n**蚌æ.**ã$S(x+1) - z\\_k S(x)$ ãèããã°ïŒ$k$ ã«é¢ããåž°çŽæ³ã§ç€ºãããïŒ\r\n___\r\nãéè² æŽæ° $k$ ã«å¯ŸãïŒæ°å $\\left\\\\{ t\\_k \\... | ã$n\geq 3$ ãæŽæ°ãšããŸãïŒ$n$ 次以äžã®å®æ°ä¿æ°å€é
åŒ $P\_n(x)$ ã
$$P\_n(0) = P\_n(1) = 0,\quad P\_n(2) = 1$$
ããã³
$$ P\_n(k + 3) = P\_n(k + 2) + P\_n(k + 1) + P\_n(k) \qquad (k = 0, 1, \ldots , n - 3) $$
ãã¿ãããšãïŒå $n$ ã«å¯Ÿã㊠$P\_n(x)$ ãäžæã«å®ãŸããŸãïŒ\
ãããŸïŒæ°å $\left\\{ a\_n \right\\}\_{n=3,4,\ldots}$ ã $a\_n = P\_n(n + 1)$ ã§å®ããŸãïŒãã®ãšãïŒä»»æã® $n=... |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/2917 | A | OMC080(A) | 100 | 358 | 360 | [
{
"content": "ãé ç¹ã蚪ããé çªãèããã°è¯ã, æ¡ä»¶ãæºãããã®ã¯æ¬¡ã® $\\mathbf 4$ éãã®ã¿ã§ãã: \r\n- $A \\rightarrow B \\rightarrow C$\r\n- $A \\rightarrow D \\rightarrow C$\r\n- $A \\rightarrow B \\rightarrow D \\rightarrow C$\r\n- $A \\rightarrow D \\rightarrow B \\rightarrow C$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest... | ãæ£æ¹åœ¢ $ABCD$ ã«ç·å $BD$ ãåŒãããŠããŸã.\
ãç·ã®äžã®ã¿ããã©ã£ãŠ $A$ ãã $C$ ãžè¡ãæ¹æ³ã§ãã£ãŠ, åãå Žæãäºåéããªããã®ã¯ããã€ãããŸãã. |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/200 | B | OMC080(B) | 200 | 293 | 323 | [
{
"content": "ãæ¹ã¹ãã®å®çãã $PA^2=PB\\times PC$ ã§ãããã $PC=24$ ã§ãã. ãã㧠$BC$ ã®äžç¹ã $M$, $ABC$ ã®å€å¿ã $O$ ãšããã° $OAPM$ ã¯é·æ¹åœ¢ã§ãããã, æ±ããååŸã¯\r\n$$OA=PM=\\dfrac{PB+PC}{2}=\\textbf{15}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc080/editorial/200"
}
] | ãå€æ¥åã $\Omega$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, $A$ ã«ããã $\Omega$ ã®æ¥ç·ãš $BC$ ã®äº€ç¹ã $P$ ãšããŸã. 以äžã®æ¡ä»¶ãæç«ãããšã, $\Omega$ ã®ååŸãæ±ããŠãã ãã.
$$\angle APB=90^\circ,\ \ PA=12,\ \ PB=6$$ |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/1900 | C | OMC080(C) | 200 | 273 | 287 | [
{
"content": "ãæ±ããæ£æŽæ° $n$ ã«ã€ããŠ, ãã®æ£ã®çŽæ°ãå°ããé ã« $a_{1}, a_{2},\\cdots, a_{m}$ ãšããã°,\r\n$$1170=a_{1}+a_{2}+\\cdots a_{m}=\\frac{n}{a_m}+\\frac{n}{a_{m-1}}+\\cdots+\\frac{n}{a_1}=3.25n$$\r\nãããã£ãŠ $n=\\textbf{360}$ ãå¿
èŠã§ãã, éã«ãã®ãšãæ¡ä»¶ãã¿ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc080/... | ããã¹ãŠã®æ£ã®çŽæ°ã«ã€ããŠ, ãããã®ç·åã $1170$ ã§ãã, ãããã®éæ°åã $3.25$ ã§ãããããªæ£æŽæ°ããã¹ãŠæ±ã, ãããã®ç·åã解çããŠãã ãã. |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/2095 | D | OMC080(D) | 300 | 105 | 170 | [
{
"content": "$$AB:BC:CA=\\dfrac{1}{CH_C}:\\dfrac{1}{AH_A}:\\dfrac{1}{BH_B}=13:14:15$$\r\nãã㧠$PQ=13,QR=14,RP=15$ ãªãäžè§åœ¢ $PQR$ ã«ãããŠ, $P$ ãã察蟺ã«ããããåç·ã®é·ã㯠$12$ ã§ããããšãããããã, $ABC$ ãš $PQR$ ã®çžäŒŒæ¯ã¯ $780:12=65:1$ ã§ãã, æ±ããå€ã¯\r\n$$AB+BC+CA=65(13+14+15)=\\textbf{2730}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathc... | ãäžè§åœ¢ $ABC$ ã«ã€ããŠ, $A$ ãã察蟺ã«ããããåç·ã®è¶³ã $H_A$ ãªã©ãšããã°, 以äžãæç«ããŸããïŒ
$$AH_A=780,\quad BH_B=728,\quad CH_C=840$$
ãã®ãšã, $AB+BC+CA$ ãæ±ããŠãã ãã. |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/3196 | E | OMC080(E) | 300 | 130 | 233 | [
{
"content": "ãå¿
èŠååæ¡ä»¶ãšããŠä»¥äžã®ããã«è¡šçŸã§ããããšã¯å®¹æã«ç¢ºèªã§ãã. \r\n\r\n- å·Šäž $4\\times4$ ãã¹ã¯åé¡æã®æ¡ä»¶ãæºãã\r\n- äžãã $i$ è¡ç®, å·Šãã $j$ åç®ã®è²ã $c_{i,j}$ ãšãããšã $c_{i,j}=c_{i-4,j}=c_{i,j-4}$\r\n\r\nããªãã¡, $10\\times10$ ãã¹ã®å¡ãæ¹ã¯ å·Šäž $4\\times4$ ãã¹ã®å¡ãæ¹ãšäžå¯Ÿäžå¯Ÿå¿ãããã, $4\\times4$ ã®å¡ãæ¹ã®ç·æ°ãæ±ããã°ãã. äžäžè¡ãšå·Šäžåã®å¡ãæ¹ãåºå®ãããš, æ®ãã® $3\\times3$ ãã¹ã®å¡ãæ¹ã¯ä»¥äžã® $4$ éã... | ã$10\times10$ ã®ãã¹ç®ã®åãã¹ãèµ€, é, é», ç·ã®ããããäžè²ã§å¡ãæ¹æ³ã®ãã¡, 次ã®æ¡ä»¶ãæºãããã®ã¯äœéããããŸãã.
