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 |
|---|---|---|---|---|---|---|---|---|---|
OMC026 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc026/tasks/155 | B | OMC026(B) | 300 | 274 | 288 | [
{
"content": "ã解ãšä¿æ°ã®é¢ä¿ãã $a+b=4,ab=8$ ã§ãããã, ç¹ã« $a^2+b^2=0$ ã§ãã. ããã§\r\n$$\\begin{aligned}\r\nf(x) &\\coloneqq x^4+px^3+qx^2+rx+s \\\\\\\\\r\n&= (x-(a+2b))(x-(2a+b))\\left(x-\\frac{a}{b}\\right)\\left(x-\\frac{b}{a}\\right) \\\\\\\\\r\n&= (x^2-3(a+b)x+2(a^2+b^2)+5ab)\\left(x^2-\\frac{a^2+b^2}{ab}x+1\\right) \\\... | ãäºæ¬¡æ¹çšåŒ $x^2-4x+8=0$ ã® $2$ 解ã $x=a,b$ ãšãããšã, å次æ¹çšåŒ $x^4+px^3+qx^2+rx+s=0$ 㯠$x=a+2b,2a+b,\dfrac{a}{b},\dfrac{b}{a}$ ã $4$ 解ã«æã¡ãŸãã. $p+q+r+s$ ã®å€ãæ±ããŠãã ãã. |
OMC026 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc026/tasks/156 | C | OMC026(C) | 300 | 268 | 285 | [
{
"content": "ãæ£ã®çŽæ°ãå¥æ°åã§ããããšã¯å¹³æ¹æ°ã§ããããšãšåå€ã§ãããã,\r\n$$n^4+24n^3=n^2(n^2+24n)$$\r\nãã $n^2+24n$ ã¯å¹³æ¹æ°ã§ãã. æ£æŽæ° $a$ ã«ãã£ãŠããã $a^2$ ãšãããš,\r\n$$(n+12)^2-144=a^2 \\iff (n+a+12)(n-a+12)=144$$\r\nã$n\\pm a+12$ ã®å¶å¥ãäžèŽããããšã«çæããã°, çµ $(n,a)$ ã®åè£ã以äžã®ããã«åæã§ãã.\r\n$$(n,a)=(25,35),(8,16),(3,9),(1,5)$$\r\nãã®ãã¡ $n^4+24n^3=(an)^2$ ãæ£... | ã$n^4+24n^3$ ãæ£ã®çŽæ°ãã¡ããã© $21$ åãã€ãããª, æ£æŽæ° $n$ ãšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC026 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc026/tasks/157 | D | OMC026(D) | 500 | 149 | 199 | [
{
"content": "ã$AD$ ã®äžç¹ã $M$ ãšããã° $AB:AM=5:3=DM:CD$ ã§ãã, $\\angle BAD=\\angle ADC$ ãšåãããŠäžè§åœ¢ $ABM$ ãš $DMC$ ã¯çžäŒŒã§ãã. ããã«ãã®ãšã, $BM:CM=5:3$ ã§ãã,\r\n$$\\angle BMC=180^\\circ-\\angle AMB-\\angle CMD=180^\\circ-\\angle AMB-\\angle ABM=\\angle BAM$$\r\nããäžè§åœ¢ $MBC$ ãåããçžäŒŒã§ãã.\\\r\nããããã£ãŠ, $AB:BM=BM:BC$ ãã $BM=10\\sqrt{13... | ãåžåè§åœ¢ $ABCD$ ã以äžã®æ¡ä»¶ãã¿ãããšã, ãã®é¢ç©ãæ±ããŠãã ãã.
$$AB=25,\ BC=52,\ CD=9,\ DA=30,\ \angle{BAD}=\angle{ADC}$$
ããã ã, $XY$ ã§ç·å $XY$ ã®é·ããè¡šããã®ãšããŸã. |
OMC026 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc026/tasks/158 | E | OMC026(E) | 600 | 72 | 137 | [
{
"content": "ãäžè¬ã« $10^7$ ã $n$ ãšãã. $[i,j]$ 㧠$i$ è¡ç® $j$ åç®ã®ãã¹ãè¡šã, $s(a,b,j)$ 㧠$[a,j]$ ãš $[b,j+1]$ ã®äžå¿ãçµã¶ç·åãè¡šã. ãã®åœ¢åŒã§è¡šãããç·åã®å
šäœã $T$ ãšã, åå $f:T\\to \\lbrace 0,1\\rbrace$ ã以äžã§å®ãã.\r\n$$f(t)= \\begin{cases} 1 & (t\\text{ãš}\\ell\\ \\text{ãå
±æç¹ãæã€ãšã})\\\\\\\\ 0 & (\\text{otherwise}) \\end{cases}$$\r\nãã®ãšã, æ±ããå€ã¯ä»¥... | ã$10^7$ è¡ $10^7+1$ åã®ãã¹ç®ããã, æãå·Šäžã®é ç¹ãšæãå³äžã®é ç¹ãçµã¶çŽç·ã
$\ell$ ãšããŸã. ããŸ, ååã«ã€ããŠã¡ããã© $1$ ãã¹ãé»ãå¡ã, é£ãåãåã®é»ããã¹ã®äžå¿ãç·åã§çµã¶ããšã§æãç· $\ell^{\prime}$ ãäœããŸã. ãã®ãšã, ãã¹ãŠã®é»ãã¹ã®å¡ãæ¹ $10^{7(10^7+1)}$ éãã«ã€ããŠ, $\ell$ ãš $\ell^{\prime}$ ã®å
±æç¹ã®åæ°ã®å¹³åãæ±ããŠãã ãã.\
ããã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ã«ãã£ãŠ $\dfrac{p}{q}$ ãšè¡šããã®ã§, $p+q$ ã解çããŠãã ãã.\
ãããã§, ãã¹ã¯ãã¹ãŠæ£æ¹åœ¢ãšã... |
OMC026 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc026/tasks/159 | F | OMC026(F) | 600 | 17 | 67 | [
{
"content": "ã$AD,BC$ ã®äžç¹ããããã $M,N$ ãšããã°, äžç·å®çãã $BM=CM=\\sqrt{113}$ ã§ãããã, $MN$ 㯠$BC$ ã«åçŽã§, $MN=10$ ã§ãã. ãã㧠$BC$ ãå«ã¿ $MN$ ã«åçŽãªå¹³é¢ã $U$ ãšã, ããã« $A,D$ ããããããåç·ã®è¶³ããããã $A^\\prime,D^\\prime$ ãšããã°, $N$ 㯠$A^\\prime D^\\prime$ ã®äžç¹ã§ããããã $A^\\prime BD^\\prime C$ ã¯å¹³è¡å蟺圢ã§ãã. ããã«åè§é $M-A^\\prime BD^\\prime C$ ã®äœç©ã¯åé¢äœ... | ã$AD=2\sqrt{5},BC=2\sqrt{13}$ ãªãåé¢äœ $ABCD$ ã¯, äœç©ã $40$ ã§, ããã«ä»¥äžã®æ¡ä»¶ãã¿ãããŸã.
$$AB^2+BD^2=AC^2+CD^2=236$$
ãäžè§åœ¢ $ABC$ ã®é¢ç©ã $33$ ã§ãããšã, äžè§åœ¢ $BCD$ ã®é¢ç©ã¯æ£æŽæ° $S$ ã«ãã£ãŠ $\sqrt{S}$ ãšè¡šãããŸã.\
ã$S$ ã解çããŠãã ãã. ãã ã, $XY$ ã§èŸº $XY$ ã®é·ããè¡šããã®ãšããŸã. |
OMC025 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc025/tasks/150 | A | OMC025(A) | 100 | 278 | 298 | [
{
"content": "ã$\\lbrace 0,1,4,9\\rbrace$ ãèããã° $n\\leq 9$ ã§, ããã« $\\lbrace 1,4,9,16\\rbrace$ ãèããããšã§ $n=9$ ã¯äžé©ã§ãã.\\\r\nãéã«, å¹³æ¹æ°ã $8$ ã§å²ã£ãäœã㯠$0,1,4$ ã®ããããã§ãããã, æ±ããæ倧å€ã¯ $\\textbf{8}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc025/editorial/150"
}
] | ã次ã®æ¡ä»¶ãã¿ããæ£æŽæ° $n$ ã®æ倧å€ãæ±ããŠãã ããïŒ
- çžç°ãªã $4$ ã€ã®å¹³æ¹æ°ãä»»æã«ãšã£ããšã, $n$ ã§å²ã£ãäœããçãã $2$ ã€ãå¿
ãååšãã.
ããã ã, ããã§**å¹³æ¹æ°**ãšã¯, ããæŽæ°ã® $2$ ä¹ã«ãã£ãŠè¡šãããæŽæ°ã®ããšãæããã®ãšããŸã. |
OMC025 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc025/tasks/151 | B | OMC025(B) | 200 | 181 | 247 | [
{
"content": "ãäžè§åœ¢ $AEH$ ãš $BDH$ ã®çžäŒŒãã $DH:EH=BH:AH=4:3$ ã§ãã. åæ§ã«ããŠ,\r\n$$DH:EH:FH=4\\times5:3\\times5:3\\times4=20:15:12$$\r\nã§ãããã, æ±ããå€ã¯ $20+15+12=\\textbf{47}$ ã§ãã.\\\r\nããªã, ãã®ãããªéè§äžè§åœ¢ $ABC$ ã®ååšã¯èšŒæã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc025/editorial/151"
}
] | ãåå¿ã $H$ ãšããéè§äžè§åœ¢ $ABC$ ã«ãããŠ, ç¹ $A,B,C$ ãã察蟺ã«ããããåç·ã®è¶³ããããã $D,E,F$ ãšãããŸã. $AH:BH:CH=3:4:5$ ã®ãšã, æ倧å
¬çŽæ°ã $1$ ã§ããæ£æŽæ° $p,q,r$ ãååšã㊠$DH:EH:FH=p:q:r$ ãšè¡šããŸã. $p+q+r$ ã解çããŠãã ãã. |
OMC025 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc025/tasks/152 | C | OMC025(C) | 300 | 67 | 128 | [
{
"content": "ã$3000$ åã®ããŒã«ããã, $1000$ åãã€ãåãè²ã§å¡ãããŠããç¶æ³ãèãã. ãã®ãã¡ $1500$ åãéžã¶æ¹æ³ã¯ $\\_{3000}\\mathrm{C}\\_{1500}$ éããã. äžæ¹ã§, åè²ããšã«ç¬ç«ããŠèããããšã§, ãã㯠$M$ éãã«ãçããããšãããããã, æ±ããå€ã¯Legendreã®å®çãã $2999\\times 2^{2993-2\\times1493}=\\mathbf{383872}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/conte... | ã$i+j+k=1500$ ã〠$i,j,k\leq 1000$ ãªãéè² æŽæ°ã®é åºä»ããçµ $(i,j,k)$ ãã¹ãŠã«ã€ããŠ,
$$\_{1000}{\mathrm{C}}\_{i}\times{}\_{1000}{\mathrm{C}}\_{j}\times{}\_{1000}{\mathrm{C}}\_{k}$$
ã®ç·åã $M$ ãšããŸã. $M$ ãå²ãåãæ倧ã®çŽ æ°ãš, $M$ ãå²ãåãæ倧㮠$2$ ã¹ãã®**ç©**ãæ±ããŠãã ãã.\
ãäŸãã° $M=3080=2^3\times5\times7\times11$ ã§ãã£ããªãã°, 解çãã¹ãå€ã¯ $2^3\times11=88$ ã§ã.\
ããªã, [... |
OMC025 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc025/tasks/153 | D | OMC025(D) | 400 | 67 | 107 | [
{
"content": "ã$3$ ã€ã®æ£æŽæ°è§£ã $m\\leq n\\leq l$ ãšãããš, 解ãšä¿æ°ã®é¢ä¿ãã\r\n$$\\begin{aligned}(m+n)^2+(m+l)^2+(n+l)^2&=2[(m+n+l)^2-(mn+nl+lm)]\\\\\\\\\r\n&=2[(a+14)^2-(a^2+28a-1)]=394\\end{aligned}$$\r\nãããã®æ£æŽæ°è§£ãèã㊠$(m+n,m+l,n+l)=(5,12,15),(9,12,13)$ ãã\r\nãã$$(m,n,l)=(1,4,11),(4,5,8)$$\r\nãåŸããã, æ±ããç·åã¯åã³è§£ãšä¿æ°ã®é¢ä¿ãã $1\\tim... | ãå®æ° $a,b$ ã«ã€ããŠ, $x$ ã®äžæ¬¡æ¹çšåŒ
$$x^3+(a+14)x^2+(a^2+28a-1)x=b$$
ã®è§£ããã¹ãŠæ£æŽæ°ã§ãããšã, $b$ ãšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã.\
ããã ã, æ±ããç·åã¯éè² æŽæ°å€ã«ãªãããšã蚌æã§ããŸã. |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/144 | A | OMC024(A) | 200 | 239 | 290 | [
{
"content": "**解ç1.**ãåå³ã®åé
ãç¬ç«ã«èããã°ãã. ããå³ã®é£Žã«ã€ããŠ, A,Båã«äžã€ãåé
ããªãæ¹æ³ããããã $51$ éããããã, ãããã«ãåé
ããªãæ¹æ³ã®éè€ãèããã° $M=(2\\times 51-1)^7=101^7$ ãåŸã.\r\n\r\n**解ç2.**ã倩äžãçã ã, 以äžã®ãåé¡ããèããã. å
ã®åé¡ã¯ $k=7,a_1=\\cdots=a_7=50$ ã®å Žåã«ç䟡ã§ãã.\r\n\r\n**åé¡.**ãæ£æŽæ° $n$ ã $p_1^{a_1}p_2^{a_2}\\cdots p_k^{a_k}$ ãšçŽ å æ°å解ããããšãã. çžç°ãªããšã¯éããªãäºã€ã® $n... | ã$7$ çš®é¡ã®å³ã®é£Žããããã $50$ åãã€ãããŸã. åãå³ã®é£Žãåºå¥**ããªã**ãšã, ããã $350$ åã®é£ŽãããããAå, Bå, Cåã®ããããã«**ãã¹ãŠ**åé
ããæ¹æ³ã§ãã£ãŠ, AåãšBåãåãå³ã®é£Žãå
±æããªããã®ã¯ $M$ éããããŸã.\
ã$M$ ã解çããŠãã ãã. ãã ã, äžã€ã风ãããããªã人ãååšããããšãèš±ããã®ãšããŸã. |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/145 | B | OMC024(B) | 400 | 140 | 250 | [
{
"content": "ã$f(0)\\neq 0$ ã®ãšã, $b=0$ ããä»»æã® $a$ ã«ã€ã㊠$f(a)=1$ ã§ãããã, ä»¥äž $f(0)=0$ ãšãã. ããã« $f(1)\\neq 1$ ã®ãšã, $b=1$ ããä»»æã® $a$ ã«ã€ã㊠$f(a)=0$ ã§ãããã, ä»¥äž $f(1)=1$ ãšãã.\\\r\nã$f(2)^3=f(8)$ ããã³ $f(3)^2=f(9)$ ããããã $S$ ã«å±ããããšãã, $f(2)\\leq 2$ ããã³ $f(3)\\leq 3$ ãåŸã. ç¹ã« $f(2)=2$ ã®ãšã $f(5)$ 㯠$6$ 以äžã§ãã. éã«ãã®ãšã, $f(4),f(... | ãéå $\lbrace 0,1,2,\cdots,12\rbrace$ ã $S$ ãšãããŸã.\
ãé¢æ° $f:S\to S$ ã§ãã£ãŠ, $ab\leq 12$ ãªã $a,b\in S$ (çãããŠãè¯ã)ã«å¯ŸããŠ
$$f(ab)=f(a)f(b)$$
ãã¿ãããã®ã¯ $M$ åååšããŸã. $M$ ã解çããŠãã ãã. |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/146 | C | OMC024(C) | 500 | 19 | 79 | [
{
"content": "ãäžè§åœ¢ã®äžç·ãšçè§å
±åœ¹ã®é¢ä¿ã«ããçŽç· (ãã®åé¡ã§ã¯ $ABC$ ã«ãããçŽç· $AO$) 㯠$\\textit{symmedian}$ (æ¬äŒŒäžç·) ãšåŒã°ã, æ§ã
ãªæ§è³ªãç¥ãããŠãã. ãã®è£é¡ã¯, ãããã®åºçºç¹ãšãèšãã¹ã, ç¹ã«äž»ãããã®ã§ãã.\r\n\r\n**è£é¡.**ãäžè§åœ¢ $ABC$ ã®å€æ¥åã $\\Omega$ ãšã, $B,C$ ã§ã® $\\Omega$ ã®æ¥ç·ã®äº€ç¹ã $X$ ãšãããš, $\\angle BAX=\\angle CAM$.\r\n\r\n**蚌æ.**ã蟺 $BC$ äžã§ $\\angle BAX=\\angle CAM^\\prim... | ãç¹ $O$ ãäžå¿ãšããååŸ $20$ ã®å®å $\Gamma$ ããã³ $OA=21$ ãªãå®ç¹ $A$ ããããŸã. $AB\neq AC$ ãªã $\Gamma$ äžã®ç¹ $B,C$ ã«ã€ããŠ, ç·å $BC$ ã®äžç¹ã $M$ ãšãããš, åçŽç· $AO$ 㯠$\angle BAC$ ã®å
åŽã«ãã, ã〠$\angle BAO=\angle CAM$ ãæç«ããŸãã. ãã®ãšã, $B,C$ ã®ãšãæ¹ã«ããã, äžè§åœ¢ $ABC$ ã®å€å¿ã¯åžžã«ããçŽç· $\ell$ äžã«ããããšã蚌æã§ããŸã.\
ã$O$ ãš $\ell$ ã®è·é¢ãæ±ããŠãã ãã. ãã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $x,y$ ã«ãã£ãŠ $\dfr... |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/147 | D | OMC024(D) | 600 | 65 | 197 | [
{
"content": "ãäžè¬ã« $10^6$ ã $2n$ ãšããã°, $M$ ã¯å€é
åŒ $(x+y+z+w+v+u)^{2n}$ ã«ãããŠåæåããã¹ãŠå¶æ°ã¹ãã§ãããããªé
ã®ä¿æ°ã®åã«çããããšã«çæãã. ããã«\r\nãã$$\\displaystyle f(x,y,z,w,v,u)=\\frac{1}{2^6}\\sum_{\\lbrace1,-1\\rbrace^6} (ix+jy+kz+lw+mv+nu)^{2n}$$\r\nãšããã°, $M$ 㯠$f$ ã®åé
ã®ä¿æ°ã®åã«çãã, ããã¯ããªãã¡ $f(1,1,1,1,1,1)$ ã§äžãããã. ãããã£ãŠ,\r\nãã$$\\begin{al... | ã$13$ 以äžã®**çŽ æ°** $10^6$ åãããªã**é åºä»ãã**çµ $(a_{1},a_{2},\cdots,a_{10^6})$ ã§ãã£ãŠ, ããããã¹ãŠã®ç©ãå¹³æ¹æ°ã§ãããã®ã¯ $M$ åååšããŸã. $M$ ã $1000$ ã§å²ã£ãäœããæ±ããŠãã ãã. |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/148 | E | OMC024(E) | 700 | 41 | 73 | [
{
"content": "ç»å ŽããåŒã¯ãã¹ãŠæ次åŒã§ãããã, ãã¹ãŠã®å€æ°ã $10$ ã§å²ã£ãŠèããŠãè¯ã. ãã®ãšãæ¡ä»¶ã¯\r\nãã$$|P-Q|\\geq 1, \\quad |Q-R|\\geq 1, \\quad |R-P|\\geq 1$$\r\nãã®ãšã, æ±ããæå°å€ã¯ $4\\/3$ ã§ããããšã瀺ã. \r\n\r\n---\r\n\r\n**解ç1.**ã$(a,b,c,x,y,z)$ ãããããå®æ° $k,l$ ã«ãã£ãŠ\r\n$$(a+k,b+k,c+k,x+l,y+l,z+l)$$\r\nã«çœ®ãæããŠã $|P-Q|,|Q-R|,|R-P|$ ã¯ããããäžå€ã§ãã. ããã«çæããŠ,... | ãå®æ° $a,b,c,x,y,z$ ã«å¯Ÿã,
$$P=ax+by+cz,\quad Q=ay+bz+cx,\quad R=az+bx+cy$$
ã§å®ãŸã $3$ æ°ã
ãã$$|P-Q|\geq 100, \quad |Q-R|\geq 100, \quad |R-P|\geq 100$$
ãã¿ãããšã, 以äžã®åãåŸãæå°å€ãæ±ããŠãã ãã.
