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 |
|---|---|---|---|---|---|---|---|---|---|
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/270 | D | OMC074(D) | 600 | 96 | 142 | [
{
"content": "ã$AX$ ã¯å $ABC$ ã®çŽåŸã§ãã. ãŸã, $AYX$ ãš $XYZ$ ã®å€æ¥åååŸãçããããšãã $\\angle XAY=\\angle XZY$ ãæç«ã (ãããããšãã«éè§ã§ããããšã«çæãã), ç¹ã« $AX=XZ$ ããã³ $AY=XY$ ãæç«ããããšãããã.\\\r\nã äžè§åœ¢ $AXY$ ã®åå¿ã $H$ ãšã, äžè§åœ¢ $ABC,AHX$ ã®å€å¿ããããã $O_1,O_2$ ãšãããš, æåäºå®ãšããŠäžè§åœ¢ $AHX$ ãš $AXY$ ã®å€æ¥åååŸãçããããšãã $AO_2=HO_2=25$ ã§ãã. ãããã£ãŠ,\r\n$$O_1O_2=\\s... | ã$AB\lt AC$ ãªãéè§äžè§åœ¢ $ABC$ ã«ãããŠ, $A$ ãã察蟺ã«ããããåç·ã $s_A$, $B$ ãéã $AB$ ã«åçŽãªçŽç·ã $s_{B}$, $C$ ãéã $AC$ ã«åçŽãªçŽç·ã $s_C$ ãšããŸã. ãŸã $s_{B}$ ãš $s_{C}$ ã®äº€ç¹ã $X$, $s_{A}$ ãš $s_{B}$ ã®äº€ç¹ã $Y$, $s_{A}$ ãš $s_{C}$ ã®äº€ç¹ã $Z$ ãšããŸã. äžè§åœ¢ $ABC$ ã®å€æ¥åååŸã $24$, äžè§åœ¢ $AXY$ ããã³ $XYZ$ ã®å€æ¥åååŸããšãã« $25$ ã§ãããšã, 蟺 $BC$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{... |
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/1698 | E | OMC074(E) | 700 | 13 | 49 | [
{
"content": "ã$1$ ãã $100$ ã®çªå·ãä»ãã $100$ åã®é ç¹ã«ã€ããŠ, $pe\\overbrace{p...p}^{n}er$ ãããçŽ çŽãªãã°é ç¹ $n$ ãçœã§å¡ã, $pe\\overbrace{p...p}^{n}er$ ãããç
§ãå±ãããªãã°é ç¹ $n$ ãé»ã§å¡ã. ããã«, é ç¹ $n$ ããé ç¹ $a_n$ ãžæå蟺ã匵ã£ãã°ã©ããèãã. ãã®ã°ã©ãã«ãããŠ, ã©ã®èŸºã䞡端ã®é ç¹ã®è²ã¯ç°ãªã, ã©ã®é ç¹ããã äžã€ã® $2$ ãã倧ããªå¶æ°åã®é ç¹ãããªãéè·¯ã«å«ãŸããŠãã. ãããã£ãŠ, ç¹ã«çœãé ç¹ã®æ°ãšé»ãé ç¹ã®æ°ã¯çããããšã«æ³šæãã. \r\n\r\n---... | ã$peer$ æã«ã¯, $peper$ ãã, $pepper$ ãã, ... , $pe\overbrace{p...p}^{100}er$ ããã® $100$ 人ãäœãã§ããŸã. $peer$ æã®äœäººã®æ§æ Œã¯ã©ãã**çŽ çŽ**ã**ç
§ãå±ãã**ã§ãã, æ§æ ŒãçŽ çŽãªè
ã¯åžžã«çŽ çŽã«æ¬åœã®ããšãèšã, ç
§ãå±ããã¯åžžã«ç
§ããŠåãã€ããŠããŸããŸã. ãŸã, $peer$ æã®äœäººã¯ã©ã®äºäººãäºãã®æ§æ ŒãçŽ çŽã§ãããç
§ãå±ããã§ããããææ¡ããŠããŸã.\
ãããæ, $peer$ æã«æ
人ããã£ãŠããŸãã. æ
人ã¯, ãŸã $1,2,...,100$ ã䞊ã³æ¿ããé å $a_1,a_2,...,a_{100}$ ã§ãã£ãŠ $a... |
OMC074 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc074/tasks/2555 | F | OMC074(F) | 700 | 9 | 25 | [
{
"content": "ã$\\displaystyle f(x, y) = \\frac{x}{xy+1}$ ãšãããš, ä»»æã®æ£ã®å®æ°ã®çµ $(p, q, r)$ ã«å¯ŸããŠ\r\n$$f(f(p, q), r) = f(p, q + r)$$\r\nãæç«ãã. ãã£ãŠ, $S_n = x_1 + x_2 + \\cdots + x_n$ ãšå®ãããš, $x_{n+1}=f(x_{n},x_{n})$ ããåž°çŽçã«\r\n$$\r\nS_{n + 1} - S_{n} = x_{n + 1} = f(x_1, x_1 + \\cdots + x_n) = \\frac{1}{S_n + \\frac{592... | ãæ£ã®å®æ°ãããªãæ°å $\lbrace x_n\rbrace$ ã¯æ¬¡ãã¿ãããŸã.
$$x_1 = \frac{3141}{5926},\quad x_{n + 1} = \frac{x_n}{x_n^2+1}\quad (n = 1,2,\ldots)$$
ãã®ãšã, 以äžã®æ¥µé
$$
\lim_{n\rightarrow \infty} \frac{x_{n+1}+x_{n+2}+\cdots+x_{5358n}}{x_1+x_2+\cdots+x_n}
$$
ã¯ããå®æ° $k$ ã«åæããããšã瀺ããŸã.\
ãããã§, $k$ ã®æå°å€é
åŒã $P$ ãšã㊠$|P(10000)|$ ã解çããŠãã ãã. ... |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/1536 | A | OMC073(A) | 200 | 280 | 304 | [
{
"content": "ãäžåŒã¯ä»¥äžã®ããã«è¡šçŸã§ããïŒ\r\n$$1.575+\\frac{1}{7}+\\frac{1}{9}$$\r\nããã« $1\\/7$ ã¯å°æ°ç¹ä»¥äžã§ $142857$ ãç¹°ãè¿ãïŒ$1\\/9$ ã¯å°æ°ç¹ä»¥äžã§ $1$ ãç¹°ãè¿ãããïŒäžåŒã¯\r\n$$1.828,968,253,968,253,968,\\ldots$$\r\n$924=6\\times(1+153)\\$ ã§ããããšããïŒæ±ããå€ã¯ä»¥äžã®ããã«èšç®ã§ããïŒ\r\n$$(8+2+8+9+6+8)+153\\times(2+5+3+9+6+8)=\\textbf{5090}$$",
"text": "å
¬åŒè§£... | ã以äžã®å€ãåé²æ³è¡šèšã®å°æ°ã§è¡šãããšãïŒå°æ°ç¹ä»¥äžç¬¬ $1$ äœãã第 $924$ äœãŸã§ã®ç·åãæ±ããŠãã ããïŒ
$$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}$$ |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/1675 | B | OMC073(B) | 200 | 282 | 297 | [
{
"content": "ãäžè§åœ¢ $ABC$ ãšçŽç· $PM$ ã«å¯ŸããŠMenelausã®å®çãé©çšããããšã§, $AQ:QC=7:18$ ãåŸã.\\\r\nããŸã, äžè§åœ¢ $MBP$ ãšçŽç· $AC$ ã«Menelausã®å®çãé©çšããããšã§, $PQ:QM=14:11$ ãåŸã.\\\r\nã以äžãã, æ±ããã¹ãé¢ç©æ¯ã¯ $14Ã7:18Ã11=49:99$ ã§ãããã, 解çãã¹ãå€ã¯ $49+99=\\textbf{148}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc073/e... | ãäžè§åœ¢ $ABC$ ã«ãããŠ, $BC$ ã®äžç¹ã $M$ ãšã, 蟺 $AB$ ã® $A$ åŽã®å»¶é·ç·äžã« $PA:AB=7:11$ ãªãç¹ $P$ ããšããŸã. ãã®ãšã, ç·å $MP$ ãš $AC$ ã®äº€ç¹ã $Q$ ãšããã°, äžè§åœ¢ $APQ$ ãšäžè§åœ¢ $CMQ$ ã®é¢ç©æ¯ã¯, äºãã«çŽ ãªæ£æŽæ° $x,y$ ã«ãã£ãŠ $x:y$ ãšè¡šãããŸã. $x+y$ ã解çããŠãã ãã. |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/1679 | C | OMC073(C) | 300 | 205 | 228 | [
{
"content": "ã$x^{n-3}$ ã®ä¿æ°ã¯, $k$ 以äžã®æ£æŽæ°ãã $3$ ã€ãéžã¶æ¹æ³ãã¹ãŠã«ã€ããŠãããã®ç©ã足ãåããããã®ã«çãã,\r\n$$\\dfrac{1}{6}\\Biggl(\\left(\\sum_{k=1}^{n} k\\right)^3-3\\left(\\sum_{k=1}^{n} k^2\\right)\\left(\\sum_{k=1}^{n} k\\right)+2\\left(\\sum_{k=1}^{n} k^3\\right)\\Biggr)=\\dfrac{1}{48}(n-2)(n-1)n^2(n+1)^2$$\r\nãã㯠$n$ ã«ã€ããŠå調ã§ãã,... | ã$x$ ã®å€é
åŒ $\displaystyle\prod_{k=1}^{n}(x+k)$ ã«ãããŠ, $x^{n-3}$ ã®ä¿æ°ã $55770$ ã§ãããããª, æŽæ° $n\geq 3$ ã®ç·åãæ±ããŠãã ãã. |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/1638 | D | OMC073(D) | 400 | 63 | 157 | [
{
"content": "ãéœåžãé ç¹ïŒéã蟺ãšããããšã§é路網ã(ç¡å)ã°ã©ããšã¿ãªãïŒäžè¬ã«ïŒé£çµãªã°ã©ãã«Eulerè·¯ãååšããªã(äžçæžãäžå¯èœã§ãã)ããšã¯ïŒæ¬¡æ°ãå¥æ°ã®é ç¹ã $4$ å以äžããããšãšåå€ã§ããïŒããã§ã¯ãã¹ãŠã®é ç¹ã®æ¬¡æ°ãå¥æ°ã§ããããšãæå³ããïŒæ¬¡æ°ãå¥æ°ã®é ç¹ã $0,2,4$ åãšãªã $i$ åã®æäœã®çµã¿åããããããã $a_i,b_i,c_i$ éãã§ãããšããïŒãã ãïŒ$a_0=1,b_0=0,c_0=0$ ãšããïŒãã®ãšãïŒä»¥äžã®é¢ä¿ãæãç«ã€ïŒ\r\n$$a_{i+1}=c_{i+1}=b_i,\\quad a_{i}+b_{i}+c_{i}=6^{i}$$\r\n... | ã$4$ ã€ã®éœåž $A,B,C,D$ ãããïŒ
ã¯ãã㯠$AB,BC,CD,DA$ éã«ããããéè·¯ã $1$ æ¬ãã€ãããŸãïŒ
ãããžïŒä»¥äžã®æäœã $10^9+10$ åç¹°ãè¿ããŸãïŒ
- ç°ãªã $2$ éœåžãéžã³ïŒããããåæ¹åã«çµã¶éè·¯ãæ°ãã« $1$ æ¬å»ºèšããïŒ
ã$10^9+10$ åã®æäœã®çµã¿åãã㯠$6^{10^9+10}$ éããããŸããïŒ
ãã®ãã¡åŸãããé路網ã«ã€ããŠïŒ
ãã¹ãŠã®éè·¯ãã¡ããã© $1$ åãã€éããããªéé ãååšããªããããªæäœã®çµã¿åããã¯äœéããããŸããïŒ
çŽ æ° $10^9+7$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ
ããã ãïŒéè·¯ã¯äº€å·®ããªããã®ãšãïŒéäžã§åŒ... |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/1722 | E | OMC073(E) | 500 | 25 | 148 | [
{
"content": "ããŸã $n$ ãå¹³æ¹å åãæã€ãšã, ããªãã¡ããçŽ æ° $p$ ã«å¯Ÿã㊠$n$ ã $p^2$ ã§å²ãåãããšã, $p^k\\equiv p\\pmod{n}$ ãšãªãåŸãªããã $f(n)=-1$ ã§ãã. ãããã£ãŠ, ä»¥äž $n=p_1p_2\\cdots p_l$ ($p_1,p_2,\\dots,p_l$ ã¯çžç°ãªãçŽ æ°)ãšè¡šãããå Žåã®ã¿èããã°ãã. ãã®ãšã, Fermatã®å°å®çãã\r\n$$k^\\prime=\\textrm{lcm}(p_1-1,p_2-1,\\cdots,p_l-1)+1$$\r\nãšãããš, $k=k^\\prime$ ã¯æ¡ä»¶ãã¿ãã. ... | ãæŽæ° $n\geq 2$ ã«å¯Ÿã, 以äžã®æ¡ä»¶ãã¿ããæå°ã®æŽæ° $k\geq 2$ ã $f(n)$ ãšãããŸãïŒ
- ä»»æã®æŽæ° $m$ ã«å¯Ÿã, $m^k$ ãš $m$ ã $n$ ã§å²ã£ãäœããçãã.