- ä»»æã®é£ç¶ãã暪 $4$ ãã¹ã«å¡ãããè²ã¯çžç°ãªã
- ä»»æã®é£ç¶ãã瞊 $4$ ãã¹ã«å¡ãããè²ã¯çžç°ãªã |
OMC080 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc080/tasks/1606 | F | OMC080(F) | 400 | 100 | 172 | [
{
"content": "ã$N$ ã®çŽ å æ°å解ã $2^x3^y5^z7^wp_1^{a_1}p_2^{a_2}\\cdots p_n^{a_n}\\ (p_i\\geq 11)$ ãšãããš, \r\n$$d(N)=(x+1)(y+1)(z+1)(w+1)(a_1+1)(a_2+1)\\cdots(a_n+1)$$\r\n$M=(a_1+1)(a_2+1)\\cdots(a_n+1)$ ãšããã°\r\n$$d(N)=(x+1)(y+1)(z+1)(w+1)M$$\r\n$$d(2N)=(x+2)(y+1)(z+1)(w+1)M$$\r\n$$d(3N)=(x+1)(y+2)(z+1)(w+1)M$$\r\n$$d... | ãæ£æŽæ° $n$ ã«å¯Ÿã, $d(n)$ 㧠$n$ ã®æ£ã®çŽæ°ã®åæ°ãè¡šããã®ãšããŸã. ãã®ãšã,
$$ d(N)=d(2N)-120=d(3N)-240=d(5N)-144=d(7N)-360$$
ãæãç«ã€æå°ã®æ£æŽæ° $N$ ãæ±ããŠãã ãã. |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/1872 | A | OMC079(A) | 100 | 317 | 327 | [
{
"content": "ãé·éã»çéã¯ãããã $1$ åé㧠$6^\\circ$ ããã³ $0.5^\\circ$ é²ã. $11$ æã®æç¹ã§é·éãšçéã®ãªãè§ã¯ $30^\\circ$ ã§ãããã,\r\n$$6M=30-0.5M$$\r\nããªãã¡ $M=60\\/13$ ãåŸã. ãã£ãŠ, 解çãã¹ãå€ã¯ $\\textbf{73}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc079/editorial/1872"
}
] | ãäžè¬çãªã¢ããã°æèšã«ãããŠ, $11$ æãéããŠåããŠé·éãšçéã $12$ ãš $6$ ã®ç®çããçµãã çŽç·ã«å¯ŸããŠç·å¯Ÿç§°ãšãªãã®ã¯, $11$ æ $M$ åã§ã. ãã®ãšã, äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $M=\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/2959 | B | OMC079(B) | 200 | 283 | 295 | [
{
"content": "ãåçŽç· $AB$ äžã« $BE=9$ ãªãç¹ $E$ ããšããšïŒåè§åœ¢ $BECD$ ã¯å¹³è¡å蟺圢ã§ããïŒ\r\n$$AE=17,\\quad CE=6,\\quad \\angle AEC=60^\\circ$$\r\nãã£ãŠäžè§åœ¢ $ACE$ ã«å¯ŸããŠäœåŒŠå®çãçšããããšã§ $AC=\\sqrt{\\textbf{223}}$ ãåŸãããã",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc079/editorial/2959"
}
] | ãåžåè§åœ¢ $ABCD$ ã¯
$$AB=8,\quad BD=6,\quad CD=9,\quad \angle ABD=\angle BDC=60^\circ$$
ãæºãããŠããŸãïŒãã®ãšãïŒå¯Ÿè§ç· $AC$ ã®é·ãã® $2$ ä¹ãæ±ããŠãã ããïŒ |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/1276 | C | OMC079(C) | 300 | 207 | 283 | [
{
"content": "ã$2022^{999}$ ã®æ£ã®çŽæ°ã¯, $999$ 以äžã®éè² æŽæ° $a,b,c$ ãçšã㊠$2^a3^b337^c$ ãšè¡šãã, ç¹ã« $1$ ã®äœã $6$ ã§ããããšãã $a\\neq0$ ã«æ³šæãã. ãŸãå $b,c$ ã«ã€ã㊠$3^b$ ã $337^c$ ã® $1$ ã®äœã調ã¹ããš, $1,3,7,9$ ã§ãããã®ããããã $250$ åãã€ãã. ãããã£ãŠ, $a\\geq 1$ ã®ãšã, $b,c$ãã©ã³ãã ã«éžã¶ãšç¢ºç $1\\/4$ 㧠$1$ ã®äœã $6$ ã«ãªã. 以äžãã, 解çãã¹ãå€ã¯ $999\\times 1000^2\\/4=\\tex... | ã$2022^{999}$ ã®æ£ã®çŽæ°ã§ãã£ãŠ, ãã® $1$ ã®äœã $6$ ã§ãããã®ã¯ããã€ãããŸããïŒ |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/2671 | D | OMC079(D) | 400 | 125 | 186 | [
{
"content": "ãæ±ããç·åãžã® $p\\times 10^k$ ã®å¯äžãèããïŒ$p\\times 10^k$ ãå ç®ãããã®ã¯ïŒ$1$ ä»¥äž $p-1$ 以äžããä»»æã«çšããŠïŒ$p+1$ ä»¥äž $9$ 以äžããã¡ããã© $k$ åãçšãããšãã§ããããïŒ$2^{p-1}\\times{}\\_{9-p}\\mathrm{C}\\_{k}$ åå ç®ãããïŒãã£ãŠïŒ$p$ ãåºå®ããã°\r\n$$\r\n\\begin{aligned}\r\n\\sum_{k=0}^{9-p} \\bigl(2^{p-1}\\times{}\\_{9-p}\\mathrm{C}\\_{k}\\times(p\\times... | ãæ£æŽæ° $N$ ã**å¢å ç**ã§ãããšã¯ïŒ$N$ ã $10$ é²æ³ã§è¡šèšããéã«åäœã®æ°ãäžäœããç矩å調å¢å ãšãªãããšããããŸãïŒäŸãã°ïŒ$157$ ã $5$ ã¯å¢å çã§ããïŒ$804$ ã $421$ ã $334$ ã¯å¢å çã§ã¯ãããŸããïŒ\
ãå¢å çãªæ£æŽæ°ã¯æéåããååšããŸããïŒããããã¹ãŠã«ã€ããŠç·åãæ±ããŠãã ããïŒ |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/2672 | E | OMC079(E) | 500 | 79 | 141 | [
{
"content": "ã以äž, åååŒã¯ãã¹ãŠ $3$ ãæ³ãšãã. ãã¹ç®ã®å
äžãã $i$ åç®, å·Šãã $j$ è¡ç®ã®ãã¹ã $(i,j)$ ãšè¡šã. \\\r\nã$mn\\equiv 1$ ãã, $m\\equiv n\\equiv 1$ ãŸã㯠$m\\equiv n\\equiv 2$ ãå¿