$$(a^2+b^2+c^2)(x^2+y^2+z^2)$$
ããã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $m,n$ ã«ãã£ãŠ $\dfrac{m}{n}$ ãšè¡šãããã®ã§, $m+n$ ã解çããŠãã ãã. |
OMC024 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc024/tasks/149 | F | OMC024(F) | 800 | 4 | 18 | [
{
"content": "ãæ¡ä»¶ $AP=22$ ã¯å®ã¯äžèŠã§ãã. 以äžããã解é€ã, $P$ ã¯çŽç· $AI$ äžãä»»æã«åããšãã.\r\n\r\n**è£é¡1.**ã$I$ ã¯äžè§åœ¢ $PQR$ ã®åå¿ã§ãã.\r\n\r\n**蚌æ.**ã$\\angle IPQ=\\angle APQ=\\angle ABQ=\\angle ABI=\\angle ABC\\/2$ ãªã©ããåŸã.\r\n\r\nããŸãäžã®èšŒæãã, äžè§åœ¢ $PQR$ 㯠$P$ ã®äœçœ®ã«ãããåžžã«çžäŒŒã§ãã.\r\n\r\n**è£é¡2.**ãäžè§åœ¢ $ABC$ ã®å€å¿ã $O$ ãšãããšã, äžè§åœ¢ $PQR$ ã®ãªã€ã©ãŒç·ã¯çŽç· $... | ã$AB=20,AC=21$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, å
å¿ã $I$ ãšã, å
æ¥åãšèŸº $AB,AC$ ã®æ¥ç¹ããããã $F,E$ ãšããŸã. çŽç· $AI$ äžã® $I$ ã«ã€ã㊠$A$ ãšå察åŽã« $AP=22$ ãªãç¹ $P$ ããšã, äžè§åœ¢ $ABP$ ã®å€æ¥åãšçŽç· $BI$ ã®äº€ç¹ã $Q(\neq B)$, äžè§åœ¢ $ACP$ ã®å€æ¥åãšçŽç· $CI$ ã®äº€ç¹ã $R(\neq C)$ ãšããŸã.\
ãçŽç· $BC,EF$ ããã³äžè§åœ¢ $PQR$ ã®ãªã€ã©ãŒç·ãäžç¹ã§äº€ãããšã, $BC$ ã®é·ããæ±ããŠãã ãã. ãã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $x,y$ ã«ãã£ãŠ $\dfrac{x}{y... |
OMC023 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc023/tasks/140 | A | OMC023(A) | 100 | 308 | 308 | [
{
"content": "ãæããã«äž¡è
ã®åã€ç¢ºçã¯çãã. ãããã£ãŠ, åŒãåããšãªã確ç㯠$\\dfrac{6}{6^2}$ ã§ããããšã«çæããã°, toriiåãåã€ç¢ºç㯠$\\dfrac{1}{2}\\times\\left(1-\\dfrac{6}{6^2}\\right)=\\dfrac{5}{12}$ ã§ãã, æ±ããå€ã¯ $a+b=\\textbf{17}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc023/editorial/140"
}
] | ãtoriiåãštorioåããµã€ã³ãã§åè² ãããŸã. å
·äœçã«ã¯, $1$ ãã $6$ ã®ç®ãç確çã§åºããµã€ã³ãããããã $1$ åãã€æ¯ã, 倧ããç®ãåºããæ¹ãåã¡ãšããŸã. ãã ã, åãç®ãåºãå Žåã¯åŒãåããšãªã, åè² ã¯ã€ããŸãã.\
ããã®ãšã, toriiåãåã€ç¢ºçãæ±ããŠäžãã.\
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC023 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc023/tasks/141 | B | OMC023(B) | 200 | 306 | 308 | [
{
"content": "ãçžå ã»çžä¹å¹³åã®é¢ä¿ãã $a+b+c\\geq3\\sqrt[3]{abc}\\gt20$ ã§ãã.\\\r\nãéã« $(a,b,c)=(5,6,10)$ ã®ãšã $abc=300$ ã〠$a+b+c=21$ ãã¿ãããã, æ±ããæå°å€ã¯ $\\textbf{21}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc023/editorial/141"
}
] | ãæ£ã®æŽæ° $a,b,c$ ã $abc=300$ ãã¿ãããšã, $a+b+c$ ã®ãšãåŸãæå°å€ãæ±ããŠãã ãã. |
OMC023 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc023/tasks/142 | C | OMC023(C) | 300 | 253 | 281 | [
{
"content": "ã$\\dfrac{a_{1}}{1},\\dfrac{a_{2}}{2},\\cdots,\\dfrac{a_{n}}{n}$ ã®äžã§ã®æå°å€ã $m$ ãšãããš, å $k=1,2,\\cdots,n$ ã«ã€ã㊠$\\dfrac{a_k}{k}\\geq m$ ãã\r\nãã$$2021=a_1+a_2+\\cdots+a_n\\geq (1+2+\\cdots +n)m=\\dfrac{n(n+1)}{2}m$$\r\nããªãã¡ $m\\leq\\dfrac{4042}{n(n+1)}$ ãåŸã. çå·ã¯å $k=1,\\cdots,n$ ã«ã€ã㊠$a_k=\\dfrac{40... | ãç·åã $2021$ ã§ãããããªæ£ã®å®æ° $a_1,\dots,a_n$ ã«ã€ããŠ, $\dfrac{a_{1}}{1},\dfrac{a_{2}}{2},\dots,\dfrac{a_{n}}{n}$ ã®äžã§ã®æå°å€ãšããŠããåŸãæ倧å€ã $M(n)$ ãšãããŸã. $M(n)\leq 1$ ãªãæå°ã®æ£æŽæ° $n$ ãæ±ããŠãã ãã. |
OMC023 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc023/tasks/143 | D | OMC023(D) | 400 | 93 | 189 | [
{
"content": "ã$C_1$ ãš $C_2$ ã® $A$ 以å€ã®äº€ç¹ã $X$ ãšã, $A$ ãéã $AX$ ã«åçŽãªçŽç·ã $\\ell^\\prime$ ãšãã.\r\n\r\nã**è£é¡.**ãç·å $PQ$ ã®é·ããæ倧ãšãªãã®ã¯ $\\ell=\\ell^\\prime$ ã®ãšãã§ãã.\r\n\r\nã**蚌æ.**ã$\\ell^\\prime$ ãš$O_1,O_2$ ã®äº€ç¹ã§ãã£ãŠ $A$ ã§ãªãæ¹ããããã $P^\\prime,Q^\\prime$ ãšãããš, ååšè§ã®å®çãã $XPQ$ ãš $XP^\\prime Q^\\prime$ ã®çžäŒŒã容æã«ããã. ããã« $X$ ã... | ã$AB=2,BC=\sqrt{6},CA=1$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, ç¹ $A,B$ ãéã蟺 $BC$ ã«æ¥ããåã $O_1$, ç¹ $A,C$ ãéã蟺 $BC$ ã«æ¥ããåã $O_2$ ãšããŸã. çŽç· $\ell$ ãç¹ $A$ ãéããªããåã, ãã® $O_1,O_2$ ãšã®äº€ç¹ã®ãã¡ $A$ ã§ãªãæ¹ããããã $P,Q$ ãšãããšã, ç·å $PQ$ ã®é·ããšããŠããåŸãæ倧å€ãæ±ããŠãã ãã.\
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£ã®æŽæ° $a,c$ ãš, $1$ ãã倧ããå¹³æ¹æ°ã§å²ãåããªãæ£ã®æŽæ° $b$ ãçšã㊠$\dfrac{a\sqrt{b}}{c}$ ãšè¡šãããã®ã§... |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/134 | A | OMC022(A) | 200 | 339 | 355 | [
{
"content": "ã$S(n)=\\dfrac{n(n+1)}{2}$ ã«çæããã°, äžåŒã¯ä»¥äžã®ããã«è¡šããã.\r\nãã$$S(1)\\times S(2)\\times \\cdots \\times S(100)=\\dfrac{1\\times2}{2}\\times\\dfrac{2\\times3}{2}\\times\\cdots\\times\\dfrac{100\\times101}{2}=\\dfrac{100!\\times101!}{2^{100}}$$\r\nLegendreã®å®çãã以äžãæãç«ã€ãã, æ±ããåæ°ã¯ $97\\times2-100=\\textbf{94}... | ã$S(n)$ 㧠$1$ ä»¥äž $n$ 以äžã®æŽæ°ã®ç·åãè¡šããšãïŒ$S(1)\times S(2)\times \cdots \times S(100)$ 㯠$2$ 㧠$x$ åå²ãåããŸãïŒ$x$ ãšããŠããããæ倧ã®æŽæ°å€ã解çããŠãã ãã. |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/135 | B | OMC022(B) | 300 | 269 | 329 | [
{
"content": "**解ç1.**ãå°çã®ååŸã $r$ ãšãã. ãŸã, ç¥ç¶ã®å®¶ã®ããå°ç¹ã $A$, èµ€éäžã®è¥¿çµ $165$ 床ã®å°ç¹ã $B$, OMCåã®èªå®
ã®ããå°ç¹ã $C$ ãšãã. å°çã®äžå¿ $O$ ãåç¹ãšã, $O$ ãã $A$ ã«åããæ¹ã $x$ 軞ã®æ£ã®åã, $O$ ãã $B$ ã«åããæ¹ã $y$ 軞ã®æ£ã®åã, $O$ ããå極ã«åããæ¹ã $z$ 軞ã®æ£ã®åããšãã $3$ 次å
çŽäº€åº§æšãèãã.\\\r\nããã®åº§æšã«ãã㊠$A$ 㯠$(r,0,0)$ ã§ãã. ãŸã, èµ€éäžã®æ±çµ $150$ 床ã®å°ç¹ã $\\left(\\dfrac{r}{\\sqrt... | ãOMCåã¯, ææ島ã«ããèªå®
ããã·ã³ã¬ããŒã«ã«ããç¥ç¶ã®å®¶ãŸã§, ãã©ã€ããŒããžã§ããã䜿ã£ãŠè¡ãããšã«ããŸãã. OMCåã®èªå®
ãåç·¯ $45$ 床, æ±çµ $150$ 床ã®å°ç¹ã«ãã, ç¥ç¶ã®å®¶ãèµ€éäžã®æ±çµ $105$ 床ã®å°ç¹ã«ãããšã, $x$ kmã®è·é¢ãé£ã¶å¿
èŠããããŸã. $x$ ã解çããŠãã ãã.\
ããã ã, å°çãå®å
šãªçäœãšã¿ãªã, èµ€é $1$ åšã®é·ãã¯ã¡ããã© $40000$ kmã§ãããšããŸã. ãŸã, OMCåã¯å°çã®è¡šé¢äžãæçè·é¢(倧åã³ãŒã¹)ã§é²ããã®ãšã, 解çã¯**åã®äœãåæšäºå
¥ããŠçŸã®äœãŸã§ã®æŠæ°ã§**è¡ã£ãŠãã ãã. äŸãã°çãã $9876.5$ kmã§ãããšã, $990... |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/136 | C | OMC022(C) | 400 | 156 | 273 | [
{
"content": "ã空ç©ã $1$ ãšã¿ãªãããšã§, ããŸç©ºã®ç®±ãååšããããšãèš±ããŠèãã. ãŸã $2,4,6,8,10$ ã«æ³šç®ãããš, ãããã®åé
ã®å¿
èŠååæ¡ä»¶ã¯\r\n$2$ ã€ä»¥äžã®ç®±ãçšããããšã§ãããã, $3\\times 2^5-3=93$ éãã§ãã. ç¶ã㊠$3,6,9$ ãããããå¥ã®ç®±ã«å
¥ããŠã¯ãªããªãããšã«çæããã°, $3,9$ ã®å
¥ãæ¹ã¯ $7$ éãã§ãã. $1,5,7$ ã¯ã©ã®ããã«åé
ããŠããããã, 以äžãã空ã®ç®±ãèš±ããŠã®å Žåã®æ°ã¯ $93\\times 7\\times 3^3=17577$ éãã§ãã.\\\r\nããã£ãŠ, 空ã®ç®±ãé€å€ããå Žåã¯, å
... | ã$1$ ãã $10$ ã®æŽæ°ã $1$ ã€ãã€æžãããããŒã« $10$ åã, $3$ ã€ã®åºå¥ã§ããç®± $A,B,C$ ã®ããããã«, 以äžã® $2$ æ¡ä»¶ãæºããããã«å
¥ããæ¹æ³ã¯äœéããããŸããïŒ
- $3$ ã€ã®ç®±ããããã«, å°ãªããšã $1$ ã€ã®ããŒã«ãå
¥ã£ãŠãã.
- $A,B,C$ ã«å
¥ããããŒã«ã«æžãããæ°ã®ç·ç©ããããã $a, b, c$ ãšãããšã, ããã $3$ æ°ã®æ倧å
¬çŽæ°ã¯ $1$ ã§ãã. |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/137 | D | OMC022(D) | 400 | 141 | 194 | [
{
"content": "ã$l_n$ 㯠$y=nx-n^2+n$ ãšè¡šããããã, $l_a$ ãš $l_b$ ã®äº€ç¹ã¯ $(a+b-1, ab)$ ã§ãã. ãã£ãŠ $3$ ç¹\r\nãã$$(a+b-1, ab), (b+c-1, bc), (c+a-1,ca)$$\r\nãé ç¹ãšããäžè§åœ¢ã®é¢ç©ãæ±ããã°ãã, ãã®å€ã $S$ ãšããã°, 以äžã®ããã«èšç®ã§ãã.\r\nãã$$S=\\dfrac{1}{2}|(a-b)(b-c)(c-a)|$$\r\nãäžè¬æ§ã倱ãã $a\\gt b\\gt c$ ãšããŠãã, $X=a-b, Y=b-c$ ãšããã° $S=\\dfrac{1}{2}XY(X+Y)$... | ã$xy$ å¹³é¢ã«ãããŠ, ç¹ $(n, n)$ ãéãåŸã $n$ ã®çŽç·ã $l_n$ ã§è¡šããŸã. äŸãã° $l_3$ 㯠$y=3x-6$ ã§ã.\
ãçžç°ãªãæŽæ°ã®çµ $(a,b,c)$ ã«ã€ããŠ, $l_a, l_b, l_c$ ããªãäžè§åœ¢ã®é¢ç©ãšããŠããåŸãå®æ°å€ã®ãã¡, $50$ 以äžã§ãããã®ã¯ $M$ åãããŸã.\
ã$M$ ã解çããŠãã ãã. ãã ã, ãã®ãããªå€ã¯æéåã§ããããšã蚌æã§ããŸã. |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/138 | E | OMC022(E) | 600 | 128 | 166 | [
{
"content": "**解ç1.**ãç¹ $C$ ãäžå¿ãšã, ç¹ $B$ ã $A$ ã«ç§»ãããã« $\\triangle{ABC}$ ã $60^\\circ$ å転移åãã, ç¹ $P$ ã«å¯Ÿå¿ããç¹ $Q$ ãåãïŒãã®ãšã $AQ^2+PQ^2=BP^2+CP^2=AP^2$ ãã $\\angle AQP=90^\\circ$ ã§ããïŒãããã£ãŠ $\\angle BPC=\\angle AQC=90^\\circ+60^\\circ=150^\\circ$ ã§ããïŒãã㧠$\\triangle{ABC}$ ã®äžèŸºã®é·ãã $l$ ãšãããš, äœåŒŠå®çãªã©ãã\r\nãã$$AP^2+BP^2=... | ãé¢ç© $S$ ã®æ£äžè§åœ¢ $ABC$ ã«ãããŠ, ãã®å
éšã®ç¹ $P$ ã以äžã®çåŒãã¿ãããŸãã.