ãã ã, ååšããªãå Žå㯠$f(n)=-1$ ãšããŸã. $f(n)=f(2021)$ ãªã $n$ ã¯, $2021$ **ãå«ã**ããã€ãããŸããïŒ |
OMC073 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc073/tasks/298 | F | OMC073(F) | 600 | 17 | 48 | [
{
"content": "ãäžè§åœ¢ $BDF$ ãš $GFD$ ãååãšãªããããªç¹ $G$ ã $DF$ ã«é¢ã㊠$B$ ãšåãåŽã«ãšããš, æ¡ä»¶ãã $FG=BD=CE$ ã§ãã, ç°¡åãªè§åºŠèšç®ãšäœµã㊠$CEF$ ãš $GFE$ ãååã§ããããšãããã. ãã®ãšã, $BFCED$ ã®é¢ç©ã¯ $DEG$ ã®é¢ç©ã«çãã. ããã§, äžè§åœ¢ $DEG$ ã¯ä»¥äžã®æ¡ä»¶ã«ãã£ãŠç¹åŸŽä»ããããïŒ\r\n$$DE=11,\\quad DG=20,\\quad \\angle G=30^\\circ$$\r\nãã®ãããªäžè§åœ¢ã¯ $2$ éãååšããã, ãã®ãã¡éè§äžè§åœ¢ã§ãããã®ãåœãŠã¯ãŸãããšãç°¡åã«ããã.... | ã$\angle A=100^\circ$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, ãããã蟺 $AB,AC$ äžã®ç¹ $D,E$ ã $BD=CE$ ããã³ $DE=11$ ãã¿ãããŸã. ããã« $BE$ ãš $CD$ ã®äº€ç¹ã $F$ ãšããã°, $BF=20$ ããã³ $\angle BFC=130^\circ$ ãæç«ããŸãã. ãã®ãšã, äºè§åœ¢ $BFCED$ ã®é¢ç©ã¯, æ£æŽæ° $a\lt b$ ã«ãã£ãŠ $\sqrt{a}+\sqrt{b}$ ãšäžæã«è¡šããã®ã§, $a+b$ ãåçããŠãã ãã. |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/1654 | A | OMC072(A) | 100 | 311 | 321 | [
{
"content": "ã$\\textrm{A}n$ ãµã€ãºã®çŽã®åšé·ã¯, $\\textrm{A}(n+2)$ ãµã€ãºã®çŽã®åšé·ã® $2$ åã§ãããã, æ±ããå€ã¯ $2^{5}=\\textbf{32}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc072/editorial/1654"
}
] | ãä»»æã®éè² æŽæ° $n$ ã«å¯Ÿã $\textrm{A}n$ ãµã€ãºã®çŽãš $\textrm{A}(n+1)$ ãµã€ãºã®çŽã¯çžäŒŒã§ãã, ãã®é¢ç©æ¯ã¯ $2:1$ ã§ã. $\textrm{A}0$ ãµã€ãºã®çŽã®åšé·ã¯, $\textrm{A}10$ ãµã€ãºã®çŽã®åšé·ã®äœåã§ããïŒ |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/252 | B | OMC072(B) | 200 | 296 | 306 | [
{
"content": "ãèµ€è²ããã³çœè²ã®ããŒã«ã®åæã®åæ°ããããã $x,y$ ãšããã°, æ¡ä»¶ã¯ä»¥äžã® $2$ åŒã«è¡šçŸããã.\r\n$$xy=a,\\ \\ (x+17)(y+3)=a+146$$\r\nãããã蟺ã
åŒãåãããŠ, $3x+17y=95$ ãåŸã. ããã®æ£æŽæ°è§£ã¯\r\n$$(x,y)=(9,4),(26,1)$$\r\nã§ãããã, æ±ããå€ã¯ $36+26=\\textbf{62}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc072/editorial/252"
... | ãåºå¥ã§ããèµ€è²ããã³çœè²ã®ããŒã«ããããã $1$ ã€ä»¥äžãã, èµ€è²ã®ããŒã« $1$ã€ãšçœè²ã®ããŒã« $1$ ã€ã®çµã¯ $a$ éããããŸãã. ãããžåºå¥ã§ããèµ€è²ã®ããŒã«ã $17$ å, åºå¥ã§ããçœè²ã®ããŒã«ã $3$ åå ãããš, èµ€è²ã®ããŒã« $1$ ã€ãšçœè²ã®ããŒã« $1$ ã€ã®çµã¯ $a+146$ éãã«ãªããŸãã. ãã®ãšã, æ£æŽæ° $a$ ãšããŠããåŸãå€ããã¹ãŠæ±ã, ãããã®ç·åã解çããŠãã ãã. |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/1229 | C | OMC072(C) | 200 | 281 | 309 | [
{
"content": "ã$1,2,3$ ã®ãã¡äžã€ãæžãããã«ãŒãããããã $3$ æãã€ããç¶æ³ãèããŠãåãã§ãã. ããã«, éžãã ã«ãŒã $3$ æã®ç·åã $3$ ã§å²ãåãããã«ã€ããŠèããã°ãã, ç¹ã«é åºã¯ç¡èŠããŠãã. ãã®ãšã, $3$ æã®æ°ã®çµã¿åãããšããŠããåŸããã®ã¯\r\n$$\\lbrace1,1,1\\rbrace,\\lbrace1,2,3\\rbrace,\\lbrace2,2,2\\rbrace,\\lbrace3,3,3\\rbrace $$\r\nãã£ãŠ, æ±ãã確ç㯠$\\dfrac{1+3^3+1+1}{{}\\_{9}\\mathrm{C}\\_{3}}=\... | ã$1$ ãã $9$ ãŸã§ã®æŽæ°ã®ãã¡ã¡ããã© $1$ ã€ãæžãããã«ãŒãã $1$ æãã€ãããŸã. ããã®äžãã $1$ æãã€åŒã, å·Šãã䞊ã¹ãŠ $3$ æ¡ã®æŽæ°ãäœã£ããšã, ããã $3$ ã§å²ãåãã確çãæ±ããŠãã ãã. ãã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/1392 | D | OMC072(D) | 300 | 176 | 225 | [
{
"content": "ãäžè¬æ§ã倱ãã $BP\\geq PC$ ã§èããŠãã. ãã®ãšã, æ¹ã¹ãã®å®çãã $AP\\times PQ=BP\\times PC$ ã§ãããã $BP=8$ ããã³ $PC=6$ ãæç«ãã. ãã㧠$\\angle APB=\\theta$ ãšããã°, äœåŒŠå®çãã\r\n$$AB^2=208-192\\cos\\theta,\\quad AC^2=180+144\\cos\\theta$$\r\nããããã $\\cos\\theta$ ãæ¶å»ããããšã§, 以äžãåŸã. ãªã, ããã¯Stewartã®å®çãšããŠç¥ããããã®ã§ãã.\r\n$$3AB^2+4AC^2=13... | ãå€æ¥åã $\Gamma$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, $A$ ãéãçŽç· $\ell$ ã蟺 $BC$ ããã³ $\Gamma$ ãšãããã $P,Q$ ã§äº€ãããŸãã (ãã ã $Q\neq A$).
$$AP=12,\quad PQ=4,\quad BC=14$$
ã®ãšã, $AB+AC$ ãšããŠããåŸãæ倧å€ãæ±ããŠãã ãã. |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/1751 | E | OMC072(E) | 300 | 106 | 158 | [
{
"content": "ããµãŒããã $1$ åšã®é·ãã $L$ , $k$ åšç®ã®éãã $v_k$ ãšãããš, $k$ åšç®ã«ãããæé㯠$\\dfrac{L}{v_k}$ ã§ãããã, ã¬ãŒã¹å
šäœã®å¹³åã®éãã¯\r\n$$ \\dfrac{100L}{\\displaystyle\\sum_{k=1}^{100} \\dfrac{L}{v_k}}=\\dfrac{100}{\\displaystyle\\sum_{k=1}^{100}\\dfrac{4k}{4k^4+1}}$$\r\nã§ãã. åæ¯ã«ã€ããŠ\r\n$$\\begin{aligned}\r\n\\displaystyle\\sum_{k=1... | ãã¬ãŒãµãŒã®OMCåã¯, ããã¬ãŒã¹ã§ãµãŒãããã $100$ åšã, å $k=1,2,\cdots ,100$ ã«ã€ã㊠$k$ åšç®ã®éã㯠$k^3+\dfrac{1}{4k}$ ã§ãã. ãã®ãšã, ã¬ãŒã¹å
šäœã§ã®å¹³åã®éããæ±ããŠãã ãã. ãã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC072 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc072/tasks/2278 | F | OMC072(F) | 400 | 62 | 128 | [
{
"content": "ã$99, 100, 101$ ã¯ã©ã®äºã€ãäºãã«çŽ ã§ãããã, ä»»æã®éè² æŽæ° $r_1\\lt 99, r_2\\lt 100,r_3\\lt 101$ ã«å¯Ÿã, äžåœå°äœå®çãã以äžã®æ¡ä»¶ãã¿ããæŽæ° $n$ ã $99\\times 100\\times 101$ ãæ³ãšããŠäžæã«ååšãã.\r\n- $n$ ã $99$ ã§å²ã£ãäœãã $r_1$ ã§ãã.\r\n- $n$ ã $100$ ã§å²ã£ãäœãã $r_2$ ã§ãã.\r\n- $n$ ã $101$ ã§å²ã£ãäœãã $r_3$ ã§ãã.\r\n\r\nãããã£ãŠ, æ±ããã¹ãç·å $S$ ã¯, \r\n$$\r\n\... | ã$N$ ã $99, 100, 101$ ã§ããããå²ã£ãäœãã®ãã¡æå°ã®ãã® $f(N)$ ã«ã€ããŠ, 以äžã®ç·åãæ±ããŠãã ãã.