èŠã§ãã.\\\r\nããŸã $m\\equiv n\\equiv 2$ ã®æãèãã. $m=3p+2,n=3q+2\\ $ ($p,q$ ã¯éè² æŽæ°) ãšãã. \r\n\r\n----\r\n**è£é¡.**ããã¹ $(i,j)$ ã空ããã¹ãšãªãããã®å¿
èŠååæ¡ä»¶ã¯, $i\\equiv j\\e... | ã$m$ è¡ $n$ åã®ãã¹ç®ãããïŒããã«ïŒ$1\times 3$ ã®ãããã¯ãäºãã«éãªããªãããã«ïŒãã€ãã¹ç®ããã¯ã¿åºããªãããã«çœ®ããŸãïŒããã§å転ãèš±ããã®ãšããŸãïŒãã®ãšãïŒ$mn$ ãã¹ã®ãã¡ã¡ããã© $1$ ãã¹ã®ã¿ã空ããã¹ãšãªããããªçœ®ãæ¹ãååšãïŒç©ºããã¹ãšããŠããåŸããã¹ã¯ã¡ããã© $m+n$ ãæã§ããïŒãã®ãããªããšãèµ·ããæ£æŽæ°ã®é åºä»ããçµ $(m,n)$ ããã¹ãŠæ±ãïŒãããã«ã€ã㊠$mn$ ã®ç·åãæ±ããŠãã ããïŒ |
OMC079 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc079/tasks/1581 | F | OMC079(F) | 600 | 21 | 104 | [
{
"content": "ã挞ååŒãã $\\\\{a_n\\\\}$ ã¯(ç矩)å調å¢å ã§ãããã, $a_{M}\\leq 10001$ ãªãæ倧㮠$M$ ãæ±ããã°ãã.\\\r\nããã㧠$b_n=a_n^4$ ãšãããš, ããã¯ä»¥äžã®æŒžååŒã§è¡šçŸãããïŒ\r\n$$b_{n+1}=b_n+4+\\dfrac{6}{b_n}+\\dfrac{4}{b_n^{2}}+\\dfrac{1}{b_n^{3}}$$\r\nããã§åžžã« $b_n\\geq 10000^4$ ã§ããããšã«çæããã°,\r\n$$4\\lt b_{n+1}-b_n\\lt 4+\\dfrac{11}{10000^{4}}$$\r\nã... | ãæ°å $\\{a_n\\}\_{n=1,2,\cdots}$ ã以äžã§å®ãããšã, $a_n\leq 10001$ ãªãæ£æŽæ° $n$ ã¯ããã€ãããŸããïŒ
$$a_1=10000,\quad a\_{n+1}=a_n+\dfrac{1}{a_n^3}\quad (n=1,2,\cdots)$$ |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/2915 | A | OMC078(A) | 100 | 339 | 339 | [
{
"content": "ãæ, ãºãã³ã¯ãããã $3$ éãã®éžã³æ¹ããããã, ãããã®çµåããšã㊠$9$ éãèãããã.\\\r\nããã®ãã¡ $3$ éãã¯åãè²ã®çµåãã§ãããã, æ±ããå Žåã®æ°ã¯ $9-3=\\textbf6$ éãã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc078/editorial/2915"
}
] | ãOMCåã¯èµ€è², éè², é»è²ã®æãšèµ€è², éè², é»è²ã®ãºãã³ãããããäžçãã€æã£ãŠããŸã.\
ããããã®æãšãºãã³ãäžã€éžãã§çãæ¹æ³ã§ãã£ãŠ, æãšãºãã³ãç°ãªãè²ã«ãªããã®ã¯äœéããããŸããïŒ |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/2920 | B | OMC078(B) | 100 | 330 | 334 | [
{
"content": "ãäžå¹³æ¹ã®å®çãã $BH=3$ ã§ããïŒãã㧠$ABH$ ãš $CAH$ ã¯çžäŒŒã§ããããïŒ$AC=\\dfrac{20}{3}$ ã§ããïŒæ±ããé¢ç©ã¯\r\n$$\\frac{1}{2}\\times AB\\times AC=\\frac{50}{3}$$\r\nã§ããããïŒè§£çãã¹ãå€ã¯ $\\textbf{53}$ ã§ããïŒãªãïŒäžè¬ã«ä»¥äžãæãç«ã€ïŒ\r\n$$\\frac1{AB^2} + \\frac1{AC^2} = \\frac1{AH^2}$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.c... | ã$\angle A=90^\circ$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, $A$ ãã $BC$ ãžããããåç·ã®è¶³ã $H$ ãšããŸã.
$$AB=5, \quad AH=4$$
ã®ãšã, $ABC$ ã®é¢ç©ã¯äºãã«çŽ ãªæŽæ° $a,b$ ãçšã㊠$\dfrac ab$ ãšè¡šããã®ã§, $a+b$ ã解çããŠäžãã. |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/2415 | C | OMC078(C) | 200 | 162 | 285 | [
{
"content": "ãéã« $20x \\lt y$ ãŸã㯠$22y \\lt x$ ãªãçµãæ°ããã°ãã (ãããã¯éè€ããªã)ïŒ\\\r\nã$20x\\lt y$ ãªãçµã®åæ°ã¯ïŒ$x$ ãåºå®ããããšã§\r\n$$\\sum_{x=1}^{101}(2022-20x)=101202$$\r\nåæ§ã«ããŠïŒ$22y\\lt x$ ãªãçµã®åæ°ã¯ïŒ$y$ ãåºå®ããããšã§\r\n$$\\sum_{y=1}^{91}(2022-22y)=91910$$\r\n以äžããïŒæ±ããå Žåã®æ°ã¯ $2022^2-101202-91910=\\textbf{3895372}$ ã§ããïŒ",
"text": "... | ã$1$ ä»¥äž $2022$ 以äžã®æŽæ°ã®çµ $(x,y)$ ã§ãã£ãŠïŒ $20x \geq y$ ã〠$22y \geq x$ ãã¿ãããã®ã¯ããã€ãããŸããïŒ |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/1498 | D | OMC078(D) | 200 | 295 | 302 | [
{
"content": "ã第 $1$ åŒããå
¬æ¯ã $1$ ãŸã㯠$2$ ã§ããããšã容æã«ããã. ããã« $2^{4950}\\times 3^{100}$ 㯠$100$ ä¹æ°ã§ãªããã, å
¬æ¯ã¯ $2$ ã«éãã, ãã®ãšãåé
㯠$3$ ã§ãã, $x\\times 2^y-z=3\\times 2^{100}-3$ ã§ãã. ãã£ãŠ $x+y+z=3+100+3=\\textbf{106}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc078/editorial/1498"
}
... | ãåé
ãšå
¬æ¯ããšãã«æ£æŽæ°ã§ããçæ¯æ°å $\\{a_i\\}\_{i=1,2,\cdots,100}$ ã以äžã®çåŒ
$$a_1\times a_2\times \cdots\times a_{100}=2^{4950}\times 3^{100}$$