$$AP^2+BP^2=AB^2,\ \ BP^2+CP^2=AP^2$$
ããã®ãšã, äžè§åœ¢ $PAB, PBC, PCA$ ã®é¢ç©ã¯ãããã $\dfrac{a}{b}S, \dfrac{c}{d}S, \dfrac{e}{f}S$ ãšè¡šããŸã. ãã ã $a,b,c,d,e,f$ ã¯æ£æŽæ°ã§ãã, $a$ ãš $b$, $c$ ãš $d$, $e$ ãš $f$ ã¯ããããäºãã«çŽ ã§ã. $abcdef$ ($6$ æ°ã®ç©)ã解çããŠãã ãã. |
OMC022 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc022/tasks/139 | F | OMC022(F) | 600 | 14 | 75 | [
{
"content": "ã$n=2021$ ãšããããšãïŒä»¥äžã®å€åœ¢ã«çæããïŒ\r\nãã$$\\begin{aligned}x^2+4xy+8y^2=10^{n}&\\iff 4xy = 10^{n}-x^2-8y^2\\\\\\\\\r\nãã&\\implies (4xy)^2=(10^{n}-x^2-8y^2)^2\\\\\\\\\r\nãã&\\iff(x^2-10^{n})^2+(8y^2-10^{n})^2=10^{2n}\\end{aligned}$$\r\nãããã«ïŒ$(a-10^{n})^2+(8b-10^{n})^2=10^{2n}$ ã®æŽæ°è§£ $(a,b)$ ã«å¯ŸããŠïŒé¡æãã¿ããçµ $... | ã$x^2+4xy+8y^2=10^{2021}$ ãã€ïŒ$x^2, y^2$ ããšãã«æŽæ°ã§ãããããªïŒè€çŽ æ°ã®çµ $(x,y)$ 㯠$M$ åãããŸãïŒ$M$ ã解çããŠãã ããïŒ\
ããã ãïŒãã®ãããªçµã¯æéåã§ããããšã蚌æã§ããŸãïŒ |
OMC021 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc021/tasks/130 | A | OMC021(A) | 100 | 403 | 417 | [
{
"content": "ã$x^{4}+y^{4}\\gt0$ ã«çæããã°,\r\nãã$$x^{4}+y^{4}=\\sqrt{(x^{4}-y^{4})^{2}+4(xy)^{4}}=\\sqrt{68}=2\\sqrt{17}$$\r\nãã $x^{4}=\\dfrac{1}{2}\\left[(x^{4}-y^{4})+(x^{4}+y^{4})\\right]=1+\\sqrt{17}$ ããã ã¡ã«ããã. ãããã£ãŠ, æ±ããå€ã¯ $\\textbf{18}$ ã§ãã.\\\r\nããªã $x^4y^4=16$ ãšå©çšããããšã«æ°ä»ãã°, äºæ¬¡æ¹çšåŒã解ããŠããã.",
"text": "... | ãå®æ° $x,y$ ã以äžãã¿ãããŠããŸã.
$$x^{4}-y^{4}=xy=2$$
ãã®ãšã, $x^{4}$ ã¯æ£ã®æŽæ° $a,b$ ãçšã㊠$a+\sqrt{b}$ ãšè¡šãããŸã. $a+b$ ã解çããŠãã ãã. |
OMC021 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc021/tasks/131 | B | OMC021(B) | 200 | 319 | 407 | [
{
"content": "ã$a,b$ 㯠$10^{10}$ ã®çŽæ°ã§ããããšãã, 以äžã®ããã«è¡šãã.\r\nãã$$a=2^p5^q,\\ b=2^r5^s\\ (0\\leq p,q,r,s\\leq10)$$\r\nããã®ãšã, æå°å
¬åæ°ã®æ¡ä»¶ã¯\r\nãã$$\\max\\lbrace p,r\\rbrace=\\max\\lbrace q,s\\rbrace=10$$\r\nãšè¡šçŸã§ã, $a\\leq b$ ãç¡èŠããã°ãã®ãã㪠$(p,q,r,s)$ ã®çµã¯ $(11\\times 2-1)^2=441$ éããã. ãã®ãã¡ $a=b$ ãªãçµã¯ã¡ããã©äžã€ååšããããšã«çæããã°, æ±ã... | ã$a\leq b$ ãªãæ£æŽæ°ã®çµ $(a,b)$ ã§ãã£ãŠ, $a$ ãš $b$ ã®æå°å
¬åæ°ã $10^{10}$ ãšãªããã®ã¯ããã€ãããŸããïŒ |
OMC021 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc021/tasks/132 | C | OMC021(C) | 300 | 295 | 373 | [
{
"content": "ã$x$ ãŸã㯠$y$ ã $n$ æå䞊ã¹ãæ¹æ³ã§ãã£ãŠ, åãæåã $3$ ã€é£ç¶ããªããããªãã®ãèãã. ããã«æ«ç«¯ã® $2$ æåãåãã§ãããã®ã®ç·æ°ã $a_n$ ãšãã, ããã§ãªããã®ã®ç·æ°ã $b_n$ ãšãããš, $a_2=b_2=2$ ã§ãã, æŽæ° $n\\geq 3$ ã«å¯ŸããŠä»¥äžã®æŒžååŒãæç«ããããšã容æã«ããã.\r\nãã$$a_n=b_{n-1},\\ \\ b_n=a_{n-1}+b_{n-1}$$\r\nããã®ãšã, æ±ãã確ç㯠$\\dfrac{a_{10}+b_{10}}{2^{10}}$ ã§äžãããããã, ãã㯠$\\dfrac{1... | ãè¡šãšè£ãç確çã«åºãã³ã€ã³ã $10$ åæã, äžåºŠãè¡šãŸãã¯è£ã $3$ å以äžé£ç¶ããŠåºãªã確çãæ±ããŠãã ãã.\
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle\frac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC021 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc021/tasks/133 | D | OMC021(D) | 400 | 11 | 133 | [
{
"content": "ãäžè§åœ¢ $ABC$ ã® $\\angle A$ å
ã®åå¿ã $J$ ãšããã°, $\\angle IBJ=90^{\\circ}=\\angle ICJ$ ãã $4$ ç¹ $I,B,J,C$ ã¯å
±åã§ãã. ãã®ãšã, æ¹ã¹ãã®å®çãã以äžãæãç«ã€.\r\nãã$$ID\\times DJ=BD\\times DC=XD\\times YD$$\r\næ¡ä»¶ãã $ID=DX$ ã§ãããã $DJ=DY$ ãåŸã, ç¹ã« $IJ=XY=11$ ã§ãã. ããã«æ£åŒŠå®çãã\r\nãã$$\\sin\\angle BIC=\\dfrac{BC}{IJ}=\\dfrac{10}{11}$$... | ãå
å¿ã $I$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, çŽç· $AI$ ãšèŸº $BC$ ã®äº€ç¹ã $D$ ãšããã°, äžè§åœ¢ $ABC$ ã®å€æ¥åäžã®ç¹ $X$ ã $IDïŒDX$ ãã¿ãããŸãã. ãã®ãšã, çŽç· $DX$ ãšäžè§åœ¢ $ABC$ ã®å€æ¥åã®äº€ç¹ã®ãã¡ $X$ ã§ãªãæ¹ã $Y$ ãšããã°, 以äžãæãç«ã¡ãŸãã.
$$IXïŒ5,\ BCïŒ10,\ XYïŒ11$$
ããã®ãšã, äžè§åœ¢ $ABC$ ã®å€æ¥åã®ååŸã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle\sqrt{\frac{a}{b}}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/124 | A | OMC020(A) | 100 | 450 | 459 | [
{
"content": "**解ç1.**ãäžåŒãå€åœ¢ããã° $0=4a^4-4a^2+1=(2a^2-1)^2$ ãåŸããã, ç¹ã« $a=\\dfrac{1}{\\sqrt{2}}$ ã§ãã. ãã£ãŠ\r\nãã$$M=\\left(8a^3+\\dfrac{1}{a^3}\\right)^2=\\left(\\dfrac{8}{(\\sqrt{2})^3}+(\\sqrt{2})^3\\right)^2=\\textbf{32}$$\r\n**解ç2.**ã$\\displaystyle S=2a+\\frac{1}{a}$ ãšããã°, $\\displaystyle 4=4a^2+\\frac{1}{a^2... | ãæ£ã®å®æ° $a$ ã $4a^2+\dfrac{1}{a^2}=4$ ãã¿ãããšã, $M=\left(8a^3+\dfrac{1}{a^3}\right)^2$ ã¯æŽæ°å€ã§ã. $M$ ã解çããŠãã ãã. |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/125 | B | OMC020(B) | 200 | 322 | 434 | [
{
"content": "ãäžè¬ã« $2020$ ã $N$ ã«çœ®ãæããŠèãã. ãã®ãšã, åã«å
æ¥ããæ£ $N+1$ è§åœ¢ã«å¯Ÿã, æèšåãã« $x$ åé£ã®é ç¹ãé ã«çµãã§ãã£ããã®ãå
ç·ã®çµè·¯ãšããŠåŸããã. ãã ã, $x$ 㯠$N+1$ 以äžã§ $N+1$ ãšäºãã«çŽ ãªæ£æŽæ°ã§ãã. ããã«å $x$ ã«å¯Ÿã, $x$ ã $N+1-x$ ãšçœ®ãæãããã®ã¯åäžã®æš¡æ§ãšãªãããšã«çæããã°, çã㯠$\\varphi$ ããªã€ã©ãŒã®ããŒã·ã§ã³ããšããã° $\\varphi(N+1)\\/2$ ã§ãã, ç¹ã« $N=2020$ ã®ãšã $\\textbf{966}$ ã§ãã.",
"text... | ãååšäžã®äžç¹ããå
éšã«åãã£ãŠå
ç·ãçºãããšãã, å
ç·ã¯ååšã§ $2020$ ååå°ããŠ, **åããŠ**å
ã®äœçœ®ã«æ»ã£ãŠããŸãã. å
ç·ã®çµè·¯ãã€ããæš¡æ§ãšããŠããåŸããã®ã¯ $M$ éããããŸã. $M$ ã解çããŠãã ãã.\
ããã ãå転ããŠäžèŽãããã®ã¯åäžèŠããŸã.\
ãäŸãšããŠ, æäžã® $2020$ ã $4$ ã«çœ®ãæããå Žå, ããåŸãæš¡æ§ã¯ä»¥äžã® $2$ éãã§ã.
 |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/126 | C | OMC020(C) | 300 | 181 | 338 | [
{
"content": "ã$a_N\\leq7$ ã§ããè¯ãæ°ã¯é«ã
$14$ æ¡ã§ãããã, $15$ æ¡ã®è¯ãæ°ã«ã€ã㊠$a_N=8$ ãŸã㯠$9$ ã§ãã.\r\n\r\n(i) $a_N=9$ ã®ãšã\r\n\r\nã$123456789876543210$ ãã $9$ 以å€ã® $3$ ã€ã®æ°åãæ¶ãããšãèããã°ãã. ããã®æ¡å㯠$81$ ã§ãããã, $3$ ã®åæ°ãåŸãã«ã¯, æ¶ã $3$ ã€ã®æ°åã®åã $3$ ã®åæ°ã§ããå¿
èŠããã. ããªãã¡, æ¶ãæ°åã $3$ ã§å²ã£ãäœãã\r\nãã$$\\lbrace0,0,0\\rbrace,\\lbrace1,1,1\\rbrace,\... | ãæ£ã®æŽæ° $n$ ã $k$ æ¡ã®**è¯ãæ°**ã§ãããšã¯, 次ã®æ¡ä»¶ãã¿ããããšãæããŸã.
- $n$ ã $10$ é²æ³è¡šèšã§ $\overline{a_1a_2\cdots a_{k-1}a_k}$ ãšè¡šãããšã, ããæŽæ° $N(1\lt N\lt k)$ ãååšããŠä»¥äžãæç«ãã.
$$a_1\lt a_2\lt \cdots\lt a_{N-1}\lt a_N\gt a_{N+1}\gt\cdots\gt a_{k-1}\gt a_k$$
ãã ãå $i=1,\cdots,k$ ã«ã€ã㊠$0\leq a_i\leq 9$ ã§ãã, ç¹ã« $a_1\neq 0$ ã§ãã.
ãäŸãã° $12321$ ... |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/127 | D | OMC020(D) | 400 | 87 | 237 | [
{
"content": "ãã$$N=\\displaystyle \\sum_{k=a}^bk=\\frac{1}{2}\\{b(b+1)-(a-1)a\\}$$\r\nããæ¡ä»¶ã¯ $(b+a)(b-a+1)=2N$ ãšè¡šçŸã§ããïŒãã㧠$b+a$ ãš $b-a+1 $ã®å¶å¥ã¯ç°ãªãïŒã〠$b+a\\gt b-a+1\\gt1$ ã§ããããïŒä»¥äžã®æ¡ä»¶\r\nãã$$\\begin{cases}\\alpha\\ \\text{ã¯æ£ã®å¶æ°} \\\\\\\\ \\beta\\ \\text{㯠3 以äžã®å¥æ°} \\\\\\\\ \\alpha\\beta=2N \\\\\\\\ \\end{cases}$$... | ã次ã®æ¡ä»¶ãæºããæ£æŽæ° $N$ ã®ãã¡ïŒ$5$ çªç®ã«å°ãããã®ã $1000$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ
- $a\lt b$ ã〠$\displaystyle \sum_{k=a}^bk=N$ ãªãçžç°ãªãæ£æŽæ°ã®çµ $(a,b)$ ã, ã¡ããã© $2021$ çµååšããïŒ |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/128 | E | OMC020(E) | 500 | 4 | 38 | [
{
"content": "ã$B_{i+2}B_{i+3}$ ã®äžç¹ã $M_{i}$ ãšããã° $A_{i}M_{i}$ ã¯ãã¹ãŠäžç¹ $X$ ã§äº€ãã, $XA_{i}:XM_{i}=4:1$ ã§ãã. ãŸã, \r\nãã$$A_{1}C_{1}+A_{2}C_{2}:A_{3}C_{3}+A_{4}C_{4}+A_{5}C_{5}=7:18$$\r\nãã以äžãæç«ããããšã«çæãã.\r\nãã$$\\triangle XB_{3}B_{4}+\\triangle XB_{4}B_{5}:\\triangle XB_{5}B_{1}+\\triangle XB_{1}B_{2}+\\triangle XB... | ãé¢ç©ã $1$ ã®æ£äºè§åœ¢ $A_{1}A_{2}A_{3}A_{4}A_{5}$ ã®å
éšã«æ£äºè§åœ¢ $B_{1}B_{2}B_{3}B_{4}B_{5}$ ããã, $i=1,2,3,4,5$ ã«ã€ããŠèŸº $A_{i}A_{i+1}$ ãš $B_{i}B_{i+1}$ ã¯å¹³è¡ã§ã. ãŸã, $A_{1}$ ã¯çŽç· $B_{2}B_{5}$ ã«é¢ã㊠$B_{1}$ ã®å察åŽã«ãããã®ãšããŸã.\
ã$i=1,2,3,4,5$ ã«ã€ããŠçŽç· $A_{i}A_{i+1}$ ãšçŽç· $B_{i+2}B_{i+3}$ ã®äº€ç¹ã $C_{i}$ ãšããã°, 以äžãæãç«ã¡ãŸãã.
$$A_{1}A_{2}:B_{1}B_{3}=2:... |
OMC020 (ãšããæ°åŠã®ã³ã³ãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc020/tasks/129 | F | OMC020(F) | 600 | 0 | 0 | [
{
"content": "ãäžåŒãé©åã«å€åœ¢ããããšã§ä»¥äžãåŸã.\r\nãã$$\\dfrac{(1-x^2)(1-y^2)+2x\\times 2y}{(1+x^2)(1+y^2)}\\gt 2m-1$$\r\nããã㧠$t=\\tan(\\theta\\/2)$ ã«å¯ŸããŠ\r\nãã$$\\cos\\theta=\\dfrac{1-t^2}{1+t^2},\\ \\ \\sin\\theta=\\dfrac{2t}{1+t^2}$$\r\nã§ããããšã«çæããã°, $x=\\tan(\\alpha\\/2),\\ y=\\tan(\\beta\\/2)$ ãšããã°ä»¥äžãæç«ãã.\r\nãã$$\\cos(... | ãçžç°ãªã $100$ 以äžã®å®æ° $4$ ã€ãããªãä»»æã®éåã«ã€ããŠ, é©åã« $2$ å
$x,y$ ãéžã¶ããšã§ä»¥äžã®äžçåŒãæç«ãããããª, å®æ° $m$ ãšããŠããåŸãæå€§å€ $M$ ãèããŸã.
$$(xy+1)^2\gt m(x^2+1)(y^2+1)$$
ããã®ãšã, $aM^3+bM^2+cM+d=0$ ãã¿ãããããªäºãã«çŽ ãªæŽæ° $a,b,c,d$ (ãã ã $a\gt 0$) ãäžæã«ååšããããšã蚌æã§ããã®ã§, $|a|+|b|+|c|+|d|$ ãæ±ããŠãã ãã. |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/118 | A | OMC019(A) | 100 | 216 | 220 | [
{
"content": "ãæ¡ä»¶ã¯ $N(N+1)$ ã $2048=2^{11}$ ã®åæ°ã§ããããšãšåå€ã§ãã. ãã®ãšã, $N$ ãš $N+1$ ãäºãã«çŽ ã§ããããšãã, $N$ ãŸã㯠$N+1$ ã $2048$ ã®åæ°ã§ããããšãå¿
èŠååæ¡ä»¶ã§, ç¹ã«æ±ããæå°å€ã¯ $\\textbf{2047}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc019/editorial/118"
}
] | ã$1$ ä»¥äž $N$ 以äžã®æŽæ°ã®ç·åã $1024$ ã®åæ°ãšãªããããªæå°ã®æ£ã®æŽæ° $N$ ãæ±ããŠãã ãã. |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/119 | B | OMC019(B) | 200 | 173 | 211 | [
{
"content": "**解ç1.**ããŸã $1$ ç¹ãåºå®ã, ä»ã® $2$ é ç¹ãéžã¶æ¹æ³ã¯ ${}\\_{299}{\\rm C}\\_2$ éãã§ãã. ããã§, æ£äžè§åœ¢ã¯ $1$ å, æ£äžè§åœ¢ã§ãªãäºç蟺äžè§åœ¢ã¯ $3$ å, äžç蟺äžè§åœ¢ã¯ $6$ åæ°ããããŠããããšãèæ
®ãã. æ£äžè§åœ¢ã¯ $1$ å, æ£äžè§åœ¢ã§ãªãäºç蟺äžè§åœ¢ã¯ $148$ åã§ãããã, 以äžã®ããã«èšç®ã§ãã.\r\nãã$$N=\\dfrac{{}\\_{299}{\\rm C}\\_2 + 1\\times 5 + 148\\times 3}{6} = \\textbf{7500}$$\r\n**解ç2.**ã... | ãæ£ $300$ è§åœ¢ããçžç°ãªã $3$ é ç¹ãéžãã§ã§ããäžè§åœ¢ã¯, å転ã»è£è¿ãããŠäžèŽãããã®ã¯**åããã®**ãšããŠæ°ãããšã, $N$ åã§ã. $N$ ã解çããŠãã ãã. |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/120 | C | OMC019(C) | 300 | 113 | 137 | [
{
"content": "ã$P$ ã®åº§æšã $(p,q)$, $C$ ã®ååŸã $r$ ãšããã°, $C$ ã®åŒã¯ $(x-p)^2+(y-q)^2=r^2$ ãšè¡šããã. ããã«, $A_1, A_2$ ã® $x$ 座æšããããã $a_1, a_2$ãšããã°, $a_1, a_2$ 㯠$x$ ã«ã€ããŠã®äºæ¬¡æ¹çšåŒ $(x-p)^2+q^2=r^2$ ã® $2$ 解ã§ãããã, $a_1+a_2=2p$ ãåŸã. åæ§ã« $B_1, B_2$ ã® $y$ 座æšããããã $b_1, b_2$ ãšããã° $b_1+b_2=2q$ ã§ãã.\\\r\nããã㧠$a_2, b_2$ ã $0$ 以äžã§ããããšã«æ³š... | ã$xy$ å¹³é¢ã«ãããŠ, ã°ã©ã $y=5+\dfrac{4}{x^3}\ (x\gt 0)$ äžã®ç¹ $P$ ãäžå¿ãšã, åç¹ $O$ ãå
éš(åšäžã¯å«ãŸãªã)ã«å«ãå $C$ ãèããŸã. $C$ ãš $x$ 軞ãšã®äº€ç¹ã $x$ 座æšã倧ããé ã« $A_1, A_2$ ãšã, $C$ ãš $y$ 軞ãšã®äº€ç¹ã $y$ 座æšã倧ããé ã« $B_1, B_2$ ãšãããšã, ç¹ $P$ ããã³ å $C$ ãåãããŠä»¥äžã®åŒããšãåŸãæå°å€ã $m$ ãšããŸã.