$$f(1)+f(2)+\cdots+f(99\times100\times101)$$ |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/2204 | A | OMC071(A) | 200 | 264 | 290 | [
{
"content": "ã$2204$ 以äžã®æ£æŽæ°ãå
é²æ³ã§è¡šèšã, $0,1,2,3,4,5$ ããããã $0,1,4,6,8,9$ ã«çœ®ãæããŠåé²æ³ã§è§£éããã°, æ±ããæ£æŽæ°ãšã®éã«äžå¯Ÿäžå¯Ÿå¿ãåŸããã. ç¹ã« $2204_{(10)}=14112_{(6)}$ ãã, æ±ããå€ã¯ $\\bm{18114}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc071/editorial/2204"
}
] | ãåé²è¡šèšã§ $2,3,5,7$ ã®ããããçŸããªãæ£æŽæ°ã®ãã¡, $2204$ çªç®ã«å°ãããã®ãæ±ããŠãã ãã. |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/2455 | B | OMC071(B) | 300 | 233 | 278 | [
{
"content": "ãåå³ã®èµ€ç·éšãããã¿ã€ã«ã®å€åšãšãªããšã, å·Šå³ã®ããã« $6$ ã€ã®æ£å
è§åœ¢ã«åå²ããããã, $3^6=729$ éãã§ãã. ããã§ãªããšã, äžå³ãŸãã¯å³å³ã® $2$ éããããããã, 以äžããæ±ããå Žåã®æ°ã¯ $\\textbf{731}$ ã§ãã.\r\n\r\n||| ||
|-... |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/2866 | C | OMC071(C) | 300 | 137 | 250 | [
{
"content": "ã$f$ ã®å€åã $T$ ãšãããš, 以äžã®åœ¢åŒã§ããããšãå¿
èŠååæ¡ä»¶ãšãªã.\r\n$$\\begin{cases} f(x)=x & (x\\in T)\\\\\\\\ f(x)\\in T & (x\\notin T)\\end{cases}$$\r\nãã£ãŠ, $k=|T|$ ãšããŠéå $T$ ã®èŠçŽ ã®éžã³æ¹ãèããã°,\r\n$$M= \\sum _{k=1} ^6 {6 \\choose k} k^{6-k}= \\bm{1057}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/con... | ãéå $\lbrace 1,2,3,4,5,6 \rbrace$ ã $S$ ãšãããŸã. é¢æ° $f:S \rightarrow S$ ã§ãã£ãŠ, ä»»æã® $x \in S$ ã«å¯ŸããŠ
$$f(f(x))=f(x)$$
ãã¿ãããã®ã¯ $M$ åååšããŸã. $M$ ã解çããŠãã ãã. |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/2497 | D | OMC071(D) | 300 | 151 | 199 | [
{
"content": "ã$BC$ ã«ã€ã㊠$A$ ãšå¯Ÿç§°ãªç¹ã $A^\\prime$, $CA^\\prime$ ã«ã€ã㊠$B$ ãšå¯Ÿç§°ãªç¹ã $B^\\prime$ ãšã, ç·å $CA^\\prime,A^\\prime B^\\prime$ äžã«ãããã $CR=CR^\\prime,AP=A^\\prime P^\\prime$ ãšãªãç¹ $R^\\prime,P^\\prime$ ããšããš,\r\n$$PQ+QR+RP=PQ+QR^\\prime+R^\\prime P^\\prime \\geq PP^\\prime$$\r\nãã, çå·æç«æ¡ä»¶ãèããããšã§ $4$ ç¹ $P,Q,R^\\... | ã$\angle{A}=90^\circ,\angle{B}=30^\circ$ ãªãçŽè§äžè§åœ¢ $ABC$ ã®èŸº $AB$ äžïŒäž¡ç«¯ãé€ãïŒã®å®ç¹ $P$ ã«ã€ããŠïŒäžè§åœ¢ $PQR$ ã®åšé·ãæå°ãšãªãããã« $2$ ç¹ $Q,R$ ããããã蟺 $BC,CA$ ã«ãšããŸãïŒäžè§åœ¢ $ABC$ ã®éå¿ãç·å $QR$ ã«ååšãããšãïŒ$AP:PB$ ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$a:b$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/1929 | E | OMC071(E) | 400 | 52 | 140 | [
{
"content": "ãäžè§åœ¢ $IBC,ICA,IAB$ ã®å€å¿ããããã $X,Y,Z$ ãšãããš, well-known factãšããŠäžè¬ã«ãããã¯ãã¹ãŠ $ABC$ ã®å€æ¥åäžã«ãã. äžè§åœ¢ $ABC$ ã«ãããŠ, å€å¿ã $O$ , å€æ¥åã®ååŸã $R$ ãšããã°, $OXâ¥BC$ ããåè§åœ¢ $OBXC$ ã®é¢ç©ã¯ $\\dfrac{1}{2}ÃOXÃBC=\\dfrac{5}{2}R$ ã§ãã. \r\nç¹ $Y,Z$ ã«ã€ããŠãåæ§ã«èããããšã§, å
è§åœ¢ $AZBXCY$ ã®é¢ç©ã¯ $\\dfrac{15}{2}R$ ã§ãã, $\\triangle IYZ\\equiv \\trian... | ã$AB=4,BC=5,CA=6$ ãªãäžè§åœ¢ $ABC$ ã®å
å¿ã $I$ ãšããŸã. äžè§åœ¢ $IBC,ICA,IAB$ ã®å€å¿ã $3$ é ç¹ãšããäžè§åœ¢ã®é¢ç©ã¯, äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\sqrt{\dfrac{a}{b}}$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC071 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc071/tasks/1965 | F | OMC071(F) | 500 | 49 | 119 | [
{
"content": "ã察称æ§ãã $\\lfloor y \\rfloor = \\lfloor x \\rfloor +a, \\lfloor z \\rfloor = \\lfloor x \\rfloor +b$ $(0 \\lt a \\lt b)$ ãšããŠãäžè¬æ§ã倱ããªã. ãã®ãšãäžåŒã¯\r\n$$6 \\lfloor x \\rfloor = ab+2354-a^2-b^2-2a-2b=3(785+ab)-(a+b+1)^2$$\r\næå³èŸºã $6$ ã®åæ°ãšãªããã㪠$(a,b)$ ã®æ¡ä»¶ãèããã. $a+b+1$ ã $6$ ã§å²ãåãããšã, $ab$ ã¯å¶æ°ã§ããã, ããã¯äžé©... | ãããããæŽæ°ã§ãªãå®æ° $x,y,z$ ã
$$ \lceil x \rceil ^2 + \lceil y \rceil ^2 + \lceil z \rceil^2 = \lfloor x \rfloor \lfloor y \rfloor + \lfloor y \rfloor \lfloor z \rfloor +\lfloor z \rfloor \lfloor x \rfloor +2357$$
ãã¿ãã, ãã€, $ \lfloor x \rfloor , \lfloor y \rfloor , \lfloor z \rfloor $ ã®å€ããã¹ãŠç°ãªããšã, $ \lfloor x \rfloor + \lfl... |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/237 | A | OMC070(A) | 100 | 212 | 214 | [
{
"content": "**解æ³1.**ãæ±ããé¢ç©ã¯ $\\dfrac{1}{2}\\times AB\\times BC\\times\\sin\\angle B=30\\sin\\angle B$ ãšè¡šã, æããã«ãã㯠$\\angle B=90^\\circ$ ã®ãšãæå€§å€ $\\textbf{30}$ ããšã.\r\n\r\n**解æ³2.**ã$A$ ãã $BC$ ã«ããããåç·ã®è¶³ $H$ ã«ãã£ãŠäžè§åœ¢ã®é¢ç©ã¯ $AH\\times BC\\/2$ ãšè¡šã, $AH\\leq AB$ ãããã㯠$30$ 以äžã§ããã, éã« $\\angle B=90^\\circ$ ã®ãšã $AH=AB$... | ãäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $AB,BC$ ã®é·ãããããã $5,12$ ã§ãã£ããšã, ãã®é¢ç©ãšããŠããåŸãæ倧å€ãæ±ããŠãã ãã. |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/2306 | B | OMC070(B) | 100 | 170 | 180 | [
{
"content": "ã$n$ 人ãåã€ç¢ºçãš $100 - n$ 人ãåã€ïŒ$n$ 人ãè² ããïŒç¢ºçã¯çããã®ã§, æ±ããæåŸ
å€ã¯ $100\\/2 = \\bf{50}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc070/editorial/2306"
}
] | ã$100$ 人ã§ããããããè¡ã, ãããã§ãªããªããŸã§ç¶ãããšã, ãã®ãããããã«åå©ãã人æ°ã®æåŸ
å€ãæ±ããŠãã ãã. |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/2298 | C | OMC070(C) | 200 | 120 | 181 | [
{
"content": "**解æ³1.**ã$i \\lt j$ ã§ãããšãã$a_i$ 㯠$a_j$ ã®å·Šã«äžŠãã§ãããã$a_j$ 㯠$a_i$ ã®å³ã«äžŠãã§ããããšè¡šçŸããã°, $1$ ã®å·Šã«äžŠãã§ããæŽæ°ãš $5$ ã®å³ã«äžŠãã§ããæŽæ°ã¯åèšã§ $2$ å以äžã§ãããã, 以äžã®å Žååãããæ±ããçã㯠$\\bf{9}$ ã§ãã. \r\n\r\n- $a_1 = 1, a_3 = 5$ ã®å Žå$\\\\\\\\$\r\nã$a_2 \\lt a_4 \\lt a_5$ ã§ããå¿
èŠããããã, $(1,2,5,3,4)$ ã®ã¿. \r\n- $a_1 = 1, a_4 = 5$ ã®å Žå$\\\\\\\... | ã$1,2,3,4,5$ ã®äžŠã¹æ¿ã $a_1,a_2,a_3,a_4,a_5$ ã§ãã£ãŠ, $1\le i \lt j \le 5$ ã〠$a_i \gt a_j$ ãæºãã $(i,j)$ ã®çµãã¡ããã© $2$ ã€ååšãããã®ã¯äœéããããŸãã. |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/2297 | D | OMC070(D) | 200 | 190 | 194 | [
{
"content": "ãå³ã®ããã«ããããšã§, äžèŸºã®é·ãã $1$ ã®æ£æ¹åœ¢ $6$ åãšäžèŸºã®é·ãã $1$ ã®æ£äžè§åœ¢ $12$ åã«åå²ã§ãããã, æ±ããé¢ç©ã¯ $6+\\sqrt{27}$ ã§ãã, ç¹ã«è§£çãã¹ãå€ã¯ $\\bf{33}$ ã§ãã. \r\n\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc070/editorial/2297"