ãã¿ãããŸããïŒãã®ãšãïŒ
$$a_1+a_2+\cdots+a_{100}$$
ã¯æ£ã®æŽæ° $x,y,z$ ïŒãã ã $x$ 㯠$1$ æ¡ã®å¥æ°ïŒ$z$ 㯠$1$ æ¡ã®æ£æŽæ°ïŒãçšã㊠$x\times 2^y-z$ ãšäžæã«è¡šãããšãã§ããã®ã§ïŒ$x+y+z$ ãæ±ããŠãã ããïŒ |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/2743 | E | OMC078(E) | 300 | 117 | 207 | [
{
"content": "ãå
±éã®éå¿ã $G$ ãšãïŒ$AM$ ãš $DF$ ã®äº€ç¹ã $H$ ïŒ$AC=x$ ãšãããšïŒ\r\n$$AG:GM=EG:GH=2:1$$\r\nããã³Menelausã®å®çãã $AE:EM=12:(x-6)$ ããïŒ\r\n$$AH : HG : GE : EM = 3x : (12-x) : (24-2x) : (3x-18)$$\r\näžæ¹ïŒCevaã®å®çãã $BC\\parallel DF$ ãåŸãããïŒä»¥äžãã\r\n$$x:6=AH:HM=AF:FC=6:x-6$$\r\nããã解ã㊠$x=3+3\\sqrt{5}$ ãåŸãããïŒè§£çãã¹ãå€ã¯ $3+45=\\te... | ãäžè§åœ¢ $ABC$ ã«ãããŠïŒ$BC$ ã®äžç¹ã $M$ ãšããŸãïŒç·å $AB$ äžã®ç¹ $D$ ã«ã€ããŠïŒç·å $AM$ ãšç·å $CD$ ã®äº€ç¹ã $E$ ãšãïŒç·å $AC$ ãšçŽç· $BE$ ã®äº€ç¹ã $F$ ãšãããšïŒäžè§åœ¢ $ABC$ ãš $DEF$ ã®éå¿ãäžèŽããŸããïŒããã« $AF=6$ ã§ãããšãïŒ$AC$ ã®é·ãã¯æ£æŽæ° $a,b$ ã«ãã£ãŠ $a+\sqrt{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC078 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc078/tasks/1870 | F | OMC078(F) | 400 | 16 | 99 | [
{
"content": "ããŸã, $P(i,j)$ ã¯ä»¥äžã®ããã«èšç®ã§ããããšã容æã«ããã. ããã§, $l$ ã¯æœéžãè¡ãããåæ°ã«å¯Ÿå¿ãã.\r\n$$P(i,j)=\\frac{1}{{}\\_{ij}\\mathrm{C}\\_i}\\sum_{l=i}^{ij}\\left(l\\times{}\\_{l-1}\\mathrm{C}\\_{i-1}\\right)\r\n=\\frac{i}{{}\\_{ij}\\mathrm{C}\\_i}\\sum_{l=i}^{ij}{}\\_{l}\\mathrm{C}\\_{i}\r\n=\\frac{i}{{}\\_{ij}\\mathrm{C}\\_i... | ãæ£æŽæ° $i,j$ ã«ã€ããŠ, $j$ çš®é¡ã®çããããã $i$ åãã€å
¥ã£ãŠãã, ããããäžã€ãã€ã©ã³ãã ã«æåºããæœéžæ©ããããŸã. $j$ çš®é¡ã®äžããã©ã³ãã ã« $1$ çš®é¡ãéžãã ãšã, ãã®çã $i$ åãã¹ãŠåºãŠãããŸã§ã«, æœéžãè¡ãããåæ°ã®æåŸ
å€ã $P(i,j)$ ãšããŸã. ãã ã, äžåºŠæåºãããçã¯æ»ããªããã®ãšããŸã.\
ã$P(i,j)=210$ ãªãçµ $(i,j)$ ãã¹ãŠã«ã€ããŠ, $i+j$ ã®ç·åãæ±ããŠãã ãã. |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2043 | A | OMC077(A) | 200 | 199 | 219 | [
{
"content": "ãæ£æŽæ° $m$ ã«å¯Ÿã㊠$6^m$ ã®äž $2$ æ¡ãèšç®ãããš $6,36,16,96,76,56,36,\\cdots$ ãšãªã, $m=2$ 以éã¯åšæ $5$ ã§åŸªç°ãã. å¹³æ¹æ°ã $5$ ã§å²ã£ãäœã㯠$0,1,4$ ã®ããããã§ãããã, æ±ããç·å㯠$6+56+96+76=\\textbf{234}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc077/editorial/2043"
}
] | ãæ£æŽæ° $n$ ã«å¯ŸããŠ, $6^{(n^2)}$ ã®äž $2$ æ¡ ($100$ ã§å²ã£ãäœã) ãšããŠããåŸãæ°ã®ç·åãæ±ããŠãã ãã. |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2047 | B | OMC077(B) | 300 | 84 | 117 | [
{
"content": "ã$f(x)=x^4+ax^3+bx^2$ ã«ã€ããŠ, $f^\\prime(x)=x(4x^2+3ax+2b)$ ã§ããããšãã, æ¡ä»¶ã¯ $x$ ã®äºæ¬¡æ¹çšåŒ\r\n$$4x^2+3ax+2b=0$$\r\nã $0$ ã§ãªãçžç°ãªãå®æ°è§£ $x=s,t$ ããã¡, ãã€ä»¥äžãã¿ããããšãšåå€ã§ããïŒ\r\n$$\\dfrac{f(s)}{s}\\times\\dfrac{f(t)}{t}=-1$$\r\n解ãšä¿æ°ã®é¢ä¿ãã $s+t=-\\dfrac{3}{4}a$ ããã³ $st=\\dfrac{b}{2}$ ã§, ãããçšããŠæ¡ä»¶ãæžãæããã°\r\n$$b^2(a^2-4b)=... | ãæ£æŽæ° $a,b$ ã«å¯ŸããŠ, 以äžã§äžããããã°ã©ã
$$y=x^4+ax^3+bx^2$$
ã«æ¥µå€§å€ãŸãã¯æ¥µå°å€ããšãç¹ãåç¹ $(0,0)$ ãå«ãã¡ããã© $3$ ã€ãã, ããããé ç¹ãšããäžè§åœ¢ã¯åç¹ãçŽè§ãšããçŽè§äžè§åœ¢ã§ã. ãã®ãšã, çµ $(a,b)$ ãšããŠããåŸããã®ãã¹ãŠã«ã€ããŠ, $a+b$ ã®ç·åãæ±ããŠãã ãã. |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2397 | C | OMC077(C) | 300 | 65 | 120 | [
{
"content": "ã$f(x)$ ã¯åžžã« $x=100$ ã§å®çŸ©ãããããšã«æ³šæããïŒåºã®å€æå
¬åŒãçšãããš\r\n$$\r\nf(x)= \\log_a{x}+1+\\frac{\\log_a{b}}{\\log_a{x}+1} -1 \\geq 2\\sqrt{ \\log_a{b}} -1\r\n$$\r\nãšãªãïŒæåŸã®äžçåŒã¯çžå ã»çžä¹å¹³åã®é¢ä¿ããåŸãïŒçå·ã¯\r\n$$x=a^{\\sqrt{\\log_a{b}}-1}\\gt\\frac{1}{a}$$\r\nã§æç«ããïŒ\\\r\nãããŸïŒ$x=100$ ã§çå·ãæç«ãïŒ$f(100)$ ãæŽæ°ãšãªãããã«ã¯ïŒ\r\n$$(\\log_a... | ã$2$ 以äžã®æŽæ° $a,b$ ã«ã€ããŠïŒ$x \gt 1\/a$ ã§å®çŸ©ãããé¢æ°