$$\triangle OA_1B_1-\triangle OA_1B_2-\triangle OA_2B_1+\triangle OA_2B_2$$
ãã®ãšã ... |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/121 | D | OMC019(D) | 400 | 42 | 68 | [
{
"content": "ã$AB$ ã«é¢ã㊠$H$ ãšå¯Ÿç§°ãªç¹ã $H^\\prime$ ãšãããš,\r\nãã$$\\angle ACB+\\angle AH^\\prime B=\\angle ACB+\\angle AHB=180^\\circ$$\r\nã§ãããã, $H^\\prime$ 㯠$\\triangle ABC$ ã®å€æ¥åäžã«ãã. ãã®ãšã\r\nãã$$AH^\\prime=AH=AO=OH^\\prime$$\r\nãã, ç¹ã« $\\triangle AH^\\prime O$ ã¯æ£äžè§åœ¢ã§ãããã, \r\nãã$$\\angle H^\\prime AB=\\angle HAB... | ãéè§äžè§åœ¢ $ABC$ ã«ã€ããŠïŒãã®åå¿ã $H$ , å€å¿ã $O$ ãšããŸãïŒ
$$AH=AO, \quad OH=12, \quad BH=5$$
ã§ãããšãïŒç·å $CH$ ã®é·ãã¯æŽæ° $a,b,c$ ãçšã㊠$a\sqrt{b}+c$ ãšè¡šãããŸãïŒ$a^2b+c$ ã解çããŠãã ããïŒ |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/122 | E | OMC019(E) | 500 | 15 | 39 | [
{
"content": "ããŸãåœã $2$ ã€ããç¡ãã£ããšã, åœ $1,2$ ããããã $a,b$ åã®å³¶ãä¿æããå Žåãèãã. ãã®ãšã, ååœã®å³¶ã®éã«ã¯æ©ãæ¶ããªããšããå¶çŽã課ããŠ, æ¡ä»¶ãã¿ããæ©ã®æ¶ãæ¹ã®ç·æ° $f(a,b)$ ãæ±ããã. \r\n\r\n**解ç1.**ãåœ $1$ ã®å³¶ $a$ ãšæ¥ç¶ããæ©ãååšããªããšã $f(a-1,b)$ éãã§, åœ $1$ ã®å³¶ $a$ ãšåœ $2$ ã®å³¶ $i$ ãçµã¶æ©ãååšãããšã $f(a-1,i-1)+1$ éãã§ãããã, 以äžã®æŒžååŒãæç«ãã.\r\nãã$$f(a,b)=f(a-1,b)+f(a-1,b-1)+\\cdots+... | ã$2021\times 999$ åã®å³¶ããã, åœ $1$, åœ $2$, $\cdots$, åœ $2021$ ããããã $999$ åãã€å³¶ãä¿æããŠããŸã. ååœãä¿æãã島ã«ã¯ãããã $1$ ãã $999$ ãŸã§ã®çªå·ãæ¯ãããŠããŸã. ãããã®éã«, 以äžã®æ¡ä»¶ãã¿ããããã«æ©ãäœæ¬ãæ¶ããŸã.
- ã©ã®æ©ã, **çžç°ãªã**åœãä¿æãã $2$ åã®å³¶ãçŽæ¥çµã¶.
- ã©ã® $2$ åã®å³¶ã«ã€ããŠã, ãããã®éãçŽæ¥çµã¶æ©ã¯é«ã
$1$ æ¬ã§ãã.
- ä»»æã® $2$ ä»¥äž $2021$ 以äžã®æŽæ° $K$ ã«ã€ããŠ, ããæ£ã®æŽæ° $K^{\prime}\lt K$ ãååšã, åœ $K^{\... |
OMC019 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc019/tasks/123 | F | OMC019(F) | 600 | 7 | 22 | [
{
"content": "ã圢åŒçã¹ãçŽæ°ãçšãããš, $f(A)$ ã¯ä»¥äžã«ããã $x^{A}$ ã®ä¿æ°ã«çããããšã容æã«ç¢ºèªããã.\r\nãã$$\\displaystyle \\prod_{i=1}^{2021}(1-x+x^{2^i}-x^{2^i+1}+x^{2\\times 2^i}-x^{2\\times 2^i+1}+\\cdots)=\\prod_{i=1}^{2021}\\frac{1-x}{1-x^{2^i}}$$\r\nãããã«, $A$ ã $2022$ åã®éè² æŽæ°ã®åãšããŠè¡šã(é åºãèæ
®ãã)æ¹æ³ã¯ ${}\\_{A+2021}\\mathrm{C}\\_{2021}$ éãã§... | ã$2021$ æã®ã«ãŒãããã, $i$ çªç®ã®ã«ãŒã($i=1,2,\cdots,2021$)ã«ã¯ $2^i$ ã§å²ã£ãŠ $0$ ãŸã㯠$1$ äœãéè² æŽæ°ãäžã€ãã€æžã蟌ã¿ãŸã. ããã§, å¥æ°ãæžãããã«ãŒãã $S$ æååšãããšã, **ã³ã¹ã**ã $(-1)^S$ ã§å®ããŸã.\
ãéè² æŽæ° $A$ ã«ã€ããŠ, $2021$ åã®æ°ã®åã $A$ ãšãªããããªæžã蟌ã¿æ¹ãšããŠããåŸããã®ãã¹ãŠã«ã€ããŠã³ã¹ãã®ç·åã $f(A)$ ãšã, ããã« $k(A)$ ã以äžã§å®ããŸã.
$$\displaystyle k(A)=\sum_{i=0}^{A}f(i)\times{}\_{A-i+2021}\mathrm{... |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/112 | A | OMC018(A) | 100 | 231 | 245 | [
{
"content": "ã$f(n)$ ã以äžã§å®ãããš, $n^2-8n+17=(n-4)^2+1\\gt 0$ ããããã¯åžžã«æ£ã§ãããã, $a_n$ ãåžžã«æ£ã§ãã.\r\nãã$$\\displaystyle f(n)=\\frac{9n+1}{n^2-8n+17}\\ \\ (n=1,2,\\cdots)$$\r\nããã®ãšã, $f(n)$ ãš $1$ ã®å€§å°ãæ¯èŒããããšã§å®¹æã«ä»¥äžãåŸããã, ç¹ã«æ±ããå€ã¯ $33$ ã§ãã.\r\nãã$$a_1=a_2\\lt a_3\\lt a_4\\lt \\cdots\\lt a_{16}=a_{17}\\gt a_{18}\\gt a_{19}\\... | ã以äžãã¿ããæ°å $ \\{a_{n}\\}\_{n=1,2,\cdots} $ ã«ãããŠ, $a_n$ ãæ倧å€ããšããããªæ£ã®æŽæ° $n$ ã®ç·åãæ±ããŠãã ãã.
$$a_{1} =1,\quad a\_{n+1}=\dfrac{9n+1}{n^{2} -8n+17} a_{n}\ \ (n=1,2,\cdots)$$ |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/113 | B | OMC018(B) | 200 | 165 | 219 | [
{
"content": "ãæ±ããé åã¯ä»¥äžã§è¡šããã. ãã ã $D$ 㯠$A$ ãã $BC$ ã«ããããåç·ã®è¶³ã§ãã. ãã®é¢ç©ã¯é©åœãªæ圢ãšäžè§åœ¢ã®çµã¿åããã«ãã£ãŠå®¹æã«èšç®ã§ãã. å
·äœçã«ã¯ $\\displaystyle \\frac{35}{12}\\pi - \\sqrt{3}$ ã§ãã, æ±ããå€ã¯ $\\textbf{50}$ ã§ãã.\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontes... | ã$AB=2,AC=2\sqrt{3},BC=4$ ãªãäžè§åœ¢ $ABC$ ã, ç¹ $A$ ãäžå¿ã«å¹³é¢äžã§ $90^\circ$ å転ããããšã, 蟺 $BC$ ã®ééããé åã®é¢ç©ã¯ $\displaystyle\frac{a}{b}\pi - \sqrt{c}$ ãšè¡šããŸã. ãã ã $a,b$ ã¯æ倧å
¬çŽæ°ã $1$ ã®æ£ã®æŽæ°, $c$ ã¯æ£ã®æŽæ°ã§ã.\
ããã®ãšã, $a+b+c$ ã解çããŠãã ãã. |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/114 | C | OMC018(C) | 300 | 166 | 210 | [
{
"content": "ãäžè¬ã«æ Œåç¹ã $n\\times n$ ã§ããå Žåã«ã€ããŠèãã.\\\r\nãå蟺ã軞ãšå¹³è¡ã§ãããããªæ£æ¹åœ¢ã«ã€ããŠ, äžèŸºã®é·ãã $k\\leq n-1$ ã§ãããããªãã®ã¯ $(n-k)^2$ åååšãããã, ãã®ãããªãã®ã®ç·æ°ã¯ä»¥äžã§äžãããã.\r\nãã$$ \\displaystyle \\sum _{k=1}^{n-1}(n-k)^2$$\r\nããã以å€ã®æ£æ¹åœ¢ã«ã€ããŠ, $a+b\\leq n-1$ ã§ãããšã, ããäžèŸºã®åŸãã $b\\/a$ ã§, é·ãã $\\sqrt{a^2+b^2}$ ã§ãããããªæ£æ¹åœ¢ã¯ $(n-a-b)^2$ åååšãã. $... | ã$100\times100$ ã®æ Œåç¹ã®äžããããçžç°ãªã $4$ ç¹ãéžã¶æ¹æ³ã§ãã£ãŠïŒããããé ç¹ãšããåè§åœ¢ãæ£æ¹åœ¢ãšãªããããªãã®ã¯ $M$ éããããŸã. $M$ ã解çããŠãã ãã.\
ããã ã, å転ãå転ã«ãã£ãŠäžèŽãããã®ãåºå¥ããŸã.\
ã以äžã®å³ã¯ $5\times 5$ ã®æ Œåç¹, ããã³ãããããªãæ£æ¹åœ¢ã®äŸã§ã.
 |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/115 | D | OMC018(D) | 400 | 120 | 157 | [
{
"content": "ã$n$ ã¯çŽ æ° $p$ ã«ãã£ãŠ $n=p^2$ ãšè¡šããã. ãã ã $n\\gt 15$ ã§ããããšãã $p\\geq 5$ ã§ãã. ãã®ãšã, $m$ ã¯çžç°ãªãçŽ æ° $a,b$ ã«ãã£ãŠ $a^{p^2-1}$ ãŸã㯠$(ab)^{p-1}$ ãšè¡šãã, äœãã®æ¡ä»¶ãã $a,b$ 㯠$p$ ã§ã¯ãªã. ãããã®å Žåã, Fermatã®å°å®çãã $m\\equiv 1\\pmod p$ ã§ãããã, $15\\equiv 1\\pmod p$ ãã $p=7$ ãšãªãã»ããªã.\\\r\nãéã« $2^{48}\\equiv 15\\pmod{49}$ ã§ãããã, $... | ãæ£ã®æŽæ° $m,n$ ã«ã€ããŠïŒããããã®çžç°ãªãæ£ã®çŽæ°ã¯ $n$ åïŒ$3$ åååšããŸããïŒ\
ãããã«ïŒ$m$ ã $n$ ã§å²ã£ãäœãã $15$ ã§ãããšãïŒ$n$ ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/116 | E | OMC018(E) | 500 | 26 | 91 | [
{
"content": "ã$\\angle A=2a$ ãªã©ãšãããš, $a+b+c=90^\\circ$ ã§ãã. $\\angle A$ ã®äºçåç·ãšèŸº $BC$ ã®äº€ç¹ã $F$ ãšãããš,\r\nãã$$\\angle FIC=\\angle IAC+\\angle ACI=a+c=90^\\circ-b$$\r\nãã®äžæ¹ã§\r\nãã$$\\angle EIC=\\angle AEI-\\angle ACI=(180^\\circ-a-2b)-c=90^\\circ-b$$\r\nãæãç«ã€ãã $\\angle FIC=\\angle EIC$ ã§ãã, $\\angle ICE=\\angle I... | ãå
å¿ã $I$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, ãããã蟺 $AB,AC$ äžã«ããç¹ $D,E$ ã以äžãã¿ãããŸãã.
$$\angle{AID}=\angle{ACB},\ \ \angle{AIE}=\angle{ABC}$$
ã$AD=12,CD=17,CE=12$ ã§ãããšã, $BC$ ã®é·ããæ±ããŠãã ãã.\
ããã ã, çãã¯æ£ã®æŽæ°ã«ãªãããšã蚌æã§ããŸã. |
OMC018 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc018/tasks/117 | F | OMC018(F) | 600 | 35 | 70 | [
{
"content": "ã$y\\gt 0$ ã®ãšã, çžå ã»çžä¹å¹³åã®é¢ä¿ãã $x=y=1$ ãšãªãã»ããªã, ããã¯äžé©ã§ãã. ãããã£ãŠä»¥äž $y\\lt 0$ ãšããŠãã, $y$ ã $-y$ ãšçœ®ãçŽããŠèãã. ãŸã $y$ ãæ¢çŽåæ°ã§ $c\\/d$ ãšè¡šã. ãã ã $a,b,c,d\\gt 0$ ãšãã.\\\r\nãäžåŒã«ä»£å
¥ããŠæŽçããããšã§ $cd(a^2+b^2)-ab(c^2+d^2)=4abcd$ ã§ããã, $a^2+b^2$ ãš $ab$ ã¯äºãã«çŽ ã§ããããšã«çæããã° $cd$ 㯠$ab$ ã§å²ãåãã. éã« $ab$ 㯠$cd$ ã§å²ãåãããã, çµå± $a... | ããããã $0$ ã§ã¯ãªãæçæ° $x,y$ ã¯, $x\gt 0$ ãã€ä»¥äžãã¿ãããŸã.