}
] | ãäžèŸºã®é·ãã $1$ ã®æ£åäºè§åœ¢ã®é¢ç©ã¯, æ£ã®æŽæ° $a,b$ ãçšã㊠$a+\sqrt b$ ãšè¡šããã®ã§, $a+b$ ãæ±ããŠãã ãã. |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/2296 | E | OMC070(E) | 300 | 63 | 119 | [
{
"content": "$$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e ,\\quad Q(x) = x^5 + 2ax^4 + 2bx^3 + 2cx^2 + 2dx + 2e$$\r\nãšãããšã,\r\n$$\r\nQ(x) = (x - \\alpha + 1)(x - \\beta + 1)(x - \\gamma + 1)(x - \\delta + 1)(x - \\varepsilon + 1) = P(x + 1)\r\n$$\r\nã§ãããã, $P(x + 1)$ãå±éããŠä¿æ°ãæ¯èŒããããšã§\r\n$$\r\n\\begin{cases}\r\n2a ... | ãå®æ° $a,b,c,d,e$ ã«ã€ããŠ,
$$
x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0
$$
ã®è€çŽ æ°è§£ãéè€ã蟌ã㊠$x=\alpha, \beta , \gamma , \delta , \varepsilon$ ã§ãã,
$$
x^5 + 2ax^4 + 2bx^3 + 2cx^2 + 2dx + 2e = 0
$$
ã®è€çŽ æ°è§£ãéè€ã蟌ã㊠$x=\alpha - 1 , \beta - 1 , \gamma - 1 , \delta - 1 , \varepsilon - 1$ ã§ãããšã,
$$|a+b+c+d+e|$$
ãåãåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC070 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc070/tasks/2232 | F | OMC070(F) | 400 | 118 | 153 | [
{
"content": "ãæ£æŽæ° $k$ ã«å¯Ÿã㊠$\\sqrt{k}$ ã¯åæŽæ°ã«ãªãåŸãªãããšãã\r\n$$\\left\\lceil\\sqrt k\\right\\rfloor=\\left\\lceil 2\\sqrt k \\right\\rceil-\\left\\lceil \\sqrt k \\right\\rceil$$\r\nãæç«ãïŒä»¥äžã®ããã«èšç®ã§ããïŒ\r\n$$\r\n\\begin{aligned}\r\n\\sum_{k = 1}^{12345^2}\\left\\lceil\\sqrt k\\right\\rfloor &= \\sum_{k = 1}^{12345^2}... | ãæ£ã®å®æ° $x$ ãå°æ°ç¬¬äžäœã§åæšäºå
¥ããå€ã $\lceil x\rfloor$ ã§è¡šããšã,
$$\displaystyle \sum_{k=1}^{12345^2}\left\lceil\sqrt k\right\rfloor$$
ãæ±ããŠãã ãã. |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/2089 | A | OMC069(A) | 300 | 68 | 170 | [
{
"content": "ãåé¢äœã®é¢ã®ãã¡ããã€ãå¹³è¡å
é¢äœã®é¢ã«å«ãŸãããã§åããŠæ°ãããš, 以äžã®ããã«ãªãïŒ\r\n- $3$ é¢ (å³ã®(i))ïŒé¢ã®éžã³æ¹ $4$ éãããããã«å¯Ÿã $1$ éããã€ãããã $4$ éãïŒ\r\n- $2$ é¢ (å³ã®(ii))ïŒé¢ã®éžã³æ¹ $6$ éãããããã«å¯Ÿã $2$ éããã€ãããã $12$ éãïŒ\r\n- $1$ é¢ (å³ã®(iii))ïŒé¢ã®éžã³æ¹ $4$ éãããããã«å¯Ÿã $3$ éããã€ãããã $12$ éãïŒ\r\n- $0$ é¢ (å³ã®(iv))ïŒ$1$ éãïŒ\r\n\r\nãéã«, é©åœãªå¹³è¡å
é¢äœãåºå®ããŠåé¢äœãããã€åãããèããŠ... | ãåé¢äœ $ABCD$ ããããŸã. å¹³è¡å
é¢äœã§ãã£ãŠ, é ç¹éåã« $A,B,C,D$ ããã¹ãŠå«ãŸãããããªãã®ã¯ããã€ãããŸããïŒãã ã, ååã§ãã£ãŠãé ç¹éåãäžèŽããªããã®ã¯åºå¥ããŠæ°ããŸã. ãŸã**å¹³è¡å
é¢äœ**ãšã¯, äžçµã®å¹³è¡ãªäºå¹³é¢ã«ãã£ãŠå²ãŸããç«äœã§ã. |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/2090 | B | OMC069(B) | 300 | 130 | 156 | [
{
"content": "ãäžåŒã§ $y$ ãåºå®ããã° $f$ ã¯å
šå°ã§ããããšãããã, ç¹ã« $f(a)=0$ ãªã $a$ ããšãã. ãã®ãšãäžåŒã§ $x=a$ ãšããã° $a=1$ ã§ãã, ããããäžåŒã§ $y=1$ ãšããã°ä»¥äžãæãç«ã€ïŒ\r\n$$f(f(x)+1)=x-1$$\r\nãã£ãŠäžåŒã§ $(x,y)=(f(n)+1,0)$ ãšããã°\r\n$$f(n-1)=f(n)+f(0)$$\r\nãåŸãã, ãããã $k$ ãæŽæ°ãšã㊠$f(n)=k(n-1)$ ãšãã圢ã§ããããšãåãã. ãããäžåŒã«ä»£å
¥ããããšã§ $k=\\pm 1$ ãåŸããããã, ç¹ã«è§£çãã¹ãå€ã¯ $\\bm... | ãæŽæ°ã«å¯ŸããŠå®çŸ©ããæŽæ°å€ããšãé¢æ° $f$ ã¯, ä»»æã®æŽæ° $x,y$ ã«å¯ŸããŠä»¥äžãã¿ãããŸãïŒ
$$f(f(x)+y)+1=x+f(y)$$
ãã®ãã㪠$f$ ãšããŠèãããããã®ãã¹ãŠã«å¯Ÿã, å€ $|f(2022)|$ ã®ç·åã解çããŠãã ããïŒ |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/1378 | C | OMC069(C) | 400 | 93 | 138 | [
{
"content": "ãäžè¬ã« $4n=400000$ ãšãã, $l$ ã $2^{l-1}\\leq 4n\\lt 2^l$ ãã¿ããæŽæ°ãšãã. ãã®ãšã, æããã« $T\\lt 2^l$ ã§ãã, äžæ¹ã§ $m=l$ 㧠$x_i=2^{i-1}$ ãšããã° $T=2^l-1$ ã§ããããšãã, ããã $T$ ã®ãšãåŸãæ倧å€ã§ãã.\\\r\nãããªãã¡, èããã¹ãæå€§å€ $M(n)$ ã¯, 以äžã®ããã«è¡šçŸã§ãã.\r\n- $4n$ 以äžã®æ£æŽæ°ããä»»æã«äžã€ä»¥äžãéžã³, ãã®ãã¡ $2^k$ ã®äœã $1$ ã§ãããã®ã $k=0,1,\\cdots,l-1$ ã«ã€ããŠããããå¥æ°åã§ãã... | $$m\leq 400000,\quad x_1\lt x_2\lt\cdots\lt x_m\leq 400000$$
ãã¿ããæ£æŽæ° $m,x_1,x_2,\ldots,x_m$ ã«å¯ŸãïŒ
$$S=x_1+x_2+\cdots+x_m,\quad \ T=x_1 \oplus x_2 \oplus \dots \oplus x_m$$
ãšãããŸãïŒ$T$ ãæ倧å€ããšããã㪠$m,x_1,x_2,\ldots,x_m$ ã«ã€ããŠïŒ$S$ ã®ãšãåŸãæ倧å€ãæ±ããŠãã ããïŒ
<details> <summary>æä»çè«çå $\oplus $ ã«ã€ããŠ<\/summary>
ãéè² æŽæ°ã«å¯Ÿã, ãããã®**æä»ç... |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/2092 | D | OMC069(D) | 600 | 10 | 48 | [
{
"content": "ã$n$ ãæ£æŽæ°ãšããïŒ$k=1,\\dots,n-1$ ã«å¯Ÿãåã®æ£æŽæ°ã®çµã®ãã¡ $z=n-k$ ãã¿ãããã®ã®æ°ã $g(k)$ ãšãããšïŒãã㯠$xy$ å¹³é¢ã§ $O(0,0),A(k,0),B(0,n)$ ãšãããšãã«äžè§åœ¢ $OAB$ ã®åšãé€ãå
éšã«ããæ Œåç¹ã®æ°ã«çããïŒ\r\näžè§åœ¢ $OAB$ ã®åšäžã«ããæ Œåç¹ã¯ $n+k+\\mathrm{gcd}(n,k)$ åã§ããããïŒPickã®å®çãã次ããããïŒ\r\n$$g(k)=\\frac{1}{2}(nk-n-k-\\mathrm{gcd}(n,k))+1$$\r\nããã $k=1,\\cdots,n-1$ ... | ãæ£æŽæ° $n$ ã«å¯ŸãïŒä»¥äžãã¿ããæ£æŽæ°ã®çµ $(x,y,z)$ ã®ç·æ°ã $f(n)$ ã§è¡šããŸãïŒ
$$nx\lt(y-n)(z-n),\quad x\lt n,\quad y\lt n,\quad z\lt n$$
ãã®ãšãïŒ$f(66^{60000})$ ã $606$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/2093 | E | OMC069(E) | 700 | 33 | 65 | [
{
"content": "#### ååïŒOMCåã®åé¡ã解ã\r\nãå®æ° $x,y,z$ ã«å¯Ÿã㊠$F=\\dfrac{ax^4+by^4+cz^4+1}{x^2+y^2+z^2+1}$ ã®æå°å€ãæ±ãããïŒãŸãCauchy-Schwarzã®äžçåŒãã\r\n$$(ax^4+by^4+cz^4)\\left(\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\\right)\\geq(x^2+y^2+z^2)^2$$\r\nãã£ãŠ $x^2+y^2+z^2+1=t$ ã®ãšã $C=\\left(\\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c}\\ri... | ãOMCåã¯æ¬¡ã®åé¡ãäœããŸããïŒãã ã $a,b,c$ 㯠$a\leq b\leq c$ ãã¿ããæ£æŽæ°ãšããŸãïŒ
- **åé¡**ïŒä»»æã®å®æ° $x,y,z$ ã«å¯ŸããŠä»¥äžãæãç«ã€ãããªïŒå®æ° $m$ ãšããŠããåŸãæå€§å€ $M$ ã¯ããã€ã§ããïŒ
$$ax^4+by^4+cz^4+1\geq m(x^2+y^2+z^2+1)$$
ãOMCå㯠$M$ ãæçæ°ãšãªãããã« $a,b,c$ ã®å€ãèšå®ãããã§ãïŒ$M=\dfrac{p}{q}$ ãªãäºãã«çŽ ãªæ£æŽæ° $p,q$ ãååšãããããªçµ $a\leq b\leq c$ ã®ãã¡ïŒ$p+q$ ã®å€ã**2çªç®ã«å°ãããã®**ãã¹ãŠã«ã€ããŠïŒ$a+b+c$... |
OMC069 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc069/tasks/2094 | F | OMC069(F) | 900 | 0 | 18 | [
{