$$f(x)=\log_a{x}+\log_{ax}b$$
ã $x=100$ ã§æå°å€ããšãïŒãã®æå°å€ããŸãæŽæ°å€ã«ãªããŸããïŒ$a+b$ ãšããŠããããå€ã®ç·åãæ±ããŠãã ããã |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2044 | D | OMC077(D) | 300 | 91 | 125 | [
{
"content": "ã$OP=x$ ãšããïŒç°¡åãªè§åºŠèšç®ããïŒ$QP=QS=1$ïŒ$PO=PR=PS=x$ ã§ããïŒãŸãæ¹ã¹ãã®å®çãã $OQ\\cdot OR=OP^2=x^2$ ã§ãããã $QR=x^2-1$ ã§ããïŒãããã£ãŠPtolemyã®å®çãã\r\n$$\\begin{aligned}\r\nPQ\\cdot SR+QS\\cdot RP&=QR\\cdot PS\\\\\\\\\r\n1\\cdot \\frac{3}{2}x+1\\cdot x&=(x^2-1)\\cdot x.\r\n\\end{aligned}$$\r\nããã解ã㊠$x^2=7\\/2$ ãåŸãããïŒç¹ã«è§£çã... | ãå $\Gamma$ äžã« $2$ ç¹ $P,Q$ ã, $\Gamma$ ã®å€éšã«ç¹ $O$ ããã, $OQ=PQ=1$ ã〠$OP$ 㯠$\Gamma$ ã«æ¥ããŠããŸã. ç·å $OQ$ ã® $Q$ åŽã®å»¶é·ç·ãš $\Gamma$ ã亀ãã£ãã®ã§ãã®äº€ç¹ã $R$ ãšã, $\Gamma$ äžã« $\angle OPQ=\angle SPQ$ ãªãç¹ $S$ ããšã£ããšã, $OP:RS=2:3$ ãæç«ããŸãã. ãã®ãšã, $OP$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\sqrt{\dfrac{a}{b}}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2320 | E | OMC077(E) | 400 | 34 | 91 | [
{
"content": "ããŸãåæããããã§ã³ã¬ãŒãã®åæ°ãæ°ããïŒ $n, 1000-n$ ã®æ倧å
¬çŽæ°ã $g$ ãšãïŒçžŠã®é·ã $p=n\\/g$ ïŒæšªã®é·ã $q=(1000-n)\\/g$ ã®é·æ¹åœ¢ã**å°é·æ¹åœ¢**ãšåŒã¶ããšã«ããïŒå°é·æ¹åœ¢ãåæããå Žå $p+q-1$ åã®ãã§ã³ã¬ãŒããåæãããããšã容æã«ãããããïŒå
šäœã§ã¯ããã $g$ åãã $1000-g$ åã®ãã§ã³ã¬ãŒããåæãããïŒ\\\r\nãããã«ïŒ$p,q$ ãããããå¥æ°ã§ãããšãã«éãïŒåå°é·æ¹åœ¢ã®äžå€®ã«ãããã§ã³ã¬ãŒããã¡ããã©é¢ç© $1\\/2$ ãã€ã«åæãããã®ã§ïŒ$p,q$ ãå¥æ°ã§ããã°é¢ç© $1\\/2$ 以... | ã$999$ 以äžã®æ£æŽæ° $n$ ã«å¯ŸãïŒèŸºã®é·ãã $1$ ã®æ£æ¹åœ¢ã®ãã§ã³ã¬ãŒãã瞊㫠$n$ åïŒæšªã« $1000-n$ åïŒããéãªã䞊ã¹ãŠåšé· $2000$ ã®é·æ¹åœ¢ãäœãïŒååé·ããã€ãã§ãã®é·æ¹åœ¢ã® $1$ æ¬ã®å¯Ÿè§ç·ã«æ²¿ã£ãŠãã§ã³ã¬ãŒããã¡ãåæããŸãïŒãã®ãšãïŒé¢ç©ã $1\/2$ 以äžã§ãããã§ã³ã¬ãŒãã®æçã $m$ åæ®ããŸããïŒ$m$ ãšããŠããããéè² æŽæ°å€ã®ç·åã解çããŠãã ããïŒãã ãïŒãã€ãã®å¹
ã¯èããªããã®ãšããŸãïŒ |
OMC077 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc077/tasks/2048 | F | OMC077(F) | 500 | 9 | 28 | [
{
"content": "ã$\\\\{h_k\\\\},\\\\{v_l\\\\}$ ãããããã $p-1,q-1$ æ¬ã®ç·åãéžæãããšãã. ãã®ãšã, ãã¹ç®ã¯ $pq$ åã®é åã«åãã, ããã $27$ ãå²ãåãå¿
èŠããããã, $(p,q)=(3,3),(3,9),(9,3)$ ã§ãã. ãã®ãšã, æ£è§£ãåŸãã«ã¯çžŠã $p$ çå, 暪ã $q$ çåããããã«ç·åãéžæããå¿
èŠããã. ããã¯äžäžã»å·Šå³ã«é£æ¥ããç·åã§å²ãŸããé åãèããã°ããã.\\\r\nãããã§, $(p,q)=(3,9)$ ãŸã㯠$(9,3)$ ã§æ£è§£ãšãªãç€é¢ã¯ $(3,3)$ ã§ãæ£è§£ã§ãããã, äžè¬æ§ã倱ã... | ã$27\times 27$ ã®ãã¹ç®ããã, $k=1,2,\cdots,26$ ã«å¯ŸããŠ, 第 $k$ è¡ãš $k+1$ è¡ã®å¢ç®ããªã暪æ¹åã®ç·åã $h_k$ ïŒç¬¬ $k$ åãš $k+1$ åã®å¢ç®ããªã瞊æ¹åã®ç·åã $v_k$ ãšããŸã. ãŸã, ã¡ããã© $27$ åã®ãã¹ã« $1$ ãæžãããŠãã, åãè¡ããã³åãåã« $2$ ã€ä»¥äžã® $1$ ãæžãããŠããããšã¯ç¡ããã®ãšããŸã.\
ãOMCå㯠$\\{h\_k\\}\_{1 \leq k\leq 26} ,\\{v\_k\\}\_{1 \leq k\leq 26} $ ãã**ããããäžã€ä»¥äž**ã®ç·åãéžã³, 以äžã®æ¡ä»¶ãã¿ããããã«ãããã§ãïŒ
... |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/1602 | A | OMC076(A) | 100 | 351 | 352 | [
{
"content": "ã$1$ ã®äœãšããŠããåŸããã®ã¯ $0,1,5,6$ ã§ãã, $\\sqrt{999}\\approx 31.6$ ãšäœµããã°åè£ãšãªããã®ã¯ååå°ãªããã, ãããã¯è©Šãã°ãã. å
·äœçã«ã¯, $\\textbf{625}$ ã®ã¿ã§ããããšã容æã«ããã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc076/editorial/1602"
}
] | ã$3$ æ¡ã®å¹³æ¹æ° $n$ ã§ãã£ãŠ, ãã®äž $2$ æ¡ã $\sqrt{n}$ ã«äžèŽãããã㪠$n$ ãæ±ããŠãã ãã. |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/2749 | B | OMC076(B) | 200 | 284 | 324 | [
{
"content": "ã$C$ ãæ£çãã $3$ 人ããããã $A,B$ ãæ£çã§ããªãã£ãå Žåãèããã°, æ®ãå