$$x+\dfrac{1}{x}+y+\dfrac{1}{y}=4$$
ããã« $x$ ãäºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ã«ãã£ãŠ $\displaystyle\frac{a}{b}$ ãšè¡šãããšã, $a$ ãš $b$ ã®å·®ã¯ $10^{10}+41421^2$ ã§ãã.\
ã$a$ ãšããŠããåŸãæ倧å€ãæ±ããŠãã ãã. |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/106 | A | OMC017(A) | 100 | 262 | 263 | [
{
"content": "ãåé²æ³ãçµç±ãã, çŽæ¥äžãã $3$ æ¡ããšã«å€æããã®ãæãç°¡åã§ããã. æ±ããçã㯠$\\textbf{47532}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc017/editorial/106"
}
] | ãäºé²æ³ã§ $100111101011010$ ãšè¡šèšãããæŽæ°ãïŒå
«é²æ³ã§è¡šèšãããã®ã解çããŠãã ããïŒ\
ããã ãïŒæé«äœã®æ°å㯠$0$ ã«ããªãã§ãã ããïŒ |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/107 | B | OMC017(B) | 200 | 251 | 257 | [
{
"content": "ã$B$ ã®å£ãçªãæãããšãã®ããŒã«ã®éãã¯æ£ã®æŽæ° $n$ ã«ãã£ãŠç§é $2^n {\\rm cm}$ ãšããããã, æ±ããæéã¯\r\nãã$$\\displaystyle t=\\sum_{k=0}^n \\frac{1000}{2^k}+\\frac{1000}{2^n}=1000\\left(2-\\frac{1}{2^n}\\right)+\\frac{1000}{2^n}=\\textbf{2000}(\\text{ç§})$$\r\nã§ãã. åäœã«æ³šæãã. ãªãå®éã«ã¯ $n=18$ ã§ããã, ãããå
·äœçã«æ±ããå¿
èŠã¯ç¡ã.",
"text": "å
¬åŒè§£... | ãçŽç·ç¶ã®ã³ãŒã¹ã« $3$ ç¹ $A,B,C$ ããã®é ã§äžŠãã§ãã, $A$ ãš $B$, $B$ ãš $C$ ã®éã®è·é¢ã¯ãããã $10{\rm m}$ ã§ã. ãŸã, $A$ ãš $B$ ã«ã¯ããããäžæè°ãªå£ãç«ã£ãŠããŸã. ã³ãŒã¹ãé²ãã§ããããŒã«ããããã®å£ã«åœãããšããŒã«ã¯è·³ãè¿ã, éæ¹åã«åãã£ãŠçŽåã® $2$ åã®éãã§é²ã¿ãŸã. \
ãããã§, $A$ ã®å£ã¯éåžžã«é äžãªã®ã§æ±ºããŠå£ããŸããã, $B$ ç¹ã®å£ã¯æé $5000{\rm km}$ 以äžã§ããŒã«ãåœãããšå£ããŠããŸã, ããŒã«ã¯ãã®ãŸãŸã®éãã§å£ãçªãæããŸã. \
ãç§é $1{\rm cm}$ 㧠$A$ ãã $B$ ã«åãã£ãŠæŸãã... |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/108 | C | OMC017(C) | 300 | 130 | 173 | [
{
"content": "ã$P$ ãè¡šãåŒã $y=kx^2$ ãšã ($k\\gt 0$), $A,B,C$ ã® $x$ 座æšã $a\\lt b\\lt c$ ãšãã. ãã®ãšã, $A,B$ ã«ãããæ¥ç·ãš $P$ ã§å²ãŸããéšåã®é¢ç©ã $S_{AB}$, çŽç· $AB$ ãš $P$ ã§å²ãŸããéšåã®é¢ç©ã $T_{AB}$ ãªã©ãšããã°, æåäºå®ãšããŠ\r\nãã$$ S_{AB}=\\dfrac{1}{12}k(b-a)^3,\\ \\ T_{AB}=\\dfrac{1}{6}k(b-a)^3 $$\r\nãæç«ãã(æçŽã«ç©åãå®è¡ããã°ç¢ºèªã§ãã). ããªãã¡, ç¹ã« $2S_{AB}=T_{A... | ãé¢ç©ã $24$ ã§ããäžè§åœ¢ã«ã€ããŠ, $3$ é ç¹ããã¹ãŠéãæŸç©ç· $P$ ãèã, ããããã®é ç¹ã«ããã $3$ æ¥ç·ã®ãªãäžè§åœ¢ã®é¢ç©ã $S$ ãšãããŸã.\
ããã®ãšã, $S$ ãšããŠããåŸãæ倧å€ãšæå°å€ã®**ç©**㯠$M$ ãšãªããŸã. $M$ ã解çããŠãã ãã. |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/109 | D | OMC017(D) | 400 | 108 | 142 | [
{
"content": "ã$BC$ ã®äžç¹ã $M$ ãšã, $C$ ãã $AB$ ã«ããããåç·ã®è¶³ã $H$ ãšãã.\r\n\r\n**解ç1.**ãã¡ãã©ãŠã¹ã®å®çãã $AH=HP$ ãããã. ããã $x$ ãšããã°, äžè§åœ¢ $BCH$ ã«ãããŠ\r\nãã$$ 8x^2=(2\\sqrt{2}x)^2=(2AP)^2=BC^2=BH^2+CH^2=(\\sqrt{3}-x)^2+(x+\\sqrt{3})^2=2x^2+6$$\r\nãã $x=1$ ã§ãã. ãã®ãšã, äžè§åœ¢ $BHP$ ã«ãããŠ\r\nãã$$BP^2=BH^2+PH^2=(\\sqrt{3}-1)^2+1^2=5-2\... | ãäžè§åœ¢ $ABC$ ã«ãããŠ, $A$ ãã察蟺ãžãããã**äžç·**ãš, $C$ ãã察蟺ãžãããã**åç·**ã®äº€ç¹ã $P$ ãšããŸã.
$$2AP=BC,\ \ AB=CP=\sqrt{3}$$
ã§ãããšã, æŽæ° $a,b,c$ ãçšããŠ, $BP^2=a+b\sqrt{c}$ ãšè¡šãããŸã. $a^2+b^2c$ ã解çããŠãã ãã.\
ããã ã, $XY$ ã§ç·å $XY$ ã®é·ããè¡šããã®ãšããŸã. |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/110 | E | OMC017(E) | 500 | 38 | 147 | [
{
"content": "ãçµè«ããè¿°ã¹ããš, $N$ ãå¶æ°ã®ãšã $f(N)=N^3\\/4$ ã§ãã, $N$ ãå¥æ°ã®ãšã $f(N)=(N^3-N)\\/4$ ã§ãã. ãã®ãšã, æ±ããç·å㯠$\\textbf{26364}$ ã§ãã. å¶æ°ã®ãšãæããã§ãããã, ä»¥äž $N$ ãå¥æ°ã§ããå Žåãèãã.\\\r\nããŸã, å¶æ°æ®µç®ã¯å·Šã®ããã«, å¥æ°æ®µç®ã¯å³ã®ããã«ç¹å®ã®ãã¹ã«å°ãã€ãããš, åŠäœãªãé
眮ã«ã€ããŠãåãããã¯ã¯ã¡ããã©äžã€å°ã®ã€ãããã¹ãå«ããã, ãã®å°ãæ°ããããšã§ $f(N)\\leq (N^3-N)\\/4$ ãããã.\r\n\r\n
ãsiosioåã¯ãã¹ç®ã«æ²¿ã£ãŠç®±ã«ãããã¯ãåºæ¥ãã ãããããå
¥ãããã§ã. ããã§, äžæ¹ã®ãããã¯ã®ã¿ãçšããŠãæ§ããŸãã. siosioåãçšãããããã¯ã®åæ°ãšããŠããåŸãæ倧å€ã $f(N)$ ãšãããšã,
$$f(2)+f(3)+\cdots+f(25)$$
ã解çããŠãã ã... |
OMC017 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc017/tasks/111 | F | OMC017(F) | 600 | 39 | 86 | [
{
"content": "ãåé¡ã¯æ¬¡ã®ããã«æžãæããããïŒ\r\n\r\n- ä»»æã® $n$ ã«ã€ã㊠$\\varphi(a_n)=a_{n-1}$ ãæç«ããç¡éæ°å $\\\\{a_n\\\\}$ ãååšããåé
$a_1$ ããã¹ãŠæ±ãã.\r\n\r\n\r\n ã$\\varphi$ ã®è¿ãåŸãå€ã¯ $1$ ãŸãã¯å¶æ°ã§ããããšã«çæãã. æããã«ãã¹ãŠã® $2$ ã¹ãã¯æ¡ä»¶ãã¿ãã. $a_n$ ã $2$ ã¹ãã§ãããšã, ãã以åã®é
ã¯ãã¹ãŠ $2$ ã¹ãã§ãããã, 以äžãã $x$ ã«ã€ã㊠$a_x$ ã $2$ ã¹ãã§ãªããšãã.\\\r\nããã®ãšã, $n\\geq x$ ã«ãããŠ... | ãæ£ã®æŽæ° $n$ ã«å¯ŸãïŒ$n$ ãšäºãã«çŽ 㪠$n$ 以äžã®æ£ã®æŽæ°ã®åæ°ã $\varphi(n)$ ã§è¡šããŸãïŒä»»æã®æ£ã®æŽæ° $k$ ã«å¯ŸããŠ, ããæ£ã®æŽæ° $n_k$ ãååšããŠä»¥äžãæç«ãããããªïŒæ£ã®æŽæ°ã®å®æ° $m\leq 300$ ãèããŸã.
$$ \underbrace{\varphi( \varphi( \dots \varphi}_{kå}(n_k) \dots ))=m $$
ãã®ãã㪠$m$ ãã¹ãŠã«ã€ããŠïŒç·ç©ã¯ $M$ ãšãªããŸãïŒ$M$ ããã€æ£ã®çŽæ°ã®åæ°ãæ±ããŠãã ããïŒ |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/100 | A | OMC016(A) | 100 | 179 | 245 | [
{
"content": "ãäºãã«çŽ ãªæ£æŽæ° $m\\gt n$ ãçšã㊠$a=m\\/n$ ãšè¡šãããšãã. ãã®ãšã, æ¡ä»¶ãã $1000m\\/n$ ãš $1000n\\/m$ ã¯ãšãã«æŽæ°ã§ãã, $100m\\/n$ ãš $100n\\/m$ ã¯ãšãã«æŽæ°ã§ãªã.\r\nãã㧠$m$ ãš $n$ ãäºãã«çŽ ã§ããããšãã, $m,n$ ã¯ãšãã« $1000$ ã®çŽæ°ã§ãã, $100$ ã®çŽæ°ã§ãªã.\\\r\nããããã£ãŠ $(m,n)=(125,8)$ ãšãªãã»ããªã, æ±ããå€ã¯ $125+8=\\textbf{133}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url... | ã$a$ ãš $\displaystyle \frac{1}{a}$ ãããããå°æ°ç¬¬ $3$ äœãŸã§ã®æéå°æ°ãšããŠè¡šããããã㪠$1$ 以äžã®æçæ° $a$ ã®ç·åãæ±ããŠãã ãã.\
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£æŽæ° $b,c$ ãçšã㊠$\displaystyle \frac{b}{c}$ ãšè¡šãããã®ã§, $b+c$ ã解çããŠãã ãã.\
ãããã§, $N$ ãå°æ°ç¬¬ $n$ äœãŸã§ã®æéå°æ°ã§ãããšã¯, $N$ ãåé²æ°ã®å°æ°ãšããŠè¡šãããšãå°æ°ç¬¬ $n$ äœã $0$ ã§ãªã, ãã€ä»»æã®æ£æŽæ° $k$ ã«ã€ããŠå°æ°ç¬¬ $n+k$ äœã $0$ ã§ããããšãæããŸã. |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/101 | B | OMC016(B) | 200 | 136 | 234 | [
{
"content": "ãæççµè·¯ã蟿ãã°ã¡ããã© $8$ åºéæ©ãããšã«ãªããã, toriiåã¯ã¡ããã© $1$ åºéã ãå·ŠãŸãã¯äžã«åããããšãåããïŒå¯Ÿç§°æ§ãã, ãããå·Šã§ããå Žåã®ã¿èããã°ååã§ãã. ããªãã¡, $\\rightarrow$ ã« $5$ åïŒ$\\leftarrow$ ã« $1$åïŒ$\\uparrow$ ã« $4$ ååããããšã«ãªããã, ãããã®ç¢å°ãäžåã«äžŠã¹ã䞊ã¹æ¹ã®ç·æ°ãèããã°ãããïŒä»¥äžã®ãããªå Žåãé€å€ããªããã°ãªããªãããšã«çæãã.\r\n\r\n- $\\rightarrow$ ãš $\\leftarrow$ ã®ã¿ãåãåºãããšãïŒãã®é çªã $\\left... | ãäžå³ã®ãããªç¢ç€ã®ç®ç¶ã®éããããŸã. å³ã§ç€ºããã40åºéãé€ããŠéã¯ååšããŸãã. \
ãtoriiåã¯å°ç¹ $A$ ããå°ç¹ $B$ ãŸã§æ©ããŠè¡ãããšã«ãªããŸããã, éäžã§è¿·åã«ãªã£ãŠããŸã, åã㊠$B$ å°ç¹ã«å°éãããŸã§ã¡ããã© $10$ åºéæ©ããŸãã. toriiåãæ©ããçµè·¯ãšããŠèãããããã®ã $M$ éãã§ãããšã, $M$ ã解çããŠãã ãã.\
ããã ã, toriiåã¯çŽåã«éã£ãåºéãéåãã«åŒãè¿ããŠãè¯ãã§ãã, åºéã®äžéã§ã¯åŒãè¿ããŸãã.
 |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/102 | C | OMC016(C) | 300 | 117 | 138 | [
{
"content": "ã$\\angle ABC=\\angle FDE=2\\angle FBD$ ãã, $BF$ 㯠$\\angle ABC$ ãäºçåãã. åæ§ã« $CF$ 㯠$\\angle ACB$ ãäºçåãããã, $F$ ã¯äžè§åœ¢ $ABC$ ã®å
å¿ã§ããããšãããã. $AF$ ãš $BC$ ã®äº€ç¹ã $G$ ãšãããš, äœåŒŠå®çãã\r\nãã$$ \\dfrac{AB^2+AG^2-BG^2}{2\\times AB\\times AG}=\\dfrac{AC^2+AG^2-CG^2}{2\\times AC\\times AG} $$\r\nãããã«è§ã®äºçåç·å®çãã $BG:... | ãäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ äžã®ç¹ $D,E$ ã«ã€ã㊠$B,D,E,C$ ã¯ãã®é ã«äžŠãã§ãã, $D$ ãéã $AB$ ã«å¹³è¡ãªçŽç·ãš $E$ ãéã $AC$ ã«å¹³è¡ãªçŽç·ã®äº€ç¹ã $F$ ãšããŸã. 以äžãæç«ãããšã, $BC$ ã®é·ããæ±ããŠãã ãã.
$$BD=DF,\ CE=EF,\ AB=3,\ AC=6,\ AF=\sqrt{6}$$
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle \frac{a}{b}$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/103 | D | OMC016(D) | 400 | 108 | 134 | [
{
"content": "ã$f$ ã®æ¬¡æ°ã $d$ ãšãããš, æ¡ä»¶ããæããã« $d\\geq 4$ ã§ãã, ãã®ãšãäžåŒã®äž¡èŸºã®æ¬¡æ°ãæ¯èŒããããšã§\r\nãã$$d+(d-1)+(d-2)+(d-3)=10$$\r\nãã $d=4$ ãåŸã. ããã«, $4$ 次ã®ä¿æ°ã $a\\gt 0$ ãšãããš, æé«æ¬¡ã®ä¿æ°ãæ¯èŒããŠ\r\nãã$$a\\times 4a\\times 12a\\times 24a=1152$$\r\nãã $a=1$ ãåŸã.\\\r\nã$f$ ã $x+4$ ã§å²ãåããªããšã, $f^{\\prime}(x)f^{\\prime\\prime}(x)f^{\\prime... | ãå®æ°ãä¿æ°ãšã, æé«æ¬¡ã®ä¿æ°ã¯æ£ã§ããå€é
åŒ $f(x)$ ã, 以äžãã¿ãããŸã.
$$f(x)f^{\prime}(x)f^{\prime\prime}(x)f^{\prime\prime\prime}(x)=1152x(x+1)(x+2)(x+3)(x+4)^6$$
ãã®ãããªãã®ããã¹ãŠæ±ã, ããããã«ã€ã㊠$0$ ã§ãªãä¿æ°ã®ç·ç©ã®ç·åãæ±ããŠãã ãã. \
ãäŸãã°, $f(x)=2x^2-3,3x^3+5x$ ã§ãããšã, æ±ããå€ã¯ $2\times(-3) + 3\times 5=9$ ã§ã. |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/104 | E | OMC016(E) | 500 | 52 | 60 | [
{
"content": "ãé»æ¿ã«çŸããæ° $x$ ããããã $1\\/(x-1)$ ã«çœ®ãæãããšïŒåãã«ã¯ $1\\/(k^2-1)\\ (k=2,3,...,10000)$ ãæžãããŠããïŒåé¡æã®æäœã«ãã£ãŠ $1\\/(a-1)$ ãš $1\\/(b-1)$ ãæ¶ããŠæžã足ãæ°ã¯\r\nãã$$\\displaystyle \\frac{1}{\\frac{ab-1}{a+b-2}-1}=\\frac{a+b-2}{ab-a-b+1}=\\frac{1}{a-1}+\\frac{1}{b-1}$$\r\nã§ãããã, æäœã®æ¹æ³ã«ãããé»æ¿ã«æžãããæ°ã®ç·åã¯äžå®ã§ãã, ç¹ã«æåŸã«æ®ãæ°ã$S$ãšãã... | ãé»æ¿ã« $9999$ åã®æŽæ° $2^2,3^2,4^2,...,10000^{2}$ ãããããäžã€ãã€æžãããŠããŸã. ããã§, é»æ¿ã«æžãããŠããæ°ãã¡ããã© $1$ ã€ã«ãªããŸã§, 次ã®æäœãç¹°ãè¿ãè¡ããšã, æåŸã«é»æ¿ã«æ®ãæ°ãšããŠããåŸããã®ã®ç·åãæ±ããŠãã ãã.
- æäœïŒé»æ¿ãã $2$ ã€ã®æ° $a,b$ ãéžãã§æ¶ã, æ°ãã« $\displaystyle \frac{ab-1}{a+b-2}$ ãæžã足ãïŒ
ããã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle \frac{a}{b}$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc016/tasks/105 | F | OMC016(F) | 600 | 15 | 44 | [
{
"content": "ãåé¡æã«ãããŠ, ç®æ¡æžãã§ç€ºããã $3$ æ¡ä»¶ãããããæ¡ä»¶ $1$, æ¡ä»¶ $2$, æ¡ä»¶ $3$ ãšåŒã¶.\\\r\nããŸãå·Šäžã®ãã¹ã« $1$ ãå
¥ããå Žåã®ã¿ãèã, äžè¬ã« $8$ ã $n$ ãšãããšã, é¡æãã¿ããæ°ã®æžã蟌ã¿æ¹ã $2^{n-1}-n$ éãã§ããããšãæ°åŠçåž°çŽæ³ã«ãã£ãŠç€ºã. $n=1,2$ ã®ãšãæããã« $0$ éãã§ãã.\\\r\nããã $k\\geq 2$ ã«ã€ã㊠$n\\leq k$ ã§æç«ãä»®å®ãã. $xy$ å¹³é¢äžã«ãããŠ, $0\\leq x\\leq k$ ã〠$-1\\leq y\\leq 0$ ãªãæ Œåç¹å
šäœãé ç¹... | ã$2\times 8$ ã®ãã¹ç®ã®åãã¹ã«, $1$ ä»¥äž $16$ 以äžã®æŽæ°ããããã $1$ ã€ãã€, 以äžã®æ¡ä»¶ãã¿ããããã«éè€ãªãæžã蟌ã¿ãŸã.
- $1$ ä»¥äž $15$ 以äžã®ä»»æã®æŽæ° $k$ ã«ã€ããŠ, ãã¹ $k$ ãšãã¹ $k+1$ ã¯é ç¹ãå
±æãã.
- $1$ ä»¥äž $15$ 以äžã®ããæŽæ° $l$ ãååšã, ãã¹ $l$ ãšãã¹ $l+1$ ã¯é ç¹ã®ã¿ãå
±æãã.
- $1$ ä»¥äž $14$ 以äžã®ä»»æã®æŽæ° $m$ ã«ã€ããŠ, ãã¹ $m$ ãšãã¹ $m+2$ ã¯èŸºãå
±æããªã.
ããã®ãšã, æ°ã®æžã蟌ã¿æ¹ã¯ $M$ éããããŸã. $M$ ã解çããŠãã ãã.