"content": "ã$ABCD$ ã®å€æ¥çã®ååŸã $r$ ãšããïŒç¹ $A$ ãäžå¿ãšããååŸ $1$ ã®çé¢ã«é¢ããå転ãèããïŒãã®ãšã $ABCD$ ã®å€æ¥ç㯠$A$ ããè·é¢ $\\dfrac{1}{2r}$ ã®å¹³é¢ã«ç§»ãïŒãŸãç¹ $B,C,D$ ã®ç§»ãå
ã®ç¹ããããã $P,Q,R$ ãšããã°ïŒ\r\n$$AB\\times AP=AC\\times AQ=AD\\times AR=1$$\r\n$$PQ=\\frac{BC}{AB\\times AC},\\quad QR=\\frac{CD}{AC\\times AD},\\quad RP=\\frac{DB}{AD\\times AB}$... | ãäœç© $2022$ ã®åé¢äœ $ABCD$ ã以äžã®æ¡ä»¶ãã¿ãããšãïŒãã®å€æ¥çã®ååŸã¯æ£æŽæ° $a,b,c$ (ãã ã $a,c$ ã¯äºãã«çŽ ã§ïŒ$b$ ã¯å¹³æ¹å åããããªã) ã«ãã£ãŠ $\dfrac{a\sqrt{b}}{c}$ ãšè¡šãããã®ã§ïŒ$a+b+c$ ã解çããŠãã ããïŒ
$$AB\times CD=2021,\quad AC=40,\quad AD=43,\quad BC=47,\quad BD=50$$
ããã ãåé¢äœã®**å€æ¥ç**ãšã¯ïŒãã® $4$ ã€ã®é ç¹ãå
šãŠéãçã®ããšãæããŸãïŒ |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/2929 | A | OMC068(A) | 100 | 307 | 312 | [
{
"content": "ã$0\\leq\\lbrace x\\rbrace\\lt1$ ãã $\\lfloor x\\rfloor=\\dfrac{1}{\\lbrace x\\rbrace}\\gt 1$ ã§ãã, $\\lfloor x\\rfloor$ ã¯æŽæ°ã§ãããã $\\lfloor x\\rfloor\\geq 2$ ãå¿
èŠã§ãã.\\\r\nãæ¡ä»¶ãæºãã $x$ ã®æå°å€ãèããŠãããã $\\lfloor x\\rfloor=2$ ã®ãšããèããã°ãã. ãã®ãšã $\\lbrace x\\rbrace=0.5$ ã§ãã, $x=\\dfrac{5}{2}$ ãçåŒãæºããæå°ã® $x$ ... | ãæ£ã®å®æ° $x$ ã«å¯Ÿã, $\lfloor x\rfloor$ 㧠$x$ ã®æŽæ°éšå, $\lbrace x\rbrace$ 㧠$x$ ã®å°æ°éšåãè¡šããŸã. äŸãã°
$$\lfloor 3.14\rfloor =3, \quad \lbrace 3.14\rbrace=0.14$$
ã§ã. ãã®ãšã,
$$ \lfloor x\rfloor\times\lbrace x\rbrace=1$$
ãæºããæå°ã®æ£ã®å®æ° $x$ ãæ±ããŠãã ãã.\
ããã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/2246 | B | OMC068(B) | 200 | 277 | 291 | [
{
"content": "ãæ£ã®æŽæ° $a,b\\lt\\min(10,N)$ ã«ã€ããŠæ¡ä»¶ãã\r\n$$10a+b=Nb+a \\iff 9a=(N-1)b$$\r\n$N=2,3$ ã§ã¯ãã®ãããªããšã¯ããåŸã, $N=4$ ã§ã¯ $(a,b)=(1,3)$ ãé©ãããã, æ±ããæå°å€ã¯ $\\textbf{4}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc068/editorial/2246"
}
] | ãããæ£ã®æŽæ°ã $10$ é²æ³ã§è¡šèšãããš $\overline{ab}$ ã§ãã, $N$ é²æ³ã§è¡šèšãããš $\overline{ba}$ ã§ãã. ãã®ãããªããšãèµ·ããåŸãæ£ã®æŽæ° $N(\geq2)$ ã®æå°å€ãæ±ããŠãã ãã. ãã ã, $a,b$ 㯠$1\leq a,b\lt \min(N,10)$ ãæºãããã®ãšããŸã. |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/2275 | C | OMC068(C) | 200 | 215 | 248 | [
{
"content": "ãæ±ããåã®ååŸã $r$ ãšãããšïŒ$\\triangle ABC \\sim \\triangle HBA \\sim \\triangle HAC$ãã\r\n$$BC : BA : AC = 505 : r : 100$$\r\nããããïŒ$BC = 505t, BA = rt, AC = 100t$ ãšãããïŒäžæ¹ã§ïŒ$â³ABC$ã«ãããŠäžå¹³æ¹ã®å®çãçšããã°\r\n$$(505t)^2 = (rt)^2 + (100t)^2$$\r\nãããã£ãŠ $r^2 = 505^2 - 100^2 = 495^2$ ã§ãã, æ±ããåã®ååŸã¯ $\\textbf{495}$ ã§ããïŒ"... | ãè§ $A$ ãçŽè§ãšããäžè§åœ¢ $ABC$ ã«ãããŠïŒ$A$ ãã $BC$ ã«äžãããåç·ã®è¶³ã $H$ ãšãããšïŒäžè§åœ¢ $ABC$ ã®å
æ¥åã®ååŸã¯ $505$ïŒäžè§åœ¢ $ACH$ ã®å
æ¥åã®ååŸã¯ $100$ ã§ããïŒäžè§åœ¢ $ABH$ ã®å
æ¥åã®ååŸãæ±ããŠãã ããïŒ |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/2277 | D | OMC068(D) | 300 | 135 | 246 | [
{
"content": "$$\\begin{aligned}\r\nP &= 1! à (1! à 2) à 3! à (3! à 4) à \\cdots à 599! à (599! à 600) \\\\\\\\\r\n&= (1! à 3! à \\cdots à 599!)^2 à (2 à 4 à \\cdots à 600) \\\\\\\\\r\n&= (1! à 3! à \\cdots à 599!)^2 à (2^{150})^2 à 300!\r\n\\end{aligned}$$\r\nã§ããããïŒ$n = 300$ ã¯é¡æãæºããããšã確èªã§ããïŒ\\\r\nãéã«ïŒ$n = 299$... | $$P = 1! à 2! à 3! à \cdots à 600!,\quad Q = \dfrac{P}{n!}$$
ã«ã€ããŠïŒ$Q$ ãå¹³æ¹æ°ãšãªããããªæå°ã®æ£æŽæ° $n$ ãæ±ããŠãã ããïŒ |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/1676 | E | OMC068(E) | 300 | 134 | 199 | [
{
"content": "ãæé ã®å®ãæ¹ãã, $i$ çªç®ã«æžãããæ° $a_i$ 㯠$i$ ã®äºé²æ°è¡šç€ºã§ã®åæ¡ã®åã«çãã. \r\n\r\n**蚌æ.** æ£æŽæ° $n$ ãäºé²æ°è¡šèšããæã® $1$ ã®åæ°ã $\\mathrm{popcount}(n)$ ãšãããš, $\\mathrm{popcount}(n)$ ã¯\r\n\r\n- $\\mathrm{popcount}(1)=1$\r\n- $\\mathrm{popcount}(2n)=\\mathrm{popcount}(n)$\r\n- $\\mathrm{popcount}(2n+1)=\\mathrm{popcount}(2n)+1$\... | ãOMCåã¯ä»¥äžã®æé ã«ãã£ãŠé»æ¿ã« $1000$ åã®æ°ãæžãããšã«ããŸããïŒ
- $1$ çªç®ã«ã¯ $1$ ãæžã.
- æ£æŽæ° $n$ ã«å¯Ÿã, $2n$ çªç®ã«ã¯ $n$ çªç®ã«æžããæ°ãšåãæ°ãæžã.
- æ£æŽæ° $n$ ã«å¯Ÿã, $2n+1$ çªç®ã«ã¯ $2n$ çªç®ã«æžããæ°ããã¡ããã© $1$ 倧ããæ°ãæžã.
ãäžé£ã®æäœã§é»æ¿ã«æžãããæ°ã®ãã¡æ倧ã®ãã®ã $M$ ãšãããšã, $i$ çªç®ã« $M$ ãæžãããã㪠$1000$ 以äžã®æ£æŽæ° $i$ ã®ç·åãæ±ããŠãã ãã. |
OMC068 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc068/tasks/2240 | F | OMC068(F) | 400 | 52 | 126 | [
{
"content": "ãåçŽç· $PA$ äžã« $PC=PQ$ ãªãç¹ $Q$ ããšããš,äžè§åœ¢ $BCP$ ãšäžè§åœ¢ $BQP$ ã¯ååã§ããããç¹ã« $BC=BQ$ ã§ãã, ãããš $â CBQ = â CBP + â PBQ = 60^\\circ$ ããäžè§åœ¢ $BCQ$ ã¯æ£äžè§åœ¢ã§ããïŒäžæ¹ïŒ$â ABQ=â AQB$ ãã $AB = AQ$ ã§ããããïŒãããããäžè§åœ¢ $ABC$ ãšäžè§åœ¢ $AQC$ ã¯åå, ç¹ã« $\\angle ACB=30^\\circ$ ã§ããïŒåŸã£ãŠïŒ\r\n$$\\begin{aligned}\r\nâ CAP &= \\angle{BAC}-\\angle{BAP} \... | ãäžè§åœ¢ $ABC$ ããã³ãã®å
éšã®ç¹ $P$ ã以äžã®æ¡ä»¶ãã¿ãããŸãïŒ
$$â BPA = â BPC = â ABP+120^\circ=141.4^\circ,\quad â PBC = 30^\circ$$
ãã®ãšã, $â PAC$ ã®å€§ããã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$a\/b$ 床ãšè¡šãããã®ã§ïŒ$a + b$ ã解çããŠãã ããïŒ |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2286 | A | OMC067(A) | 100 | 275 | 278 | [
{
"content": "ãäžè§åœ¢ ${PAD}$ ãšäžè§åœ¢ ${PBC}$ ã® ${AD},{BC}$ ãåºèŸºãšã¿ããšãã®é«ãããããã $x,y$ ãšã, æ£æ¹åœ¢ ${ABCD}$ ã®äžèŸºã $a$ ãšãããš, äžè§åœ¢ ${PAD}$ ãšäžè§åœ¢ ${PBC}$ ããšãã«éè§äžè§åœ¢ã§ããããšãã $|x-y|=a$ ãæç«ãã.\\\r\nããã®ãšã ${PAD}$ ãš ${PBC}$ ã®é¢ç©ã®å·®ã¯\r\n$$\\left|\\frac{ax}{2}-\\frac{ay}{2}\\right|=\\frac{a^2}{2}$$\r\nãšãªã, ããã $377-233=144$ ã«çããã®ã§, æ£æ¹åœ¢ ${ABC... | ãæ£æ¹åœ¢ $ABCD$ ã®å€åŽã«ç¹ $P$ ããšã£ããšãã, äžè§åœ¢ $PAD$ ãšäžè§åœ¢ ${PBC}$ ã¯ã©ã¡ããéè§äžè§åœ¢ãšãªã, ãã®é¢ç©ã¯ãããã $233,377$ ãšãªããŸãã. ãã®ãšã, æ£æ¹åœ¢ ${ABCD}$ ã®é¢ç©ãæ±ããŠãã ãã. |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2288 | B | OMC067(B) | 300 | 199 | 252 | [
{