šå¡ã $A$ ãæ£çããŠããããšãã, $100$ ç¹ã $2$ å, $300$ ç¹ã $8$ åãšãªã. ãããã£ãŠ, CMOå㯠$1$ äœãã $8$ äœã®ãã¹ãŠããšãåŸã.\\\r\nãäžæ¹ã§, å
šå¡ã®ç²åŸããç¹æ°ãåèšã§ $2600$ ç¹ã§ããããšãã, $300$ ç¹ä»¥äžç²åŸãã人ã¯é«ã
$8$ åã§ãã.\\\r\nã以äžãã, 解çãã¹ãå€ã¯ $1+2+\\cdots+8=\\textbf{36}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": ... | ãCMOåã¯OMCã®ããã³ã³ãã¹ãã«åå ããŸãã. ãã®ã³ã³ãã¹ã㯠$A,B,C$ ã® $3$ ã€ã®åé¡ã§æ§æãã, é
ç¹ã¯ãããã $100,200,300$ ç¹ã§ãã. ãŸãåå è
㯠$10$ åã§, $A,B,C$ ã®æ£çè
ã¯ãããã $7,5,3$ åã§ãã. CMOåã $A$ ãš $B$ ã® $2$ åã®ã¿æ£çãããšã, CMOåã®ãšãåŸãé äœãšããŠããåŸããã®ã®ç·åãæ±ããŠãã ãã. \
ããã ã, é äœã¯ $1$ ä»¥äž $10$ 以äžã®æŽæ°å€ãšããŠè¡šçŸãã, åçã¯ãªãã£ããã®ãšããŸã. ãŸã, åŸç¹ã®ç·åã倧ããæ¹ãäžäœãšãªãïŒåç¹ã®å Žåã«ã¯æåŸã®æ£çã®æ©ãæ¹ãäžäœãšãªããŸã. ãŸã, 誀çã«ããããã«ãã£ã¯ãª... |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/2932 | C | OMC076(C) | 200 | 289 | 308 | [
{
"content": "ã$n$ å $31$ ç§åŸã«æ®ã£ãŠããæ ãŸããã
ãã®åæ° $a_n$ ã«ã€ããŠïŒ$a_0=3$ ããã³ $n=1,2,\\ldots$ 㧠$a_n=2a_{n-1}-n$\r\nãæç«ãïŒããã解ããš $a_n=2^n+n+2$ ã§ããïŒãã£ãŠ $a_n=2061$ ã®ãšã $n=11$ ã§ããïŒãããŸã§ã«OMCåãé£ã¹ãæ ãŸããã
ã㯠$1+2+\\cdots+11=\\textbf{66}$ åã§ããïŒ \\\r\nããŸãïŒ$n$ å $1$ ç§åŸã«æ®ã£ãŠããæ ãŸããã
ãã®åæ° $b_n$ ã«ã€ããŠïŒ$b_n=a_n+n=2^n+2n+2$ ã§ãããïŒ$b_n=2061$ ãæºãã... | ãæ ãŸããã
ãã $3$ åããïŒä»ãã $1$ åãçµéããããšã«æ®ã£ãŠãããã®ã¯ãã¹ãŠ $2$ ã€ã«åè£ããŸãïŒããªãã¡ïŒä»ãã $1$ ååŸã«ã¯ $6$ åã«ãªããŸãïŒOMCåã¯ãã®æ ãŸããã
ããïŒä»¥äžã®æ¡ä»¶ãã¿ããããã«é£ã¹ãŠãããŸãïŒ
- $n=1,2,\ldots$ ã«ã€ããŠïŒä»ãã $n$ å $30$ ç§åŸã«ã¡ããã© $n$ åã®æ ãŸããã
ããé£ã¹ãïŒ
ããæç¹ã§æ®ã£ãŠããæ ãŸããã
ãã $2061$ åãã£ããšãïŒãããŸã§ã«OMCåãé£ã¹ãæ ãŸããã
ãã¯ããã€ã§ããïŒ\
ããã ãïŒOMCåãæ ãŸããã
ããé£ã¹ãã®ã«èŠããæéã¯èæ
®ããªããã®ãšããŸãïŒ |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/1794 | D | OMC076(D) | 300 | 117 | 169 | [
{
"content": "äžè§åœ¢ $ABC$ ã®å
æ¥åã®ååŸã $r$ ãšãããšã, 以äžã®æç«ã容æã«ããã.\r\n$$\\begin{aligned}r=\\frac{AB+AC-BC}{2}\r\n\\end{aligned}$$\r\nããã§\r\n$$DE=DF=AB+AC-BC$$\r\nã§ãããã, $BFGC$ ã®é¢ç©ã«ã€ããŠ\r\n$$\\begin{aligned}\r\n\\frac{1}{2}(BC+GF)DF=(AB+AC)r\r\n\\end{aligned}$$\r\nãããš $ABC$ ãšã®é¢ç©ã®å·®ãèšç®ããã°,\r\n$$\\begin{aligned}\r\n(AB+AC)r-... | ãè§ $A$ ãçŽè§ã§ãããããªé¢ç© $420$ ã®çŽè§äžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ äžã« $D, E$ ã
$$AC=CD,\quad AB=BE$$
ãšãªãããã«ãšããŸã. ãŸã, 蟺 $BC$ ã«é¢ã㊠$A$ ãšå察åŽã«åè§åœ¢ $DFGE$ ãæ£æ¹åœ¢ã«ãªããã㪠$F, G$ ããšã£ããšãã, åè§åœ¢ $BFGC$ ã®é¢ç©ã $492$ ãšãªããŸãã. ãã®ãšã, äžè§åœ¢ $ABC$ ã®å
æ¥åã®ååŸã® $2$ ä¹ãæ±ããŠãã ãã. |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/1746 | E | OMC076(E) | 300 | 116 | 219 | [
{
"content": "ã声ãè£è¿ããªãæŽæ°, ããªãã¡ã$3$ ã®åæ°ã§ãªã, ã〠$3$ ãä»ããªãããã®ãæ°ãäžããã°ãã. ããã«, $3$ ã®åæ°ã§ãªãããšã¯, åäœã®æ°ã®åã $3$ ã®åæ°ãšãªããªãããšã§ãã, $3$ ãé€ã $0$ ãã $9$ ã®æ°åã, 以äžã®ããã«åé¡ãã.\r\n$$A:\\\\{0,6,9\\\\},\\quad B:\\\\{1,4,7\\\\},\\quad C:\\\\{2,5,8\\\\}$$\r\n\r\n(i) $999$ 以äžã®å ŽåïŒåæ¡ã®çµã¿åãããšããŠããåŸããã®ã¯ä»¥äžã§ãã, ${}_3\\mathrm{C}_1Ã3^3Ã6=486$ åãã.\r... | ãOMCåã¯æ°ãæ°ãããšã, ããã $3$ ã®åæ°ã§ãããšã, ãŸãã¯åé²æ³è¡šèšã§ $3$ ãçŸãããšãã«å£°ãè£è¿ã£ãŠããŸããŸã. OMCåã $1$ ãã $2021$ ãŸã§æŽæ°ãé ã«æ°ãããšã, 声ãè£è¿ã£ãŠããŸãæŽæ°ã¯ããã€ãããŸããïŒ |
OMC076 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc076/tasks/2669 | F | OMC076(F) | 400 | 105 | 159 | [
{
"content": "ãä»»æã® $N \\geq 2$ ã«å¯ŸãïŒ\r\n$$\\sum\\limits_{n = 2^{N - 1} + 1}^{2^{N}}a_n =\r\n \\sum\\limits_{n = 2^{N - 2}+1}^{2^{N - 1}}(a_{2n - 1} + a_{2n}) =\r\n \\sum\\limits_{n = 2^{N - 2} + 1}^{2^{N - 1}}(n +a_n) $$\r\nã§ããããšããïŒåž°çŽçã« $N\\geq 2$ ã«å¯Ÿã\r\n$$\\sum\\limits_{n = 2^{N - 1} + 1}^{2^{N}}a_n = a_1+\\sum... | ãæ°å $\\{a_n\\}_{n=1,2,\ldots}$ ã以äžã§å®ããŸãïŒ