ãã... |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/94 | A | OMC015(A) | 100 | 252 | 252 | [
{
"content": "ã2人ã®å¹Žéœ¢å·®ã¯ $3$ æ³ã§äžå®ã§ããããšã«çæããã°, æ¡ä»¶ãã¿ããã®ã¯ $C$ ããã $3$ æ³, $Y$ ããã $6$ æ³ã®ãšãã§ãã, ãã㯠$\\textbf{12}$ 幎åã§ãã.\\\r\nããªã, æ¹çšåŒ $2(15-M)=18-M$ ã解ããŠããã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc015/editorial/94"
}
] | ãã㟠$C$ ãã㯠$15$ æ³, $Y$ ãã㯠$18$ æ³ã§ã. $C$ ããã®å¹Žéœ¢ã $Y$ ããã®å¹Žéœ¢ã®ååã ã£ãã®ã¯ $M$ 幎åã§ã.\
ãäºäººã®èªçæ¥ãåãã§ãããšã, $M$ ãšããŠé©ããæ£æŽæ°ã解çããŠãã ãã. |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/95 | B | OMC015(B) | 200 | 204 | 236 | [
{
"content": "**解ç1.**ã$3$ æ¬ã®å¯Ÿè§ç·ãäºãã«ç«¯ç¹ä»¥å€ã§äº€ç¹ãæã¡, ã〠$1$ ç¹ã§äº€ãããªãããšãå¿
èŠååæ¡ä»¶ã§ãã. ããã§, åè
ã®æ¡ä»¶ã®ã¿ãã¿ããéžã³æ¹ã¯, æ£å
«è§åœ¢ã®é ç¹ãã $6$ ã€ãéžã¶æ¹æ³ $28$ éããšäžå¯Ÿäžã«å¯Ÿå¿ãã. ããã«, æ£å
«è§åœ¢å
㧠$3$ æ¬ä»¥äžã®å¯Ÿè§ç·ã亀ããåŸãç¹ã¯ $9$ åãã, ãã®ãã¡äžå¿ã®ã¿ $4$ æ¬ã®å¯Ÿè§ç·ãéããã, åè
ãã¿ããåŸè
ãã¿ãããªããã®ã¯ $12$ éãã§ãã. 以äžãã, æ±ããå Žåã®æ°ã¯ $\\textbf{16}$ éãã§ãã.\r\n\r\n**解ç2.**ã$3$ æ¬ã®å¯Ÿè§ç·ãçŽçã $7$ ã€ã«åãã€ãšã,... | ãæ£å
«è§åœ¢ã®çŽçããããŸã. $3$ æ¬ã®å¯Ÿè§ç·ã®éžã³æ¹ã§ãã£ãŠ, ãããã«æ²¿ã£ãŠçŽçãåããš $7$ æã«åããããããªãã®ã¯ $M$ éããããŸã.\
ã$M$ ã解çããŠãã ãã. ãã ã, $8$ ã€ã®é ç¹ã¯åºå¥ãããã®ãšããŸã. |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/96 | C | OMC015(C) | 300 | 37 | 85 | [
{
"content": "ãç¹ $A$ ãäžå¿ãšããŠç¹ $D,E$ ãéãå, ç¹ $B$ ãäžå¿ãšããŠç¹ $D$ ãéãå, ç¹ $C$ ãäžå¿ãšããŠç¹ $E$ ãéãåãããããèãããš, $F$ ã¯ããã $3$ åã®æ ¹å¿ã§ãããã, 以äžã®çåŒãæç«ããããšãããã.\r\n$$BP^2-BD^2=BF^2-BD^2-FP^2=CF^2-CE^2-FP^2=CP^2-CE^2$$\r\næ¡ä»¶ãçšããŠããã解ãããšã§ $BP=\\sqrt{\\dfrac{49}{8}}$ ãåŸããã, ($a,b$ ã®ãšãæ¹ã«äŸãã)æ±ããå€ã¯ $\\textbf{392}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬... | ã$BC=5\sqrt{2}$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, å
éšã®ç¹ $D,E$ ããšããš
$$AD=AE, \quad BD=3,\quad CE=2\sqrt{6},\quad DE=3$$
ãæãç«ã¡ãŸãã. $D$ ãéã $AB$ ã«åçŽãªçŽç·ãš $E$ ãéã $AC$ ã«åçŽãªçŽç·ã®äº€ç¹ã $F$ ãšã, $F$ ãã $BC$ ã«ããããåç·ã®è¶³ã $P$ ãšãããšã, $BP$ ã®é·ã㯠$ab$ ãš $c$ ã®æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£ã®æŽæ° $a,b,c$ ãçšã㊠$\displaystyle a\sqrt{\frac{b}{c}}$ ãšè¡šãããŸã. $a^{2} bc$ ã解çããŠãã ... |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/97 | D | OMC015(D) | 400 | 20 | 100 | [
{
"content": "ãäžåŒãæå°å€ãåããšã, $a_2,\\cdots,a_{2020}$ ã $0$ ä»¥äž $1$ 以äžã§ããããšã¯æããã§ãã. ãã®ãšã, $xy$ å¹³é¢äžã«ãããŠ, 以äžã®ç¹ãé ã«ç¹ãã æãç·ãèãããš, äžåŒã¯ããå
šäœã®é·ãã«çããïŒ\r\n$$(0,0),(1,1-a_2),(2-a_3,1),(2,2-a_4),(3-a_5,2),\\cdots,(1010,1010-a_{2020}),(1011,1010)$$\r\nãã£ãŠ, æããã«ãããäžçŽç·ãšãªãå Žåã®ã¿ãæå°å€ããšã, ãã®ãšã $a_{1000}=500\\/1011$ ãšèšç®ã§ãããã, æ±ããå€ã¯ $\\te... | ãå®æ°ãããªãæ°å $a_{1} ,a_{2} ,\ldots,a_{2021}$ 㯠$a_{1} =1, a_{2021} =2$ ãæºãããšããŸã.
$$\displaystyle \sum_{k=1}^{2020} \sqrt{a_{k}^{2}+(a_{k+1}-1)^{2}}$$
ãã®ãšã, äžåŒã«ã¯æå°å€ãååšããããšã蚌æã§ããŸã. äžåŒãæå°å€ãåããšã, $a_{1000}$ ãšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã. ãã ã, æ±ããç·åã¯æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£æŽæ° $a,b$ ãçšã㊠$\displaystyle \frac{a}{b}$ ãšè¡šãããã®ã§, $ab$ ã解çããŠãã ãã. |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/98 | E | OMC015(E) | 500 | 10 | 49 | [
{
"content": "ãäŸãã° $\\\\lbrace 2,3,5,1,1,4\\rbrace$ ã $0011100000101111$ ãšããèŠé ã§, æ°åããã€ããªåã«å€æãã. ãã®ãšã, æäœã¯ãé£ãåã $0$ ãš $1$ ãå
¥ãæ¿ãã(䞡端ãé€ã)ããšè¡šçŸã§ãã. æäœãçµäºããã®ã¯ $0$ ããã¹ãŠå·ŠåŽã«, $1$ ããã¹ãŠå³åŽã«å¯ã£ãç¶æ
ã§ãããã, $M$ ã¯åæã®ãã€ããªåã®è»¢åæ°ã«çãã, ãã㯠$\\textbf{376}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/om... | ãæ£ã®æŽæ°ãããªãæéå $X$ ããããŸã. $X$ ã«å¯ŸããŠ, 以äžã®æé ãããªãæäœãç¹°ãè¿ãè¡ããŸã.
- ãŸã, é£ãåã $2$ æ°ãéžæãã. ãã ã, 䞡端ã«äœçœ®ããæ°ãå«ãã§ã¯ãªããªã.
- é£ãåã $2$ æ°ããšãã« $2$ 以äžã®ãšã, éžæãã $2$ æ°ããããã $1$ æžãã, éã« $1$ ã $2$ ã€æ¿å
¥ãã.
- é£ãåã $2$ æ°ããšãã« $1$ ã®ãšã, ãããã®äž¡é£ã«äœçœ®ãã $2$ æ°ããããã $1$ å¢ãã, éžæãã $2$ æ°ãåé€ãã.
- é£ãåã $2$ æ°ã®äžæ¹ã®ã¿ã $1$ ã®ãšã, $1$ ã§ãªãæ¹ã $1$ æžãã, $1$ ã§ããæ¹ã«é£ãåã£ãŠã... |
OMC015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc015/tasks/99 | F | OMC015(F) | 600 | 2 | 25 | [
{
"content": "ã$AB$ ã®é·ãã $171.4x$ ãšãã. ãŸã, äžåŒã®å·ŠèŸºã $f(P)$ ãšãã.\\\r\nã$\\triangle ABC-P$ ãæŽæ°ãšãªãç¹ $P$ ãååšãããã㪠$171$ é¢ã§æ£åé¢äœãåå²ãã. åæ§ã«, ä» $3$ é¢ã«æ²¿ã£ãŠããã«åå²ãããš, $P$ ãåãé åã«ããã° $f(P)$ ã¯äžå®ã§ãã, é¢ãè·šãã°å€ãã¡ããã© $1$ å€åãã.\\\r\nãäžäŸãšããŠ, 以äžã« $\\triangle ABC-P$ ããããã $167+\\delta,168-\\delta$ ã§ãããããªé¢ã§ã®æé¢ãæ瀺ãã. ç¹ $P$ ãèµ€, é, é», ç·ã®é åã«ã... | ãæ£åé¢äœ $ABC-D$ ããã, ãã®äœç©ã¯ $171.4$ ã§ã.\
ã$\triangle WXY-Z$ ã§åé¢äœ $WXY-Z$ ã®äœç©ãè¡šã, $\lfloor x\rfloor$ 㧠$x$ ãè¶
ããªãæ倧ã®æŽæ°ãè¡šããšã,
$$\lfloor \triangle ABC-P\rfloor +\lfloor \triangle ABD-P\rfloor +\lfloor \triangle ACD-P\rfloor +\lfloor \triangle BCD-P\rfloor =169$$
ãªãæ£åé¢äœã®å
éšã®ç¹ $P$ ãååšãåŸãé åã®äœç©ãæ±ããŠãã ãã.\
ããã ã, æ±ããäœç©ã¯æ倧å
¬çŽæ°ã $1$... |
OMC014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc014/tasks/88 | A | OMC014(A) | 100 | 230 | 230 | [
{
"content": "ãç®ã®åºãé çªãèæ
®ãããšãç®ã®åºæ¹ã¯å
šéšã§ $6\\times6\\times6=216$ éããã, ãã®ãã¡ $6$ ãäžåºŠãåºãŠããªããããªãã®ã¯ $5\\times5\\times5=125$ éããã. ãã£ãŠ, äžåºŠã§ã $6$ ãåºããããªç®ã®åºæ¹ã¯ $216-125=91$ éããªã®ã§çã㯠$\\dfrac{91}{216}$ ã§ãã, ããã¯æ¢çŽåæ°ã§ããããæ±ããå€ã¯ $a+b=216+91=\\textbf{307}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/cont... | ã$1,2,3,4,5,6$ ã®ç®ãç確çã«åºãããããã $3$ åæ¯ããšã, $6$ ã®ç®ãå°ãªããšã $1$ ååºã確çãæ±ããŠãã ãã. ãã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£ã®æŽæ° $a,b$ ãçšã㊠$\frac{b}{a}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc014/tasks/89 | B | OMC014(B) | 200 | 202 | 206 | [
{
"content": "ãäžã€ç®ã®æ¡ä»¶ãã¿ãã $4$ æ¡ã®æ£æŽæ°ã¯ïŒ$2$ æ¡ã®æ£æŽæ° $n$ ãçšã㊠$100n+(n+1)=101n+1$ ãšè¡šããïŒããã«, ãããäºã€ç®ã®æ¡ä»¶ãã¿ãããšãïŒããæ£æŽæ° $m$ ãçšããŠ\r\nãã$$101n+1=(m+2)(m-2)=m^2-4$$\r\nãšæžããïŒ$101\\times20+1=45^2-4(=2021)$ ã蟺ã
åŒãããšã§\r\nãã$$101(n-20)=m^2-45^2=(m+45)(m-45)$$\r\nããã㧠$101$ ã¯çŽ æ°ã§ãããã, $m+45,m-45$ ã®å°ãªããšãäžæ¹ã¯ $101$ ã®åæ°ã§ããïŒããªãã¡, $m$ 㯠$9... | ã$2021$ ã¯æ¬¡ã® $2$ ã€ã®æ§è³ªãæ〠$4$ æ¡ã®æ£æŽæ°ã§ãïŒ
- $100$ ã§å²ã£ãäœã㯠$100$ ã§å²ã£ãåãã $1$ 倧ãã
- $2021=43\times47$ ã®ããã«ïŒå·®ã $4$ ã§ãã $2$ ã€ã®æ£æŽæ°ã®ç©ã§è¡šãããšãã§ãã
ãã® $2$ ã€ã®æ§è³ªãæ〠$4$ æ¡ã®æ£æŽæ°ã¯ $2021$ 以å€ã«ãååšããŸãïŒãã®ãããªãã®ã®ç·åãæ±ããŠãã ãã.\
ããã ã, $2021$ ã¯æ±ããç·åã«å«ããªããã®ãšããŸã. |
OMC014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc014/tasks/90 | C | OMC014(C) | 300 | 127 | 154 | [
{
"content": "ãäžè§åœ¢ $PCD$ ã®å€æ¥åãšçŽç· $AB$ ã®æ¥ç¹ã $Q$ ãšãã. ãã®ãšãæ¹ã¹ãã®å®çãã\r\nãã$$AQ^2=AP\\times AC=64,\\ \\ BQ^2=BP\\times BD=36$$\r\nãšãªããã,\r\nãã$$AB=AQ-BQ=8-6=2$$\r\nããããšäžå¹³æ¹ã®å®çãã, äžè§åœ¢ $PAB$ ã®é¢ç©ã¯\r\nãã$$\\dfrac{1}{2}\\times 2\\times \\sqrt{4^2-1^2}=\\sqrt{15}$$\r\nã§äžãããããã, é¢ç©æ¯ãèããããšã§äžè§åœ¢ $PCD$ ã®é¢ç©ã¯\r\nãã$$\\sqrt{15}\\t... | ãç·å $AC$ ãšç·å $BD$ ãç¹ $P$ ã§äº€ãã£ãŠããŸã. ãŸã, äžè§åœ¢ $PCD$ ã®å€æ¥åãšçŽç· $AB$ ã¯æ¥ããŠããŸã. \
$$PA=PB=4,\ PC=12,\ PD=5$$
ã®ãšã, äžè§åœ¢ $PCD$ ã®é¢ç©ãæ±ããŠãã ãã. ãã ã, çãã¯æ倧å
¬çŽæ°ã $1$ ã§ããæ£ã®æŽæ° $a, c$ ãš, $1$ ãã倧ããå¹³æ¹æ°ã§å²ãåããªãæ£ã®æŽæ° $b$ ãçšã㊠$\dfrac{a\sqrt{b}}{c}$ ãšè¡šããã®ã§, $a+b+c$ ã解çããŠãã ãã. |
OMC014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc014/tasks/91 | D | OMC014(D) | 400 | 72 | 149 | [
{
"content": "ãæ¡ä»¶ãæºãã $n$ ã®éåã $S$ ãšããïŒäžè¬æ§ã倱ãã $p\\leq q$ ãšããïŒãã®ãšã $\\gcd(p-1,q)=1$ ã§ãã.\r\n\r\n(i) $2\\lt p\\lt q$ ã〠$\\gcd(p,q-1)=1$ ã®ãšã\r\n\r\nã$p-1,q-1$ ããšãã«å¶æ°ã§ããããšã«çæãããš, Fermatã®å°å®çãã\r\nãã$$\\begin{aligned}\r\nãã(p-1)^{q-1}+(q-1)^{p-1}\\equiv(-1)^{q-1}+1\\equiv2 \\pmod p \\\\\\\\\r\nãã(p-1)^{q-1}+(q-1)^{... | ã次ã®æ¡ä»¶ãã¿ãã $100$ 以äžã®æ£ã®å¶æ° $n$ ã®ç·åãæ±ããŠãã ãã.
- æ¡ä»¶ïŒ$n\lt pq$ ã〠$(p-1)^{q-1}+(q-1)^{p-1}\equiv n \pmod{pq}$ ãã¿ããçŽ æ°ã®çµ $(p,q)$ ãååšãã. |
OMC014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc014/tasks/92 | E | OMC014(E) | 500 | 20 | 61 | [
{
"content": "ãéæ³é£ã®äžå¿ã«å
¥ãæ£æŽæ°ã $n$ ãšãããš, æåäºå®ãšããŠéæ³é£ã®åè¡,åå,å察è§ç·äžã«ãã $3$ ã€ã®æ°ã®å㯠$3n$ ã§ãã. ãããã£ãŠ, 以äžã®ããã«èšç®ã§ãã.\r\n\r\nããã®ãšã, $b$ ãåºå®ã, ãã¹ã«å
¥ããã¹ãŠã®æ°ãæ£ãšãªãæ¡ä»¶ã $an$ å¹³é¢ã«å³ç€ºãããš, 以äžã®ããã«ãªã(å¢çç·äžãå«ãŸãªã).\r\n\r\n$ 㯠$M$ åãããŸã. $M$ ã解çããŠãã ãã.
 | 600 | 45 | 98 | [
{
"content": "ã$a$ ã $b$ ã§å²ã£ãããŸãã $a\\\\%b$ ã§è¡šãããšãšãã. \\\r\nãåé¡ã®æäœã®ãéãã«ãããæäœ $B$ ãèãã. ããªãã¡, äžååãå¶æ°çªç®ã«ãã®é ã«äžŠã¹, äžååãå¥æ°çªç®ã«éé ã«äžŠã¹ãæäœã $B$ ãšãã. ããã $20210106$ åç¹°ãè¿ãããšã, $1$ ãšæžãããã«ãŒããäžããäœçªç®ã«ããããèããã°ãã. \\\r\nãããã§, $2n$ æã®ã«ãŒããçšæã, äžååãå¶æ°çªç®ã«ãã®é ã«, äžååãå¥æ°çªç®ã«ãã®é ã«äžŠã¹ããšããæäœ $B^\\prime$ ãèãã. ãããš, äžãã $k$ æç®ã®ã«ãŒãã¯æäœ $B^\\prime$... | ã$n$ æã®ã«ãŒããç©ãŸããŠãã, äžããé ã« $1,2,\dots,n$ ãæžãããŠããŸã. ããã§, 次ã®ãæäœããèããŸãïŒ
- (æäœ)ïŒãŸãäžããå¥æ°çªç®ã®ã«ãŒãããã¹ãŠåãåºã, éé ã«ããŠéãã. ãã®äžã«æ®ãã®ã«ãŒãããã®ãŸãŸéãã.