"content": "ãåååŒã¯ä»¥äžãã¹ãŠ $x^{10}-1$ ãæ³ãšãã. $x^{10m+n}\\equiv{x}^n$ ã«æ³šæããŠ,\r\n$$\\begin{aligned}\r\n&\\quad\\\\,\\\\,\\\\, x^{99}+2x^{98}+\\cdots+98x^2+99x+100 \\\\\\\\\r\n&\\equiv{x}^{9}+2x^{8}+\\cdots+9x+10+11x^{9}+12x^{8}+\\cdots+99x+100 \\\\\\\\\r\n&\\equiv(1+11+\\cdots+91)x^9+(2+12+\\cdots+92)x^8+\\cdots+(1... | ãå€é
åŒ $x^{99}+2x^{98}+\cdots+98x^2+99x+100$ ãå€é
åŒ $x^9+x^8+\cdots+x^2+x+1$ ã§å²ã£ãäœãã¯, å®æ° $a_0,a_1,\ldots,a_8$ ãçšã㊠$a_8x^8+\cdots+a_1x+a_0$ ãšè¡šãããã®ã§, $a_0+a_1+\cdots+a_8$ ã解çããŠãã ãã. |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2287 | C | OMC067(C) | 300 | 150 | 197 | [
{
"content": "ã$a\\lt{b}\\lt{c}$ ãšä»®å®ããŠãäžè¬æ§ã倱ããªã. ãã®ãšã, $6$ æ°ã®ãã¡å°ããæ¹ $3$ ã€ã¯ $a\\/b,b\\/c,a\\/c$ ã§ãããã,\r\n$$\\frac{a}{b}+\\frac{b}{c}+\\frac{a}{c}=\\frac{3}{5},\\quad \\frac{b}{a}+\\frac{c}{b}+\\frac{c}{a}=39$$\r\nãã㧠$p=b\\/a,\\\\,q=c\\/b$ ãšãããš, 以äžã®ããã«æžãæãããã.\r\n$$p+q+pq=39,\\quad \\frac{1}{p}+\\frac{1}{q}+\\f... | ãçžç°ãªãæ£ã®å®æ° $a,b,c$ ã«å¯Ÿã,
$$\frac{b}{a},\quad \frac{a}{b},\quad \frac{c}{b},\quad \frac{b}{c},\quad \frac{a}{c},\quad \frac{c}{a}$$
ãå€ãå°ããæ¹ããé ã« $x_1\leq{x_2}\leq\cdots\leq{x_6}$ ãšãããŸã.\
ã$x_1+x_2+x_3=\dfrac{3}{5},\\,x_4+x_5+x_6=39$ ãšãªã£ããšãã$x_4^2+x_5^2+x_6^2$ ã®å€ãæ±ããŠãã ãã. |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2289 | D | OMC067(D) | 400 | 32 | 105 | [
{
"content": "ãäžè¬ã«æ£ $n$ è§åœ¢ $A_1A_2\\cdots A_{n}$ ãšããŠèã, ãã®å€æ¥åã®åšé·ã $n$ ã§ãããšããŠãã.\r\nã㟠$\\angle A_iPA_j$ ã¯åŒ§ $A_iA_j$ ãšåŒ§ $A_kA_l$ ã«å¯Ÿããååšè§ã®åã§ããããšã«çæãã. ä»¥äž $A_0=A_n,A_1=A_{n+1}$ ãšãã.\\\r\nãããã§åŒ§ $A_0A_i,A_iA_j,A_jA_k,A_kA_l,A_lA_{n+1}$ ã®é·ãããããã $x_1,x_2,x_3,x_4,x_5$ ãšãããš, ãããã¯ãã¹ãŠæ£æŽæ°ã§ãã, ç·å㯠$n+1$ ã§ãã. 察称æ§ããããããã®å€ã®æåŸ
... | ãæ£ $100$ è§åœ¢ $A_1A_2\cdots A_{100}$ ã«ãããŠ, $1\leq{i}\lt{j}\lt{k}\lt{l}\leq{100}$ ãªããã¹ãŠã®çµã«å¯Ÿãåè§åœ¢ $A_iA_jA_kA_l$ ãèãããšã, ãã®å¯Ÿè§ç·ã®äº€ç¹ $P$ ã«ã€ã㊠$\angle{A}_iPA_j$ ã®å¹³åå€ $K$ ãèããŸã.\
ã$K$ ã床æ°æ³ã§è¡šããæã®å€ã¯, äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã®å€ã解çããŠãã ãã. |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2290 | E | OMC067(E) | 500 | 17 | 73 | [
{
"content": "ã$1$ ä»¥äž $p-1$ 以äžã® $a,b$ ã«å¯Ÿã, $a^{p-8}\\equiv{b}^{p-8} \\pmod{p}$ ã®ãšã, Fermatã®å°å®çãã $a^{p-1}\\equiv{b}^{p-1}\\pmod{p}$ ã§ããããã, $\\mathrm{gcd}(p-8,p-1)=\\mathrm{gcd}(7,p-1)=1$ ã«æ³šæã㊠$a\\equiv{b}\\pmod{p}$ ã§ããããšãããã. ããã¯ã$1^{p-8},2^{p-8},\\cdots,(p-1)^{p-8}$ ã $p$ ã§å²ã£ãäœãã¯ãã¹ãŠçžç°ãªãããšãæå³ããã®ã§, $a_1+a_2+\\... | ã$p=10^9+7$ ã¯çŽ æ°ã§ã. $a_1^{10^9-1}+a_2^{10^9-1}+\cdots+a_p^{10^9-1}$ ã $p$ ã®åæ°ãšãªããããª, $1$ ä»¥äž $p-1$ 以äžã®æŽæ°ã®é åºä»ããçµ $(a_1,a_2,\ldots,a_p)$ ã®ç·æ°ã $N$ ãšãããŸã. $N$ ã $10000$ ã§å²ã£ãäœããæ±ããŠãã ãã. |
OMC067 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc067/tasks/2291 | F | OMC067(F) | 600 | 5 | 25 | [
{
"content": "ã$\\angle{PAR}=\\angle{QAR}$ ãã $DR=RP,QR=RE$ ã§ãã, ç°¡åãªè§åºŠèšç®ã§ $\\angle{DRQ}=\\angle{PRE}$ ãšäœµã㊠$DRQ$ ãš $PRE$ ã¯ååã§ãã, ç¹ã« $DQ=PE$ ã§ãã. ãŸã, ååšè§ã®å®çãã $\\angle{DQC}=\\angle{PEB},\\angle{QDC}=\\angle{EPB}$ ãåŸããã, $DQC$ ãš $PEB$ ã¯ååã§ãã, $DC=13,BE=11$ ãã $BC$ 㯠$18$ ã§äžå®ã§ãã. ãããã£ãŠ, ããšã¯ $A$ ãã $BC$ ã«ããããåç·ã®é·ã ... | ãäžè§åœ¢ $ABC$ ã®èŸº $BC$ äžã« $B,D,E,C$ ã®é ã«äžŠã¶ç¹ $D,E$ ã $DE=6$ ãã¿ãããŠãã, äžè§åœ¢ $ABD,ACE$ ã®å€æ¥åããããã $O_1,O_2$ ãšããŸã. ãŸã, ç·å $AE$ ãš $O_1$ ã $A$ 以å€ã®ç¹ã§äº€ãã£ãã®ã§ããã $P$, ç·å $AD$ ãš $O_2$ ã $A$ 以å€ã®ç¹ã§äº€ãã£ãã®ã§ããã $Q$ ãšãããš, $BP=13,CQ=11$ ãæç«ããŸãã. ããã«, $O_1$ ãš $O_2$ ã¯äžè§åœ¢ $ADE$ ã®å
éšã®ç¹ $R (\neq A)$ ã§äº€ãã, $\angle{PAR}=\angle{QAR}$ ãæç«ããŸãã. ãã®ãšã, $ABC... |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/2225 | A | OMC066(A) | 100 | 296 | 306 | [
{
"content": "ãçŽåã®æ¡ãšç°ãªãæ°å $9$ çš®é¡ãã $1$ ã€ãéžã¶ããšã $4$ åç¹°ãè¿ãããšã«ãªãããïŒ$9^4=\\textbf{6561}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc066/editorial/2225"
}
] | ãOMCçåœã§çšããããéµäŸ¿çªå·ã¯ïŒãã¹ãŠïŒåé²æ³è¡šèšã§ïŒã¡ããã© $7$ æ¡ã®æ£æŽæ°ã§ããïŒãã€é£ãåãæ¡ã®æ°åã¯çžç°ãªããŸãïŒãã®ãããªçªå·ãšããŠããåŸããã®ã®ãã¡ïŒç¹ã«æåã® $3$ æ¡ã $120$ ã§ãããã®ã¯äœéããããŸããïŒ |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/1920 | B | OMC066(B) | 100 | 291 | 305 | [
{
"content": "ãããæ£æŽæ°ãæ£ã®çŽæ°ãã¡ããã© $3$ åãã€ããšã¯, çŽ æ°ã® $2$ ä¹ã®åœ¢ã«è¡šããããšãšåå€ã§ãã. $44^2 \\lt 2022 \\lt 45^2$ ããæ±ããåæ°ã¯ $44$ 以äžã®çŽ æ°ã®åæ°ã«çãã, ãã㯠$\\textbf{14}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc066/editorial/1920"
}
] | ã$2022$ 以äžã®æ£æŽæ°ã®ãã¡, æ£ã®çŽæ°ãã¡ããã© $3$ åãã€ãã®ã¯ããã€ãããŸããïŒ |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/1671 | C | OMC066(C) | 200 | 290 | 303 | [
{
"content": "ãäžæ¹çšåŒã® $2$ 解ãšããŠããåŸããã®ã¯, 解ãšä¿æ°ã®é¢ä¿ããåã $20$ ã§ããããšãã\r\n$$(1,19),(2,18),\\ldots,(9,11),(10,10)$$\r\nã§ãã. $\\alpha$ 㯠$2$ 解ã®ç©ã§ãããã, æ±ããç·å㯠$\\displaystyle\\sum_{k=1}^{10} k(20-k)=\\textbf{715}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc066/editorial/1671"
}
] | ã$x$ ã«ã€ããŠã®äºæ¬¡æ¹çšåŒ
$$x^2-20x+\alpha=0$$
ãäºã€ã®å®æ°è§£ïŒé解ã§ãããïŒããã¡ïŒãã€ãããããã¹ãŠæ£æŽæ°ã§ãããããª, å®æ° $\alpha$ ã®ç·åãæ±ããŠãã ãã. |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/1362 | D | OMC066(D) | 200 | 237 | 273 | [
{
"content": "ã$D$ ãã $AC$ ã«ããããåç·ã®è¶³ã $H$ ãšãããšã, $DH=9$ ã§ããããšã瀺ã.\r\n\r\n**蚌æ1.** ã$AC$ ãš $BD$ ã®äº€ç¹ã $E$ ãšããã°, å
è§ã®äºçåç·å®çãã\r\n$$AE:EC=34:16$$\r\nãæç«ãããã, $AH=HC$ ãšããããŠ\r\n$$AH:HE:EC=25:9:16$$\r\nãåŸã. ãã£ãŠ $BC:DH=CE:EH$ ãã $DH=9$ ã§ãã.\r\n\r\n**蚌æ2.**ã$AB$ ãš $DH$ ã®äº€ç¹ã $F$ ãšããã°, $AF=BF=17$ ã§ãã, äžæ¹ã§\r\n$$\\angle{BDF}... | ã以äžã®æ¡ä»¶ããã¹ãŠã¿ããåžåè§åœ¢ $ABCD$ ã®é¢ç©ãæ±ããŠãã ãã.
- $AB=34,BC=16,AD=CD,\angle ACB=90^\circ$.