$$a_n =
\begin{cases}
\dfrac{n+1}{2} & (n ãå¥æ°ã®ãšã) \\\\
a\_{(n\/2)} & (n ãå¶æ°ã®ãšã)
\end{cases}
$$
ãã®ãšãïŒç·å $S = \displaystyle\sum _{n=1}^{500} a _{n}$ ãæ±ããŠäžããïŒ |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/3465 | A | OMC075(A) | 100 | 257 | 260 | [
{
"content": "ã$m=n$ ãæºããçµ $(m,n)$ 㯠$9$ éãã§ããïŒãŸãïŒ$m\\gt n$ ãæºããçµ $(m,n)$ ãš $m\\lt n$ ãæºããçµ $(m,n)$ ã¯åæ°ã§ããïŒãããã $\\left(9^2-9\\right)\\div2=36$ éãã§ããïŒåŸã£ãŠïŒè§£çãã¹ãå€ã¯ $9+36=\\textbf{45}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc075/editorial/3465"
}
] | ã$1$ ä»¥äž $9$ 以äžã®æŽæ°ã®çµ $(m,n)$ ã§ãã£ãŠã $m\ge n$ ãæºãããã®ã¯äœéããããŸããïŒ |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/2900 | B | OMC075(B) | 200 | 238 | 253 | [
{
"content": "ã$m-n \\lt m+n\\le 20$ ã〠$m-n$ ãš $m+n$ ã®å¶å¥ãäžèŽããããšããïŒ$(m-n,m+n)$ ã®çµãšããŠã¯\r\n$$(m-n,m+n)=(0,4),(0,16),(4,16),(1,9)$$\r\nãããåŸãïŒããããã«ã€ããŠ\r\n$$(m,n)=(2,2),(8,8),(10,6),(5,4)$$\r\nã§ããããïŒè§£çãã¹ãå€ã¯ $\\textbf{148}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc075/editorial... | ã以äžã®æ¡ä»¶ãã¿ãã $1$ ä»¥äž $10$ 以äžã®æŽæ°ã®çµ $(m,n)$ ãã¹ãŠã«ã€ããŠïŒ$mn$ ã®ç·åãæ±ããŠãã ããïŒ
- $m+n$ ãš $m-n$ ããšãã«å¹³æ¹æ°ã§ããïŒ
ãããã§ïŒå¹³æ¹æ°ãšã¯ïŒããæŽæ°ã®äºä¹ã®åœ¢ã§è¡šãããéè² æŽæ°ã®ããšã§ïŒç¹ã« $0$ ãå«ã¿ãŸãïŒ |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/2899 | C | OMC075(C) | 300 | 175 | 232 | [
{
"content": "ã察称æ§ãã $A_1=1$ ãšã㊠$3$ åããã°ããïŒäžæŠ $A_1\\neq A_8$ ãšããæ¡ä»¶ãé€å€ãïŒ\r\n$$d_{i,j}=(A_1 ãã A_i ãŸã§æ±ºãããšãïŒA_i=j ã§ãããããªãã®ã®åæ°)$$\r\nãšããïŒãã®ãšãïŒ\r\n$$\\displaystyle d_{i,j}=\\sum_{k=1}^3d_{i-1,k}-d_{i-1,j}$$\r\nãšãã挞ååŒãæãç«ã€ããïŒãããçšããŠèšç®ãããš\r\n$$d_{8,1}=42,\\quad d_{8,2}=d_{8,3}=43$$\r\nãåããïŒãããã£ãŠïŒæ¡ä»¶ $A_1\\neq A_8$ ãèæ
®... | ã以äžã® $2$ ã€ã®æ¡ä»¶ãæºããé·ã $8$ ã®æŽæ°å $A_1,A_2,\ldots,A_8$ ã¯ããã€ãããŸããïŒ
- $1\le A_i\le 3\ \ (1\le i\le 8)$
- $A_i\neq A_{i+1}\ \ (1\le i\le 8)$ã(ãã ã $A_9=A_1$ ãšãã) |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/1766 | D | OMC075(D) | 300 | 57 | 130 | [
{
"content": "ãçµ $1\\le i\\lt j\\le 2022$ ã«å¯ŸãïŒ$i$ ãš $j$ ãé£ãåã£ãŠçŸããé å㯠$2\\times 2021\\times 2020!$ éãããïŒãã ãïŒ$2$ 㯠$i$ ãš $j$ ã®é åºïŒ$2021$ 㯠$i,j$ ã®äœçœ®ïŒ$2020!$ ã¯æ®ãã®æ°ã®äžŠã³ã«å¯Ÿå¿ããïŒããããïŒæ±ããç·åã¯\r\n$$\\displaystyle S=\\sum_{j=1}^{2022}\\sum_{i=1}^{j-1}\\left(2\\times2021!\\times (j-i)\\right)=2\\times2021!\\times\\sum_{j=1}^{... | ãé·ã $2022$ ã®æŽæ°å $A_1,A_2,\ldots,A_{2022}$ ã«å¯ŸãïŒãã®**ã¹ã³ã¢**ã以äžã§å®ããŸãïŒ
$$\displaystyle\sum_{i=1}^{2021}|A_i-A_{i+1}|$$
ã$1$ ä»¥äž $2022$ 以äžã®æŽæ°ã䞊ã³æ¿ããŠåŸãããé·ã $2022$ ã®é åå
šãŠã«å¯ŸãïŒ**ã¹ã³ã¢**ã®ç·åã $S$ ãšãããŸãïŒ$S$ ã $3$ ã§å²ããããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/2907 | E | OMC075(E) | 300 | 125 | 194 | [
{
"content": "ãæ£æŽæ° $a,b,c$ ãçšã㊠$\\displaystyle n=2^{a-1}3^{b-1}5^{c-1}$ ãšè¡šãã°ïŒ$n$ ã®æ£ã®çŽæ°ã®åæ°ã¯ $abc$ ãšæžããïŒããªãã¡ïŒ$abc=k$ ãªãæ£æŽæ°ã®çµ $(a,b,c)$ ãã¡ããã© $45$ åã§ãããããªæå°ã®æ£æŽæ° $k$ ãæ±ããåé¡ã«åž°çãããïŒ\\\r\nã$k$ ã®çŽ å æ°å解ã $k=\\prod_{i=1}^m p_i^{e_i}$ ãšãããšïŒå $i$ ã«å¯ŸãçŽ å æ° $p_i$ ã $a,b,c$ ã«åèšã§ $e_i$ ååé
ããæ¹æ³ã¯\r\n$$\\frac{(e_i+1)(e_i+2)}2$$\r\n... | ã以äžã®æ¡ä»¶ãã¿ããæ£æŽæ° $n$ ãã¡ããã© $45$ åååšãããããªïŒæå°ã®æ£æŽæ° $k$ ãæ±ããŠäžããïŒ