ãäŸãã° $n=6$ ã®å Žå, åãã«ãŒãã«ã¯äžããé ã« $1,2,3,4,5,6$ ãæžãããŠããŸãã, æäœã $1$ åè¡ããšäžããé ã« $2,4,6,5,3,1$ ãšãªããŸã. ããã« $1$ åæäœãè¡ããš $4,5,1,3,6,2$ ãšãªããŸã. \
ã$n=2^{10105050}+2$ ã®å Žåã«ãããŠ, æäœã $20210106$ åç¹°ãè¿ãããšã, ... |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/82 | A | OMC013(A) | 100 | 255 | 266 | [
{
"content": "ãäžã®äœã $1$ ã§ãããã®ã¯æããã« $6!=720$ åã§ãã.\\\r\nãäžã®äœã $3$ ã®å Žåã¯, ãŸã $1$ ãåºå¥ããŠããèããããšã§ $6!\\/2=360$ åãšããã.\\\r\nãäžã®äœã $5$ ã®å Žåãåæ§ã« $360$ åã§ãããã, ç·æ°ã¯ $720+360+360=\\textbf{1440}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc013/editorial/82"
}
] | ã$1, 1, 2, 3, 4, 5, 6$ ã䞊ã¹æ¿ããŠã§ãã $7$ æ¡ã®å¥æ°ã¯ $x$ åãããŸã. $x$ ã解çããŠãã ãã. |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/83 | B | OMC013(B) | 200 | 226 | 237 | [
{
"content": "ãäžå³ã®ããã«ç¹ãåããš, åè§åœ¢ $AEOF$ ã¯äžèŸºã®é·ãã $5$ ã®æ£æ¹åœ¢ã§, $\\triangle GPO$ ãš $\\triangle HOQ$ ã¯ååã§ãã. äžå¹³æ¹ã®å®çãã $HQ=GO=\\sqrt{21}$ ã ãã $AD=5+\\sqrt{21}$ ã§, ç¹ã«æ±ããå€ã¯ $5+21=\\textbf{26}$ ã§ãã. \r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathco... | 
ãå³ã®ããã«, é·æ¹åœ¢ $ABCD$ ã®å
éšã«ååŸ $5$ ã®ååãå
æ¥ããŠããŸã. $AB=7$ ã®ãšã, 蟺 $AD$ ã®é·ãã¯æ£æŽæ° $a, b$ ãçšã㊠$a+\sqrt{b}$ ãšè¡šããŸã. $a+b$ ã解çããŠãã ãã. |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/84 | C | OMC013(C) | 300 | 172 | 206 | [
{
"content": "ãæ¹çšåŒã® $2$ 解ã $\\alpha\\geq\\beta$ ãšãããš, 解ãšä¿æ°ã®é¢ä¿ãã $\\alpha+\\beta=2m-960$ ããã³ $\\alpha\\beta=4m+97$ ãæç«ãã. ããããã $m$ ãæ¶å»ãããš $\\alpha\\beta=2(\\alpha+\\beta)+2\\times960+97$ ããªãã¡\r\nãã$$(\\alpha-2)(\\beta-2)=2021=47\\times43$$\r\nãããã $(\\alpha, \\beta)$ ãšããŠããåŸãçµã¯ $(2023, 3)$ ããã³ $(49, 45)$ ãšããã. ã... | ãè€çŽ æ° $x$ ã«ã€ããŠã®æ¹çšåŒ $x^{2}-2(m-480)x+(4m+97)=0$ ã, æ£æŽæ°è§£ã®ã¿ããã€ãããªæŽæ° $m$ ã«ã€ããŠ, ãã®ç·å㯠$M$ ãšãªããŸã. $M$ ã解çããŠãã ãã. |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/85 | D | OMC013(D) | 400 | 91 | 154 | [
{
"content": "ã$n+3$ ã®äœçœ®ãé£ã³è¶ããšã, ãã㯠$n+2$ ã®äœçœ®ããè·é¢ $2$ 以äžãžã£ã³ãããã, $n+1$ ã®äœçœ®ããè·é¢ $3$ 以äžãžã£ã³ãããã, $n$ ã®äœçœ®ããè·é¢ $4$ ãžã£ã³ããããã®ããããã§ãããã, 以äžã®æŒžååŒãåŸã.\r\nãã$$\\displaystyle p_{n+3}=1-\\left(\\frac{3}{4}p_{n+2}+\\frac{2}{4}p_{n+1}+\\frac{1}{4}p_{n}\\right)$$\r\nããããæŽçã㊠$4p_{n+3}+3p_{n+2}+2p_{n+1}+p_n-4=0$ ã§ãããã, æ±ããå€ã¯ $\... | ãæ°çŽç·äžã«ã«ãšã«ããã, åãã«ãšã«ã¯ $0$ ã®äœçœ®ã«ããŸã. ã«ãšã«ã¯æ¬¡ã®æäœãç¡éã«ç¹°ãè¿ããŸãïŒ
- æäœïŒ$1$ ä»¥äž $4$ 以äžã®æŽæ° $m$ ãç確çã«éžã³, æ£ã®æ¹åã« $m$ ã ããžã£ã³ããã.
ããã®ãšã, æ£ã®æŽæ° $n$ ã«ã€ããŠ, $n$ ã®äœçœ®ã«çå°ããããšã®ãã確çã $p_n$ ãšãããš, ä»»æã®æ£ã®æŽæ° $n$ ã«ã€ããŠ
$$ap_{n+3}+bp_{n+2}+cp_{n+1}+dp_n+e=0$$
ãäžæã«æãç«ã¡ãŸã (ãã ã $a, b, c, d, e$ ã¯æ倧å
¬çŽæ°ã $1$ ã®æŽæ°ã§, $a$ ã¯æ£ãšãã, ). \
ã$10000a+1000b+100c+... |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/86 | E | OMC013(E) | 500 | 62 | 170 | [
{
"content": "ãé£ãåã $2$ åŒãããããæ¯èŒããããšã§ä»¥äžãåŸã.\r\nãã$$x_1^2-x_1=x_2^2-x_2=\\cdots=x_{15}^2-x_{15}$$\r\nç¹ã«, $x_1,x_2,\\cdots,x_{15}$ ã«å«ãŸããæ°ã¯é«ã
$2$ çš®é¡ã§ãã. æ¹çšåŒ $x^2+14x=1$ ã¯å®æ°è§£ã $2$ ã€ãã€ãã, ããã $1$ çš®é¡ã§ãããããªãã®ã¯ $2$ åååšãã. 以äžã¡ããã© $2$ çš®é¡ã§ããå Žåãèãã.\\\r\nã$\\alpha$ ã $n$ å, $\\beta$ ã $15-n$ åã§ãããšã, äžè¬æ§ã倱ãã $n\\leq 7$ ã§ãããš... | ã以äžã® $15$ åã®åŒããã¹ãŠã¿ããå®æ°ã®çµ $(x_{1}, x_{2}, \cdots, x_{15})$ ã¯ããã€ãããŸããïŒ
$$\begin{cases}x^{2}\_{1}+x\_{2}+x\_{3}+\cdots+x\_{15}=1 \\\ x\_{1}+x^{2}\_{2}+x\_{3}+\cdots+x\_{15}=1 \\\ x\_{1}+x\_{2}+x^{2}\_{3}+\cdots+x\_{15}=1 \\\ \quad \vdots \\\ x\_{1}+x\_{2}+x\_{3}+\cdots+x^{2}\_{15}=1\end{cases}$$ |
OMC013 (ChristMATHContest 2020) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc013/tasks/87 | F | OMC013(F) | 600 | 15 | 59 | [
{
"content": "ã$AC=AF$ ãªã $BC$ äžã® $C$ ã§ãªãç¹ $F$ ããšããš $CB=BF$ ããã³ $\\angle CDF=90^\\circ$ ã§ãã, $\\angle ADF=180^\\circ-\\angle ACB$ ãããã. ãŸã, $\\triangle ABC$ ã®å€éšã« $\\triangle APC\\equiv\\triangle ADF$ ãã¿ããç¹ $P$ ããšããš, $\\angle PAD=\\angle CAF$ ãš $AP=AD$ ãã $\\angle ADP=\\angle ACB$ ãªã®ã§, $P, D, F$ ã¯å
±ç·ã§ãã. \\\r\nãã... | ã$\angle B=90^\circ$ ãªãçŽè§äžè§åœ¢ $ABC$ ã®å
éšã«ç¹ $D$ ããã,
$$BC=BD,\quad\angle BAC+\angle ADC=180^\circ$$
ãã¿ãããŠããŸã. ç·å $CD$ äžã« $\angle BEC=\angle ACB$ ãªãç¹ $E$ ããšã£ããšãã,
$$CE=8,\quad ED=1$$
ãæç«ããŸãã. ãã®ãšã, 蟺 $BC$ ã®é·ãã¯, æ倧å
¬çŽæ°ã $1$ ã§ããæ£æŽæ° $a, c$ ãš, $1$ ãã倧ããå¹³æ¹æ°ã§å²ãåããªãæ£æŽæ° $b$ ãçšã㊠$\displaystyle \frac{a\sqrt{b}}{c}$ ãšè¡šããŸã. $... |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/76 | A | OMC012(A) | 100 | 210 | 214 | [
{
"content": "ã$10$ å硬貚ãç¡èŠããã°, æ¯æããéé¡ã¯ $0$ åãã $1700$ åã® $18$ éãã§ãã. ããããã«ã€ã㊠$10$ å硬貚ã®åºãæ¹ $0$ æãã $4$ æã§ç°ãªãéé¡ãåŸããããã, $x=18\\times 5-1=\\textbf{89}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc012/editorial/76"
}
] | ã$10$ å硬貚 $4$ æ, $100$ å硬貚 $7$ æ, $500$ å硬貚 $2$ æã®å
šéšãŸãã¯äžéšãçšããŠã¡ããã©æ¯æãããšãã§ããéé¡ã¯ $x$ éããããŸã. $x$ ãæ±ããŠãã ãã.\
ããã ã, å°ãªããšã $1$ æã¯ç¡¬è²šãçšããããšãšããŸã. |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/77 | B | OMC012(B) | 200 | 206 | 209 | [
{
"content": "ã$x=\\dfrac{n}{126}$ ãšããã°, $10x=d.333\\cdots$, $100x=d3.333\\cdots$ ãã $90x=9d+3$ ã§ãã, ããªãã¡\r\nãã$$\\displaystyle n=126x=\\frac{21}{5}(3d+1)}$$\r\nããããæŽæ°ãšãªãã®ã¯ $d=3,8$ ã®ãšãã§, ãããã $n=42,105$ ã§ãããã, æ±ããå€ã¯ $\\textbf{147}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/om... | ã$\displaystyle\frac{n}{126}$ ã $10$ é²æ³ã®å°æ°ã§è¡šãããšãã« $0.d333...$ ãšãªããããªæ£ã®æŽæ° $n$ ã®å€ãšããŠ, ããåŸããã®ã®ç·åãæ±ããŠãã ãã. ãã ã, $d$ 㯠$0$ ä»¥äž $9$ 以äžã®æŽæ°ãšããŸã. |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/78 | C | OMC012(C) | 300 | 191 | 207 | [
{
"content": "ã$p,q,r$ ã $0$ ãšãªãããšãèš±ãã°, æ±ããç·å㯠$3^{10}=59040$ ã§ãã. ãã®ãã¡, $p,q,r$ ã®ãã¡ $2$ ã€ã $0$ ã§ãããããªãã®ã®ç·å㯠$3$ ã§ãã, ã¡ããã© $1$ ã€ã $0$ ã§ãããããªãã®ã®ç·å㯠$3\\times(2^{10}-2)=3066$ ã§ãããã, 以äžããæ±ããå€ã¯ $59049-3-3066=\\textbf{55980}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc012/editor... | ã$p+q+r=10$ ãæºããæ£æŽæ°ã®çµ $(p,q,r)$ ãã¹ãŠã«å¯Ÿã, $\displaystyle\frac{10!}{p!q!r!}$ ã®ç·åãæ±ããŠãã ãã. |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/79 | D | OMC012(D) | 400 | 70 | 172 | [
{
"content": "ããŸã, $n$ åã®ååšã«ãã£ãŠçé¢ã¯æ倧㧠$n^2-n+2$ åã«åå²ã§ããããšã瀺ã. åå²ã®æ倧æ°ãå®çŸããã«ã¯, ã©ã® $2$ åãäºã€ã®äº€ç¹ããã¡, ãã€ã©ã® $3$ åãäžç¹ã§äº€ãããªããã°ãã, ãã®ãããªé
眮ã¯å¯èœã§ãã. ãã®ãšã, çé¢ã $a_n$ åã«åå²ããããšãããš, 挞ååŒ $a_{n+1}=a_{n}+2n$ ãæç«ãããã, $a_{1}=2$ ãšåãã㊠$a_{n}=n^2-n+2$ ã®æç«ãããã.\\\r\nã以äž, $n$ åã®çé¢ã«ãã£ãŠç©ºéã¯æ倧㧠$\\dfrac{n(n^2-3n+8)}{3}$ åã«åå²ã§ããããšã瀺ã. åå²ã®æ倧... | ã$10$ åã®çé¢ã«ãã£ãŠç©ºéãåå²ãããšã, 空éã¯æ倧 $x$ åã«åå²ã§ããŸã. $x$ ãæ±ããŠãã ãã.\
ããã ã, çé¢ã®ååŸã«å¶çŽã¯ãªã, ããçé¢ã¯ä»ã®çé¢ãšäº€ããããšãåºæ¥ãŸã. |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/80 | E | OMC012(E) | 500 | 75 | 105 | [
{
"content": "**è£é¡.**ãäžå³ã§ $AB=AC$ ã®ãšã, $BQ:QC=PQ:QR$ ã§ãã.\r\n\r\n**蚌æ.**ãçŽç· $AC$ äžã« $QR=QR^\\prime$ ãªã $R$ ã§ãªãç¹ $R^\\prime$ ããšããš, $\\angle BPQ=\\angle ARQ=\\angle CR^\\prime Q$ ãæç«ãã. ãããš $\\angle B=\\angle C$ãã $\\triangle BPQ$ ãš $\\triangle CR^\\prime Q$ ã¯çžäŒŒã§ãããã, $BQ:QC=PQ:QR^\\prime=PQ:QR$.\r\n. $a^2bc$ ãæ±ããŠãã ãã.