- çŽç· $BD$ 㯠$\angle ABC$ ãäºçåãã. |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/2181 | E | OMC066(E) | 300 | 82 | 203 | [
{
"content": "$$C=(x+y)(x^2+y^2-xy)=\\frac{A(3B-A^2)}{2}$$\r\nãã $A=2$ ã§ãããã, $C=3B-4$ ãªãå¥æ°ã®åææ° $B,C$ ã«ã€ããŠèããã°è¯ã, å°ããæ¹ããæ¢çŽ¢ããŠããããšã§ $(B,C)=(27,77)$ ãåŸã. ãã®ãããªå®æ° $x,y$ ã¯ç¢ºãã«ååšãããã, æ±ããæå°å€ã¯ $\\textbf{106}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc066/editorial/2181"
}
] | ãå®æ° $x,y$ ã«å¯Ÿã, 以äžã®ããã« $A,B,C$ ãå®ããŸã.
$$A=x+y,\quad B=x^2+y^2,\quad C=x^3+y^3$$
ãããã以äžã®æ¡ä»¶ãã¿ãããšã, $A+B+C$ ãšããŠããåŸãæå°å€ã解çããŠãã ãã.
- $A,B,C$ ã¯ãããã $1$ ãã倧ããæŽæ°ã§ãã.
- $A,B,C$ ã¯ã©ã®äºã€ãäºãã«çŽ ã§ãã.
- $A,B,C$ ã®ãã¡çŽ æ°ã¯é«ã
äžã€ã§ãã. |
OMC066 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc066/tasks/1681 | F | OMC066(F) | 400 | 39 | 146 | [
{
"content": "ãéå $A_n\\setminus A_{n-1}$ ã¯æ£äžè§åœ¢ã®èŸºäžã«çééã«äžŠã¶ $3n$ ç¹ãããªãããšã容æã«ããã. $n$ åã®è¡åã§ãããã®ç¹ã«è³ãã«ã¯, $3$ æ¹åãã¹ãŠã«ç§»åããªãããšãå¿
èŠååæ¡ä»¶ã§ãããã, ãã®ãããªçµè·¯ãšããŠããåŸããã®ã®æ°ã¯\r\n$$3\\times\\left(\\sum_{k=0}^{n}{}_{n}\\mathrm{C}_k-2\\right)+3=3\\times 2^{n}-3$$\r\nç¹ã« $n=10$ ã®ãšããã㯠$3069$ ã§ãã, æ±ãã確ç㯠$\\dfrac{3069}{3^{10}}=\\dfrac{341}{... | ã$xy$ 座æšå¹³é¢äžã®åç¹ã« $1$ å¹ã®ã¢ãªãããŸã. ã¯ããã«ã¢ãªã¯åç¹ã«å°ãã€ããŸã.\
ããã®ã¢ãªã¯ $x$ 軞ã®æ£æ¹åãåããŠãã, ããŸãã次ã®äžé£ã®è¡åãç¹°ãè¿ãè¡ããŸãïŒ
- é²è¡æ¹åãå€ããªãã, å·Šã« $120^{\circ}$ å転ããã, å³ã« $120^{\circ}$ å転ãã. ãããã¯ç確çã«éžæããã.
- ãã®åŸ, é²è¡æ¹åã«æ²¿ã£ãŠã¡ããã©è·é¢ $1$ ãé²ã¿, å°éããå°ç¹ã«å°ãã€ãã.
ããã®ã¢ãªãã¡ããã© $n$ åç¶ããŠè¡åãçµããæç¹ã§, ãããŸã§ã«ã¢ãªãå°ãã€ããå¯èœæ§ã®ããç¹ã®éåã $A_n$ ãšããŸã. $A_n$ ã¯ã¢ãªã®è¡åã«äŸåããŠå®ãŸããã®ã§ã¯ãªã, $... |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2874 | A | OMCT001(A) | 100 | 192 | 194 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2874"
},
{
"content": "ã$5, 8$ ã«æ³šç®ããã° $25, 81$ ãå«ãŸãïŒ$3$ ã«æ³šç®ããã°æ®ã㯠$36, 49$ ãšäžæã«æ±ºå®ãããã®ã§ïŒè§£çãã¹ã㯠$\\mathbf{25364981}$ïŒ",
"text": "ãŠãŒã¶ãŒè§£èª¬",
"url": "https://onlinemathcontest.co... | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äžæã«ååšããããšã蚌æã§ããŸãïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ**å¹³æ¹æ°ã§ãã**ïŒ
ãã®ãããªãã®ã«ã€ããŠïŒ$A,B,C,D$ ããã®é ã«ç¶ããŠè§£çããŠãã ããïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2875 | B | OMCT001(B) | 100 | 184 | 188 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2875"
},
{
"content": "ã $2$ æ¡ã®äžè§æ°ã¯ $10,15,21,28,36,45,55,66,78,91$ ã®ã¿ã§ããïŒ\\\r\nããããã®ãã¡ïŒ $3$ ãå«ããã®ã¯ $36$ ã®ã¿ã§ããïŒ $4$ ãå«ããã®ã¯ $45$ ã®ã¿ã§ããïŒ $8$ ãå«ã㧠$7$ ãå«ãŸãªããã®ã¯ $28$ ã®ã¿ã§ããïŒ $9$ ãå«ããã®ã¯ ... | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äžæã«ååšããããšã蚌æã§ããŸãïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ**äžè§æ°ã§ãã**ïŒ
ãã®ãããªãã®ã«ã€ããŠïŒ$A,B,C,D$ ããã®é ã«ç¶ããŠè§£çããŠãã ããïŒ\
ããã ãïŒ**äžè§æ°**ãšã¯ïŒããæ£æŽæ° $n$ ã«ã€ã㊠$1+2+\cdots+n$ ã®åœ¢ã«è¡šããæ£æŽæ°ã®ããšã§ãïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2876 | C | OMCT001(C) | 200 | 149 | 169 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2876"
}
] | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äœéããããŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ**ä¹ä¹è¡šã«çŸãã**ïŒ
ããã ãïŒæ£æŽæ°ã**ä¹ä¹è¡šã«çŸãã**ãšã¯ïŒ(çžç°ãªããšã¯éããªã) $9$ 以äžã®æ£æŽæ° $2$ ã€ã®ç©ã«è¡šããããšãæããŸãïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2877 | D | OMCT001(D) | 200 | 104 | 129 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2877"
},
{
"content": "ã $$(A,B,C,D)=(18,23,54,69),(18,32,54,96),(26,39,54,81),(54,62,81,93)$$ \r\nã® $4$ çµãæ¡ä»¶ãæºããã®ã§ïŒçããã¹ãæ°å€ã¯ $\\bf{54}$ ã§ããïŒ",
"text": "çãã®ã¿",
"url": "https... | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ 㯠$4$ éãååšããããšã蚌æã§ããŸãïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A\/B=C\/D$ïŒ
ããã $4$ éããã¹ãŠã«ã€ããŠïŒã€ãã« $\\{A,B,C,D\\}$ ã«å«ãŸããå¯äžã®æŽæ°ã解çããŠãã ããïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2878 | E | OMCT001(E) | 300 | 84 | 114 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2878"
}
] | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äœéããããŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A+B+C+D$ 㯠$7$ ã§å²ããããïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2879 | F | OMCT001(F) | 300 | 62 | 84 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2879"
},
{
"content": "ã$\\gcd(A,B,C,D)$ ã $2$ 以å€ã®çŽ å æ°ãæããªãããšãéã«ç€ºããŸã.\r\n\r\n- $\\gcd(A,B,C,D)$ ã $3$ ã®åæ°ã«ãªããïŒ\\\r\næ¡åãã $A+B+C+D$ ã $3$ ã®åæ°ã«ãªããªãïŒ\r\n- $\\gcd(A,B,C,D)$ ã $5$ ã®åæ°ã«ãªã... | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äœéããããŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A,B,C,D$ ã®æ倧å
¬çŽæ°ã¯ $1$ ã§ããïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2880 | G | OMCT001(G) | 400 | 44 | 64 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2880"
}
] | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $A\lt B\lt C\lt D$ ã¯äœéããããŸããïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $A-B=C-D$ïŒ |
OMCT001 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct001/tasks/2881 | H | OMCT001(H) | 400 | 31 | 38 | [
{
"content": "ã解説ã¯æªäœæã§ãïŒãäºæ¿ãã ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct001/editorial/2881"
},
{
"content": "ããã£ïŒç©ºããæ°åŒãïŒ\r\n$$(x+1)(x+2)(x+4)(x+6)=x^4+13x^3+56x^2+92x+48$$\r\nãã£ãŠïŒçã㯠$\\bf{13569248}$",
"text": "çãã®ã¿",
"url": "https://onlinemathcontest.com/co... | ã以äžã®æ¡ä»¶ãã¿ããæŽæ°ã®çµ $(A,B,C,D)$ ã¯äžæã«ååšããããšã蚌æã§ããŸãïŒ
- $A,B,C,D$ ã¯ãã¹ãŠ $2$ æ¡ã®æ£æŽæ°ã§ããïŒåæ¡ã«ã¯ $1,2,3,4,5,6,8,9$ ãã¡ããã©äžåºŠãã€çŸããïŒ
- $x$ ã®æ¹çšåŒ $x^4+Ax^3+Bx^2+Cx+D=0$ ã®è€çŽ æ°è§£ã¯ïŒãã¹ãŠæŽæ°å€ã§ããïŒ
ãã®ãããªãã®ã«ã€ããŠïŒ$A,B,C,D$ ããã®é ã«ç¶ããŠè§£çããŠãã ããïŒ\
ã**ãã®åé¡ã«éãïŒå€§å°é¢ä¿ã®èšå®ãç¡ãããšã«æ³šæããŠãã ãã**ïŒ |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/1834 | A | OMC065(A) | 100 | 260 | 261 | [
{
"content": "ããã¹ãŠã®æåãåºå¥ããã° $5!$ éãã§ãã, ãã®ãšãåãæåå㯠$2^2$ åãã€çŸãããã, æ±ããã¹ãå€ã¯ $\\textbf{30}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc065/editorial/1834"
}
] | ã$m,n,n,o,o$ ã® $5$ æåã䞊ã³æ¿ããŠã§ããæååã¯äœéããããŸããïŒ |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/1505 | B | OMC065(B) | 200 | 190 | 247 | [
{