- $n$ 㯠$7$ 以äžã®çŽ æ°ã§å²ãåããªãïŒ
- $n$ ã¯æ£ã®çŽæ°ãã¡ããã© $k$ åãã€ïŒ |
OMC075 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc075/tasks/2898 | F | OMC075(F) | 300 | 44 | 101 | [
{
"content": "ã$AC$ ãš $BD$ ã®äº€ç¹ã $P$ïŒ$AC$ ãš $DE$ ã®äº€ç¹ã $Q$ïŒ$BC$ ãš $DE$ ã®äº€ç¹ã $R$ ãšããïŒ\r\n\r\n\r\n**解æ³1.**ã以äžã®ããã«è§åºŠèšç®ã§ããããšããïŒäžè§åœ¢ $ABF$ 㯠$CBA$ ãšçžäŒŒã§ããïŒ\r\n$$\\angle{BAF}=\\angle{BAE}=\\angle{BDE}=90^\\circ-\\angle{DQP}=90^\\circ-\\angle{RQC}=\\angle{ACB}$$\r\nãã£ãŠ $\\displaystyle BF=5\\times\\frac{5}{7}=\\frac{25}7$ ... | ãå€æ¥åã $\Gamma$ ãšããäžè§åœ¢ $ABC$ ã¯ïŒ$AB=5,AC=6,BC=7$ ãã¿ãããŸãïŒ$B$ ãã $AC$ ã«ããããåç·ãš $\Gamma$ ã®äº€ç¹ã®ãã¡ $B$ ã§ãªãæ¹ã $D$ ãšãïŒ$D$ ãã $BC$ ã«ããããåç·ãš $\Gamma$ ã®äº€ç¹ã®ãã¡ $D$ ã§ãªãæ¹ã $E$ ãšããŸãïŒç·å $AE$ ãš $BC$ ã®äº€ç¹ã $F$ ãšãããšãïŒ$BF$ ã®é·ããæ±ããŠãã ããïŒ\
ããã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/1516 | A | OMC074(A) | 200 | 207 | 219 | [
{
"content": "ãåæ人ã«ã€ããŠçŽ çŽã§ããããè¡šãççå€ã**çŽ çŽåºŠ**ãšåŒã¶. $pe\\overbrace{p...p}^{n}er$ ãããš $pe\\overbrace{p...p}^{f(10^n)}er$ ããã®çŽ çŽåºŠã¯çãã. ããã§, åååŒã®æ³ã $4$ ãšããã°,\r\n$$f(10^n) = \\begin{cases} 1 &(n \\equiv 0) \\\\\\ 10 \\equiv 2 &(n \\equiv 1) \\\\\\ 100 \\equiv 0 &(n \\equiv 2) \\\\\\ 91 \\equiv 3 &(n \\equiv 3) \\end{case... | ã$peer$ æã«ã¯, $peper$ ãã, $pepper$ ãã, ... , $pe\overbrace{p...p}^{100}er$ ããã® $100$ 人ãäœãã§ããŸã. $peer$ æã®äœäººã®æ§æ Œã¯ã©ãã**çŽ çŽ**ã**ç
§ãå±ãã**ã§ãã, æ§æ ŒãçŽ çŽãªè
ã¯åžžã«çŽ çŽã«æ¬åœã®ããšãèšã, ç
§ãå±ããã¯åžžã«ç
§ããŠåãã€ããŠããŸããŸã. ãŸã, $peer$ æã®äœäººã¯ã©ã®äºäººãäºãã®æ§æ ŒãçŽ çŽã§ãããç
§ãå±ããã§ããããææ¡ããŠããŸã.\
ãããæ, $peer$ æã«æ
人ããã£ãŠããŸãã. æ
人ã¯, $100$ 以äžã®æ£ã®æŽæ° $n$ ããããã«å¯ŸããŠ, $pe\overbrace{p...p}^{n}er$... |
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/1808 | B | OMC074(B) | 400 | 148 | 197 | [
{
"content": "ãå·Šãã $k$ åç®ã®ãã¹ã«é§ããããšã, ãã®é§ã«åŸç¹ $k - 1$ ãå²ãåœãŠ, åç€é¢ã«ã€ã㊠$100$ åã®é§ã®åŸç¹ã®ç·åããã®ç€é¢ã®åŸç¹ãšãã. ãã®ãšã, åæçªã®ååŸã§ã®ç€é¢ã®åŸç¹ã®å·®ã¯ $2$ ä»¥äž $9$ 以äžã§ãã. ãŸã, ã²ãŒã éå§æã®ç€é¢ã®åŸç¹ã¯ $0$ ç¹ã§ãã, ã²ãŒã çµäºæã®ç€é¢ã®åŸç¹ã¯ $100(n -1) - 1 = 100n - 101$ ç¹ä»¥äžã§ãã. ãããã£ãŠãã®ã²ãŒã ã¯, å®éãããå
æ»ãšããŠ, å®éãããšå³¶æããã $0$ ã«äº€äºã« $2$ ä»¥äž $9$ 以äžã®å¥œããªæ°ã足ããŠãã, $100n - 101$ 以äžã«ããã»ããåã¡ãšãã... | ã$n$ ã $2$ ä»¥äž $100$ 以äžã®æŽæ°ãšããŸã. å·Šå³ã«äžŠãã $n$ åã®ãã¹ãš $100$ åã®é§ã䜿ã£ãŠ, å®éãããšå³¶æããã¯æ¬¡ã®ãããªã²ãŒã ãè¡ããŸã.\
ãã¯ãã, ãã¹ãŠã®é§ã¯å·Šç«¯ã®ãã¹ã«çœ®ãããŠããŸã. å
æ»ãå®éãããšããŠ, äºäººã¯èªåã®æçªã§æ¬¡ã®æäœã $2$ åä»¥äž $9$ å以äžã®å¥œããªåæ°ç¹°ãè¿ããŸãïŒ
- å³ç«¯ã®ãã¹ä»¥å€ã«çœ®ãããŠããé§ãäžã€éžã³, ãããçŸåšçœ®ãããŠãããã¹ã®äžã€å³ã®ãã¹ã«ç§»åãã.
å
ã«èªåã®æçªãæ£ããéè¡ããããšãã§ããªããªã£ãæ¹ãè² ããšãªã, ããäžæ¹ãåã¡ãšãªããŸã.\
ããã®ãšã, å®éãããæåãå°œããããšã§å¿
ãåãŠããã㪠$n$ ã®ç·åãæ±ã... |
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/1514 | C | OMC074(C) | 600 | 57 | 112 | [
{
"content": "ãåæ人ã«ã€ããŠçŽ çŽã§ããããè¡šãççå€ã**çŽ çŽåºŠ**ãšåŒã³, $101$ ãæ³ãšããåå§æ ¹ $r$ ã«ã€ããŠ, $pe\\overbrace{p...p}^{f(r^n)}er$ ããã®çŽ çŽåºŠã $s(n)$ ã§è¡šã. $1\\leq n\\leq 100$ ã®ç¯å²å€ã«ã€ããŠã, $s(n) = s(n - 100)$ ã§ããããšã«çæãã.\\\r\nãæ¡ä»¶ãã $pe\\overbrace{p...p}^{f(r^{n})}er$, $pe\\overbrace{p...p}^{f(r^{2n})}er$, ... , $pe\\overbrace{p...p}^{f(r^{100... | ã$peer$ æã«ã¯, $peper$ ãã, $pepper$ ãã, ... , $pe\overbrace{p...p}^{100}er$ ããã® $100$ 人ãäœãã§ããŸã. $peer$ æã®äœäººã®æ§æ Œã¯ã©ãã**çŽ çŽ**ã**ç
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