 |
OMC012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc012/tasks/81 | F | OMC012(F) | 600 | 0 | 0 | [
{
"content": "ãæ£æŽæ° $n$ ã«å¯Ÿã $n$ 次ã®çœ®æ $\\sigma$ ã§ãã£ãŠ $\\sigma(i)\\neq i\\~(i=1,\\dots,n)$ ãã¿ãããã®ã $n$ **次ã®è¯ã眮æ** ãšåŒã¶ããšã«ããïŒ$n$ 次ã®è¯ã眮æã®åæ°ïŒ**ã¢ã³ã¢ãŒã«æ°**ïŒã $a_n$ ãšããã°ïŒç°¡åãªè°è«ã«ãã£ãŠ $N=2048!\\times(a_{2047}+a_{2048})$ ãåŸãããïŒ\r\n\r\nããã㧠$a_n$ ã®äžè¬é
ã¯æ¬¡ã®åœ¢ã«è¡šãããããšãç¥ãããŠããïŒ\r\n$$a_n=\\sum_{k=0}^{n}\\frac{(-1)^kn!}{k!}$$\r\n\r\n<deta... | ã$2047$ è¡ $2048$ åã®ãã¹ç®ããã, ããã«é»ç³ãšçœç³ã $2047$ åãã€çœ®ããŸã. ãã ã, åããã¹ã«è€æ°ã®ç³ã眮ãããšã¯ã§ããŸãã. åè¡ã«çœ®ãããç³ãçœé»ããããé«ã
$1$ å, ååã«çœ®ãããç³ãçœé»ããããé«ã
$1$ åãšãªããããªç³ã®çœ®ãæ¹ã¯ $N$ éããããŸã.\
ã$N$ ã $2$ ã§å²ãåããåæ°ã $a$, $N$ ã $3$ ã§å²ãåããåæ°ã $b$ ãšãããšã, $ab$ ãæ±ããŠãã ãã.\
ãããã§, å転ãè£è¿ãã§åäžãšãªããã®ãç°ãªããã®ãšããŠæ°ããããšãšããŸã. |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/70 | A | OMC011(A) | 100 | 199 | 228 | [
{
"content": "ãæšå¹Žã®äººå£ã¯äžæšå¹Žã® $27\\/25$ å, ä»å¹Žã® $24\\/25$ åã§ããããšãã, $\\mathrm{lcm}(27,24)=216$ ã®åæ°ã§ãã. 倧å°ã®æ¡ä»¶ãã $216\\times 8=1728$ ã $216\\times 9=1944$ ãšãªãã»ããªã, ãã®ãã¡ä»ã®2幎ãåæ¡ä»¶ãã¿ããã®ã¯ $\\textbf{1728}$ ã®ãšãã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc011/editorial/70"
}
] | ãTKGçºã®äººå£ãèããŸã. äžæšå¹Žã®äººå£ãããšã«ãããšæšå¹Žã®äººå£ã¯ã¡ããã© $8\\%$ å€ã, ä»å¹Žã®äººå£ãããšã«ãããšæšå¹Žã®äººå£ã¯ã¡ããã© $4\\%$ å°ãªãã§ã. ãããã®å¹Žã®äººå£ã $1550$ äººä»¥äž $1950$ 人以äžã§ãããšã, æšå¹Žã®äººå£ãšããŠèãããããã®ã®åèšã解çããŠãã ãã. |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/71 | B | OMC011(B) | 200 | 198 | 207 | [
{
"content": "ã$3^6\\equiv3^2\\pmod{720}$ ãã, æå
ã«ããã©ã ãç¶ã®æ¬æ° $N$ ã«ã€ããŠåŒãæããè¡ã£ãŠã $3^{N}$ ã $720$ ã§å²ã£ãäœãã¯äžå€ã§ãã. ãã£ãŠ $3^n\\equiv 3^4\\equiv\\textbf{81}\\pmod{720}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc011/editorial/71"
}
] | ãããé§èåå±ã§ã¯, $\displaystyle 1$ æ¬ $\displaystyle 100$ åïŒçšèŸŒïŒã®ã©ã ãã売ãããŠããŸã. ãŸãããã§ã¯, 飲ã¿çµãã£ãã©ã ãã®ç¶ã $\displaystyle 6$ æ¬æã£ãŠãããš, ãããšåŒãæãã«æ°ãã«ã©ã ãã $\displaystyle 2$ æ¬ããã, ãã®ã©ã ãããŸãåŒãæãã«äœ¿ãããšãã§ããŸã.\
ãã㟠$\displaystyle 100n$ åæã£ãŠããŸã($n$ ã¯æ£æŽæ°). ãã®ãéã䜿ã£ãŠ, ã§ããã ãå€ãã®ã©ã ãã飲ãã ãšãã, æå
ã«ã¯ã©ã ãã®ç¶ã $\displaystyle 4$ æ¬æ®ããŸãã. ãã®ãšã, $\displaystyle 3^{n... |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/72 | C | OMC011(C) | 300 | 137 | 198 | [
{
"content": "ãäžåŒã $k$ ãšããã°, 以äžã®ããã«å€åœ¢ã§ãã.\r\nãã$$(k+a)(k-a)=2^83^{10}5^{12}$$\r\nããã㧠$k\\pm a$ ã®å¶å¥ã¯äžèŽãããã, ç¹ã«ããããå¶æ°ã§ãã. ããªãã¡, 以äžãã¿ããæ£æŽæ°ã®çµ $x\\geq y$ ã®æ°ãæ±ããããšã«åž°çããã.\r\nãã$$xy=2^63^{10}5^{12}$$\r\nããšãã㧠$2^63^{10}5^{12}$ ã¯æ£ã®çŽæ°ã $(6+1)(10+1)(12+1)=1001$ åãã€ã, å¹³æ¹æ°ã§ããããšãã $x=y$ ãªãçµãäžã€ååšããããšã«çæããã°, æ±ããå€ã¯ $\\textbf{... | ã$\displaystyle \sqrt{a^{2} +2^{8} 3^{10} 5^{12}}$ ãæŽæ°ãšãªããããªéè² æŽæ° $\displaystyle a$ ãšããŠããåŸãå€ã¯ããã€ãããæ±ããŠãã ãã. |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/73 | D | OMC011(D) | 400 | 87 | 121 | [
{
"content": "ãäžå³ã®ããã«åº§æšãèšå®ã, $ABC$ ããã³ãããšååãªäžè§åœ¢ãã¡ãåã蟌ã.\r\n\r\nããã®ãšã, $A^\\prime B^\\prime C^\\prime$ ã®å
å¿ã $I^{\\prime}$ ãšããã°, æ±ããæå°å€ã¯ç·å $II^\\prime$ ã®é·ãã«çããããšãããã. ãªãå³å¯ã«ã¯ãããç·å $BC^\\prime$ ããã³ $A^\\prime C^\\prime$ ãšäº€ããããšã瀺ãå¿
èŠãããã, ããã¯èªè
ãžã®æŒç¿ãšãã.\... | ã$AB=2,BC=1,CA=\sqrt{3}$ ã§ããäžè§åœ¢ $ABC$ ã®å
å¿ã $I$ ãšããŸã. ç¹ $P$ ã蟺 $AB$ äžã, ç¹ $Q$ ã蟺 $BC$ äžã, ç¹ $R$ ã蟺 $CA$ äžãããããåããšã, $IP+PQ+QR+RI$ ã®ãšãåŸãæå°å€ã¯æ£ã®æŽæ° $a, b, c$ ãçšã㊠$\sqrt{a-b\sqrt{c}}$ ãšè¡šãããŸã. $a+b^{2} c$ ãæ±ããŠãã ãã. |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/74 | E | OMC011(E) | 500 | 75 | 105 | [
{
"content": "ããŸã $f(2n)=f(n)$ ã§ããããšã瀺ã.\\\r\nã$(1+x)^{2n}=[(1+x)^n]^2$ ã®äž¡èŸºã® $x^{2m}$ ã®ä¿æ°ãèããããšã§\r\nãã$$\\_{2n}\\mathrm{C}\\_{2m}=(\\_{n}\\mathrm{C}\\_{m})^2+2\\_{2n}\\mathrm{C}\\_{m+1}\\cdot\\_{2n}\\mathrm{C}\\_{m-1}+\\cdots$$\r\nããªãã¡ $\\_{2n}\\mathrm{C}\\_{2m}$ ãš $\\_{n}\\mathrm{C}\\_{m}$ ã®å¶å¥ã¯äžèŽãã. åæ§ã«, åãåŒã® ... | ãæ£æŽæ° $n$ ã«å¯ŸããŠ, ${}\_{n}\mathrm{C}\_{k}$ ãå¥æ°ã§ãããããªæŽæ° $0\leq k\leq n$ ã®åæ°ã $f(n)$ ã§è¡šããŸã. äŸãã°,
$${}\_{3}\mathrm{C}\_{0} =1,\quad {}\_{3}\mathrm{C}\_{1} =3,\quad {}\_{3}\mathrm{C}\_{2} =3,\quad {}\_{3}\mathrm{C}\_{3} =1$$
ãªã®ã§, $f(3) =4$ ã§ã. ãã®ãšã, $\displaystyle \frac{f\left( 10^{16} +7\times 2^{17}\right)}{f\left( 10^{16... |
OMC011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc011/tasks/75 | F | OMC011(F) | 600 | 44 | 74 | [
{
"content": "ã$AEGF$ ãé·æ¹åœ¢ãšãªããããªç¹ $G$ ããšããš, æ±ããé¢ç©ã®å·® $S$ ã¯åè§åœ¢ $BCDG$ ã®é¢ç©ã«çããããšã容æã«ããã. ããã«, $B$ ã«ã€ã㊠$E$ ãšå¯Ÿç§°ãªç¹ $P$ ããã³ $D$ ã«ã€ã㊠$F$ ãšå¯Ÿç§°ãªç¹ $Q$ ããšããš, $F,E,P,Q$ 㯠$C$ ãäžå¿ãšããååŸ $5$ ã®ååšäžã«ãã, å
è§åœ¢ $EPRQFG$ ã®é¢ç©ã¯ $4S$ ã«çããããšãããã. ãã ã $R$ 㯠$APRQ$ ãé·æ¹åœ¢ãšãªããããªç¹ã§ãã. ããã§æ¹ã¹ãã®å®çãã\r\nãã$$AE\\times AP=AC^2-5^2=AF\\times AQ$$\r\n... | ã察è§ç·ã®é·ãã $6$ ã§ããé·æ¹åœ¢ $ABCD$ ããããŸã. 蟺 $AB$ äžã« $CE=5$ ãªãç¹ $E$ ã, 蟺 $AD$ äžã« $CF=5$ ãªãç¹ $F$ ãåã£ããšãã, äžè§åœ¢ $AEF$ ã®é¢ç©ã¯ $\displaystyle\frac{3}{2}$ ãšãªããŸãã.\
ã$EF$ ã®äžç¹ã $M$ ãšãããšã, åè§åœ¢ $ABMD\\,(=\triangle ABM+\triangle ADM)$ ãšåè§åœ¢ $BCDM$ ã®é¢ç©ã®å·®ã¯, æ倧å
¬çŽæ°ã $1$ ã§ãããããªæ£æŽæ° $m, n$ ãçšããŠ, $\displaystyle \frac{m}{n}$ ãšè¡šããŸã. $m+n$ã解çããŠãã ãã. |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/64 | A | OMC010(A) | 100 | 168 | 169 | [
{
"content": "ãã¯ããã® $12$ ç§éã®ãã¡ $A,B$ ããšãã«å
ã£ãŠããã®ã¯ $6$ ç§éã§ãã. 以éã¯ãã® $12$ ç§éã®å
ãæ¹ãåšæãšããŠç¹°ãè¿ããã, æ±ããå€ã¯ $\\textbf{60}$ ç§éã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc010/editorial/64"
}
] | ã$A, B$ ã® $2$ ã€ã®é»çããããŸã. ã¹ã€ãããå
¥ãããš $2$ ã€ã®é»çã¯åæã«å
ãã¯ãã, $A$ ã®é»ç㯠$2$ ç§éå
ã£ãŠã¯æ¬¡ã® $1$ ç§éæ¶ãããšããããšãç¹°ãè¿ã, $B$ ã®é»ç㯠$3$ ç§éå
ã£ãŠã¯æ¬¡ã® $1$ ç§éã¯æ¶ãããšããããšãç¹°ãè¿ããŸã.\
ãã¹ã€ãããå
¥ããŠãã $120$ ç§éã§, $A, B$ äž¡æ¹ã®é»çãå
ã£ãŠããã®ã¯åèš $x$ ç§éã§ã. $x$ ã解çããŠãã ãã. |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/65 | B | OMC010(B) | 200 | 162 | 163 | [
{
"content": "ãäœåŒŠå®çãããã $2$ ã€ã®è§ã®äœåŒŠã«ã€ã㊠$-\\dfrac{1}{\\sqrt{2}},\\dfrac{\\sqrt{3}}{2}$ ãšèšç®ã§ãã. ãã£ãŠæ®ãã®è§ã®å€§ãã㯠$180^\\circ-135^\\circ-30^\\circ=\\textbf{15}^\\circ$ ã§ãã, ãããæå°ã§ãããã解çãã¹ããã®ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc010/editorial/65"
}
] | ãäžèŸºã®é·ãã $\displaystyle 3, \frac{3 \sqrt{2}}{2}, \frac{-3+3 \sqrt{3}}{2}$ ã®äžè§åœ¢ã«ãããŠ, æãå°ããè§ã¯ $x$ 床ã§ã. $x$ ã解çããŠãã ãã. |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/66 | C | OMC010(C) | 300 | 88 | 108 | [
{
"content": "ãé«æšããã®èãã«åŸããããªé²ã¿æ¹ã**çµè·¯**ãšåŒã¶ããšã«ãã.\\\r\nã$n$ åã®è¡ãéããããªããçµè·¯ãåºå®ãããšã, ãã®çµè·¯ã«ã¯ $n-1$ æ¬ã®éãå«ãŸããããšãã, ãããå®çŸã§ãã確ç㯠$\\left(\\dfrac{6}{7}\\right)^{n-1}$ ã§ãã. ãŸã $n$ åã®è¡ãéããããªçµè·¯ã¯ $\\_{98}\\mathrm{C}\\_{n-2}$ éãååšãããã, æ±ããæåŸ
å€ã¯\r\n$$\\sum_{n=2}^{100}{}\\_{98}\\mathrm{C}\\_{n-2}\\left(\\frac{6}{7}\\right)^{n-1}=... | ãããåœã«ã¯ $100$ åã®è¡ããã, è¡ $1$ ããè¡ $100$ ãŸã§ã®çªå·ãä»ããããŠããŸã.
ãŸã, ãããã®ãã¡ã¡ããã© $2$ ã€ã®éãçŽæ¥ç¹ããããªéã $0$ æ¬ä»¥äžäœãããŠããŸã.\
ãè¡ $1$ ã«äœãé«æšããã¯è¡ $100$ ãžæ
è¡ã«è¡ãããã§ãã, ãã®è¡ãæ¹ã«ã€ããŠãè¡ã®çªå·ãå°ãããªã£ãŠããŸãããã«ç§»åããããã£ãšé åãã«ãªã£ãŠããŸããã, è¡ã®çªå·ã倧ãããªã£ãŠããããã«éãé²ãã§ãããããšèããŠããŸã. é«æšããã®èããæ£ãããã©ããããããŸããã, ãããä¿¡ããŠèãã®éãã«é²ããšã, è¡ $1$ ããè¡ $100$ ãŸã§è¡ãæ¹æ³ãäœéãããããç¥ãããã§ã(ãã®ãããªæ¹æ³ãååšããªãããšãã... |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/67 | D | OMC010(D) | 400 | 96 | 128 | [
{
"content": "ãéå $A$ ã® $n$ çªç®ã®èŠçŽ ã $b_n$ ãšãã, $A$ ã®èŠçŽ ã®æå°å€ã $m$ ãšãã.\\\r\nããŸã, 以äžã®äžçåŒãã $m\\leq8000$ ãããã.\r\n$$500m\\leq\\displaystyle\\sum_{n=1}^{500}b_{n}=\\left(\\sum_{n=1}^{500}a_n\\right)^2\\leq4000000$$\r\nãéã« $a_{n}=\\sqrt{8000}(\\sqrt{n}-\\sqrt{n-1})$ ãšããã°, $a_{1}+\\cdots+a_{500}=2000$ ã§ãã, ãã€ä»»æã® $n$ ã«ã€... | ãæ£ã®å®æ°ãããªãæ°å $a_{1}, a_{2}, \cdots , a_{500}$ 㯠$a_{1}+a_{2}+...+a_{500}\leq2000$ ãæºãããšããŸã. ãã®ãšã,
$$A=\lbrace a_{1}^{2}, a_{2}^{2}+2a_{2}a_{1}, a_{3}^{2}+2a_{3}(a_{1}+a_{2}), \cdots , a_{500}^{2}+2a_{500}(a_{1}+a_{2}+...+a_{499})\rbrace$$
ã«å«ãŸããæ°ã®æå°å€ãšããŠããåŸã, æ倧ã®å€ãæ±ããŠãã ãã.\
ã圢åŒçã«ã¯, éå $A$ ã«ãã㊠$n$ çªç®ã®èŠçŽ ã¯, $a_{0}=0$ ãšããŠ... |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/68 | E | OMC010(E) | 500 | 0 | 0 | [
{
"content": "ãçŸåšå·çäžã§ã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc010/editorial/68"
}
] | ã$AB=BC=4$, $\angle B=90^{\circ}$ ãã¿ããäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $AC$ äžã«ç¹ $D$, ç·å $BD$ äžã«ç¹ $E, F$ ããã, $BE=FD, EF=1$ ãæºãããŠããŸã. ãŸã $B, E, F, D$ ã¯ãã®é ã«ãããŸã.ããã§çŽç· $AE$ ãšçŽç· $CF$ ã®äº€ç¹ã $G$ ãšãããš, äžè§åœ¢ $EFG$ ã®é¢ç©ã¯ $\displaystyle\frac{1}{2}$ ã§ãã.\
ããã®ãšã, $AD\times DC$ ãšããŠããåŸãå€ã®ç·**ç©**ã¯, æ倧å
¬çŽæ°ã $1$ ã§ãããã㪠$2$ ã€ã®æ£æŽæ° $x, y$ ãçšã㊠$\displaystyl... |
OMC010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc010/tasks/69 | F | OMC010(F) | 600 | 18 | 54 | [
{
"content": "ããŸã㯠$N$ ã $3$ ã§å²ã£ãäœããå©çšã㊠$N$ ã®å€ã®åè£ãæ±ããã.\\\r\nã$N\\equiv 0\\pmod 3$ ã®ãšã, $x=N^{2}-1,n=3,m=\\dfrac{N}{3}(N^{2}-2)$ ãšããã°ä»®å®ãã $n^{2}-m^{2}$ 㯠$x$ ã§å²ãåãã. ãã£ãŠ, 以äžãæŽæ°ãšãªãããšãã, $N=3,9$ ãåè£ãšããŠåŸã.\r\nãã$$\\dfrac{9\\left(n^{2}-m^{2}\\right)}{x}=\\dfrac{81-N^{2}\\left(N^{2}-2\\right)^{2}}{N^{2}-1}=-N^4-3N^... | ãä»»æã®æŽæ° $x,n,m$ ã«å¯ŸããŠ, 次ã®æ¡ä»¶ãæç«ãããããªæ£æŽæ° $N$ ã®ç·**ç©**ãæ±ããŠãã ãã.
- æ¡ä»¶ïŒ$x\mid N^2-1$ ã〠$\displaystyle nm=\frac{x^{2}-1}{N}$ ã®ãšã, $x\mid n^{2}-m^{2}$
ããã ã, æŽæ° $k, l$ ã«ã€ããŠ, $k$ ã $l$ ãå²ãåããšã, $k\mid l$ ãšè¡šãããšãšããŸã. |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/176 | A | OMCäžæ¬æ¯(A) | 200 | 66 | 69 | [
{
"content": "ã以äžã®ãªã³ã¯ãã芧ãã ãã. å°æ¥çã«ç§»æ€ãäºå®ããŠããŸã.\r\n\r\nãhttps:\\/\\/drive.google.com\\/file\\/d\\/1bVg2wRfG8ZxUsNak8meyfOtmq_l0F23K\\/view",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/176"
}
] | äœå: 倧平ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ãä»æ¥ã¯ä»€å $\displaystyle 2$ 幎 $\displaystyle 11$ æ $\displaystyle 8$ æ¥ã§ããïŒäžã®ããã«äžéšåã空æ¬ã«ãªã£ãããç®ã®çç®ããããŸãïŒãã®ããç®ã®ãããæ°(å³ã®ã¢ã€ãŠãš)ãšããŠèãããããã®ããã¹ãŠæ±ãïŒãã®ç·åã解çããŠãã ãã.
ããã ã, åè¡ã®æäžäœã« $\displaystyle 0$ ãå
¥ãããšã¯ãªããã®ãšããŸã.
--... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/177 | B | OMCäžæ¬æ¯(B) | 200 | 57 | 78 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/177"
}
] | äœå: éŽæšã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
ããã¹ãŠ $2$ æ¡ã®æ£æŽæ° $a,b,c,d,e$ ã, 次ã®æ¡ä»¶ããã¹ãŠã¿ãããŠããŸãïŒ
- $a,b,c,d,e$ ã®åæ¡ãèŠããš, $0$ ãã $9$ ã®æ°åã $1$ åãã€çŸãã.
- $a,b,c,d,e$ ã¯ãã¹ãŠ, åã®äœã $2$ ã§ãããããªçŽæ°ãæã€.
ããã®ãšã, $\displaystyle a+b+c+d+e$ ãšããŠããåŸãå€ããã¹ãŠæ±ãïŒãã®ç·åã解çããŠãã ãã.
----
ãäžæ¬æ¯ã®åé¡ã®çãã¯ãã¹ãŠæŽæ°å€ã«ãªããšã¯éããŸãã. çããæŽæ°ã«ãªã£ãå Žåã¯, ãã®æŽæ°ãã... |
OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/178 | C | OMCäžæ¬æ¯(C) | 200 | 31 | 42 | [
{
"content": null,
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcnakamoto/editorial/178"
}
] | äœå: å¹³ç³ã**解çã«ãŒã«ãéåžžã®ã³ã³ãã¹ããšç°ãªããŸã. æ«å°Ÿãã確èªãã ãã.**
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OMCäžæ¬æ¯ | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcnakamoto/tasks/179 | D | OMCäžæ¬æ¯(D) | 200 | 73 | 74 | [
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{
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