"content": "ã$360=2^3\\times 3^2\\times 5$ ã«çæã, åçŽ å æ°ã®åé
ãèããã°, åæ°ãçžç°ãªããšã®æ¡ä»¶ãç¡èŠããã°æ±ããå Žåã®æ°ã¯ ${}_5\\mathrm{C}_2\\times{}_4\\mathrm{C}_2\\times{}_3\\mathrm{C}_2=180$ éãã§ãã. ããã§, åæã« $2$ å䜿ããåŸãæ°ã¯ $1,2,3,6$ ã§ãããã, ($3$ åã«ã¯ãªãåŸãªã), 解çãã¹ãå€ã¯ $180-3\\times 4=\\textbf{168}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://o... | ã$abc=360$ ãªã**çžç°ãªã**æ£æŽæ°ã®çµ $(a,b,c)$ ã¯ããã€ãããŸããïŒ\
ããã ã, æ°ã®å
¥ãæ¿ãã«ãã£ãŠäžèŽãããã®ãåºå¥ãããã®ãšããŸã. |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/1977 | C | OMC065(C) | 200 | 204 | 228 | [
{
"content": "ã以äžãã, æ±ãã解ã®ç·å㯠$\\dfrac{\\sqrt{89}-1}{2}$ ã§ãã, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{92}$ ã§ãã.\r\n- $x\\leq -3$ ã®ãšã, $x^2-3x-28=0$ ã解ãã°ãã, $x=-4$ ãåŸã.\r\n- $-3\\leq x\\leq 5$ ã®ãšã, $x^2-3x-8=0$ ã解ãã°ãã, $x=\\dfrac{3\\pm\\sqrt{41}}{2}$ ãåŸã.\r\n- $x\\geq 5$ ã®ãšã, $x^2-x-22=0$ ã解ãã°ãã, $x=\\dfrac{1+\\sqrt{89}}{2}$ ãåŸã.",... | ã次㮠$x$ ã®æ¹çšåŒã®å®æ°è§£ã®ç·åãæ±ããŠãã ããïŒ
$$\lvert x^2-2x-15\rvert +\lvert x+3\rvert =10$$
ãã ã, æ±ããç·åã¯æ£æŽæ° $a,b,c$ ã«ãã£ãŠ $ \displaystyle\frac{\sqrt{a}-b}{c} $ ãšè¡šããã®ã§ (ãã ã $a$ ã¯å¹³æ¹å åããããªã), $a+b+c$ ã解çããŠãã ãã. |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/2223 | D | OMC065(D) | 200 | 168 | 211 | [
{
"content": "ãæ倧å
¬çŽæ° $g$ ã«ã€ããŠ, $a=Ag,b=Bg$ ãšããã°, æå°å
¬åæ°ã¯ $ABg$ ã§ãããã, æ¡ä»¶ã¯\r\n$$(AB-1)g=2021=43\\times 47$$\r\nç¹ã« $g=1,43,47,2021$ ã®ããããã§ãã, ãããã調ã¹ãããšã§å
šäœã§ã¯ $\\textbf{9}$ çµã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc065/editorial/2223"
}
] | ãæ£æŽæ°ã®çµ $a\lt b$ ã§ãã£ãŠ, ãããã®æå°å
¬åæ°ãšæ倧å
¬çŽæ°ã®å·®ã $2021$ ã§ãããã®ã¯ããã€ãããŸããïŒ |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/2161 | E | OMC065(E) | 300 | 64 | 138 | [
{
"content": "ã$1$ ã€ã®ç®ã®æ¡ä»¶ã«ã€ããŠ, åãšããŠããåŸããã®ã¯ $7$ ä»¥äž $195$ 以äžã§ãã, ããããã«ã€ããŠå Žåã®æ°ã¯\r\n$${}\\_{3}\\mathrm{C}\\_{3},{}\\_{3}\\mathrm{C}\\_{3},{}\\_{4}\\mathrm{C}\\_{3},{}\\_{4}\\mathrm{C}\\_{3},\\cdots,{}\\_{49}\\mathrm{C}\\_{3},{}\\_{49}\\mathrm{C}\\_{3},{}\\_{50}\\mathrm{C}\\_{3},{}\\_{49}\\mathrm{C}\\_{3},{}\\_{49}\... | ã$100$ 以äžã®æ£æŽæ°ã®çµ $(a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})$ ã§ãã£ãŠ, 以äžã®æ¡ä»¶ãã¿ãããã®ã¯ããã€ãããŸããïŒ
- $a_{1}+a_{2}=a_{3}+a_{4}=a_{5}+a_{6}$
- $a_{1}\lt a_{3}\lt a_{5}\lt a_{6}\lt a_{4}\lt a_{2}$ |
OMC065 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc065/tasks/1916 | F | OMC065(F) | 400 | 68 | 139 | [
{
"content": "$$\\angle BEC=\\angle AEB=\\angle AFE+45^\\circ=\\angle BPE+45^\\circ=180^\\circ-\\angle BEP$$\r\nãã $C,E,P$ ã¯å
±ç·ã§ãã, åæ§ã« $C,F,Q$ ãå
±ç·ã§ãã. ãã£ãŠ, $\\tan$ ã®å æ³å®çãå©çšããŠ\r\n$$\\angle{BCP}+\\angle{DCQ}=45^\\circ$$\r\nãã¿ããäžèŸºã®é·ããæ±ããã° $\\textbf{20}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcont... | ãæ£æ¹åœ¢ $ABCD$ ã«ãããŠ, 察è§ç· $BD$ äžã®ç¹ $E,F$ ã $\angle EAF=45^\circ$ ãã¿ãããŠãã, $B,E,F,D$ ã®é ã«äžŠãã§ããŸã. äžè§åœ¢ $AEF$ ã®å€æ¥åãšèŸº $AB,AD$ ã®äº€ç¹ããããã $P,Q$ ãšãããšã,
$$PB=12,\quad QD=5$$
ãæç«ããŸãã. ãã®ãšã, $ABCD$ ã®äžèŸºã®é·ããæ±ããŠãã ãã. |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/1833 | A | OMC064(A) | 300 | 100 | 143 | [
{
"content": "ãæ¹çšåŒã® $4$ 解ã $x=x_1,x_2,x_3,x_4$ ãšãããš, 解ãšä¿æ°ã®é¢ä¿ãã\r\n$$\\begin{cases}\r\n2022=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4\\\\\\\\\r\nn=x_1x_2x_3x_4\r\n\\end{cases}$$\r\nãã®ãšã, çžå ã»çžä¹å¹³åã®é¢ä¿ãã\r\n$$n\\leq \\left(\\dfrac{2022}{6}\\right)^2=113569$$\r\néã«ãã®ç¯å²ã§æ¡ä»¶ãã¿ããããšã確èªããããã, 解çãã¹ãå€ã¯ $\\textbf{113569}$ ã§ãã... | ã以äžã®æ¡ä»¶ãã¿ããæ£æŽæ° $n$ ã¯ããã€ãããŸããïŒ
- ããå®æ° $p,q$ ãååšã, 以äžã® $x$ ã«ã€ããŠã®å次æ¹çšåŒã®è€çŽ æ°è§£ããã¹ãŠæ£ã®å®æ°ãšãªã.
$$x^4-px^3+2022x^2-qx+n=0$$ |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/1855 | B | OMC064(B) | 400 | 130 | 161 | [
{
"content": "ãå¹³é¢ $\\alpha ,\\beta ,\\gamma$ ããããã $xy,yz,zx$ å¹³é¢ã§ãããšããŠãã.\\\r\nããã®ãšã, $xy$ å¹³é¢ã«ãã $P$ ã®æé¢ã®ååŸã $R_{xy}$ ãªã©ãšè¡šã, $P$ ã®ååŸã $R_P$ ãšãã.\r\n\r\n**解æ³1.**ãäžè¬æ§ã倱ããçã®äžå¿ã®åº§æšã $x,y,z\\geq 0$ ã«ãã£ãŠ $(x,y,z)$ ãšè¡šããš,\r\n$$\\begin{cases}\r\nR_P^2-z^2=R_{xy}^2=20\\\\\\\\\r\nR_P^2-x^2=R_{yz}^2=27\\\\\\\\\r\nR_P^2-y^2=... | ãå¹³é¢ $\alpha ,\beta ,\gamma$ ã¯äºãã«çŽäº€ããŠãã, ãã€ãã¹ãŠç $P$ ãšå
±ééšåããã¡ãŸã. ãã®ãšã, 次ã®æ¡ä»¶ããšãã«ã¿ãããããªç $P$ ã®ååŸãšããŠããåŸãæå°å€ã $r$, æ倧å€ã $R$ ãšããŸã. $r^2+R^2$ ã解çããŠãã ãã.
- å¹³é¢ $\alpha ,\beta ,\gamma$ ã®ãã¡, ã©ã® $2$ é¢ã®äº€ç·ãç $P$ ãšå
±ééšåãæã€.
- ç $P$ ã®å¹³é¢ $\alpha ,\beta ,\gamma$ ã«ããæé¢ç©ããããã $20\pi ,27\pi ,21\pi$ ã§ãã. |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/2174 | C | OMC064(C) | 500 | 95 | 134 | [
{
"content": "ã次ã®äºå®ã«çæããïŒ$d(x)$ ãå¥æ° $\\iff$ $x$ ã¯å¹³æ¹æ°.\\\r\nã$d(n^{d(n)})=N$ ãšãããš, $n^{d(n)}$ 㯠$n$ ãå¹³æ¹æ°ã§ãããã«ãããåžžã«å¹³æ¹æ°ã§ãããã, $N$ ã¯åžžã«å¥æ°ã§ãã. ãããã£ãŠ $d(m^2+47),d(m^2+87)$ ã®ã©ã¡ããäžæ¹ã®ã¿ãå¥æ°ã§ããã®ã§, $m^2+47,m^2+87$ ã®ã©ã¡ããäžæ¹ã®ã¿ãå¹³æ¹æ°ãšãªãïŒä»¥äž, $l$ ã¯æ£æŽæ°ãšããïŒ\\\r\nã$m^2+47=l^2$ ã®ãšã, $(m,l)=(23,24)$ ã«éãããã®ã§, $N=d(24^2)+d(616)=21+16=37$ ã§... | ãæ£ã®æŽæ° $x$ ã«å¯Ÿã㊠$d(x)$ 㧠$x$ ã®æ£ã®çŽæ°ã®åæ°ãè¡šãããšã«ããŸã.\
ã次ã®åŒãã¿ããæ£æŽæ° $m$ãååšãããããªæ£æŽæ° $n$ ã®ãã¡, å°ããæ¹ãã $10$ çªç®ãŸã§ã®ç·åãæ±ããŠãã ãã.
$$d(n^{d(n)})=d(m^2+47)+d(m^2+87)$$ |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/1904 | D | OMC064(D) | 500 | 54 | 94 | [
{
"content": "ãæ°çŽç·äžã®ç§»åã«ãªããããŠèãã. ããªãã¡, åº§æš $0$ ããã¹ã¿ãŒãã, 以äžã®ç§»å $A,B$ ãç¹°ãè¿ã㊠$100$ ãžè³ã.\r\n\r\n- 移å $A$ïŒèªèº«ã®ãã座æšã $+2$ ãã.\r\n- 移å $B$ïŒèªèº«ã®ãã座æšã $-1$ ãã.\r\n\r\nå
šäœã§ç§»å $A,B$ ãè¡ã£ãåæ°ããããã $x,y$ ãšè¡šãã°, $S_n\\geq 1$ ã®ãšã $n=3m+50\\ (m\\geq 0)$ ãšè¡šã, \r\n$$(x,y)=(m+50,2m)$$\r\nãŸã, åã座æšã $2$ åéããªããšããæ¡ä»¶ã¯, ãå $B$ ã®éã«ã¯æäœã§ã $2$ ã€... | ã$n$ ãæ£æŽæ°ãšããŸã. æ°å $\\{a_i\\}_{i=0,1,\cdots,n}$ ã以äžã®æ¡ä»¶ãã¿ãããŸãïŒ
- $a_0=0,\\,a_{n}=100$
- ä»»æã® $i=1,\ldots,n$ ã«å¯Ÿã, $a_{i}-a_{i-1}$ 㯠$2$ ãŸã㯠$-1$
- $a_0,a_1,\ldots,a_{n}$ ã¯ãã¹ãŠç°ãªã.
ãã®ãããªæ°åã®åæ°ã $S_n$ ãšãã, ããã« $p_n$ ã以äžã§å®ããŸãïŒ
$$p_n=
\begin{cases}
S_nã«å«ãŸããæ倧ã®çŽ å æ°\ \ (S_n\geq 2)\\\\
0\ \ (S_n\leq 1)
\end{cases}$$
ãã®ãšã... |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/266 | E | OMC064(E) | 700 | 12 | 33 | [
{
"content": "ã$ABC$ ã®å€æ¥åã«ãããŠåŒ§ $BAC$ ã®äžç¹ã $M$ ãšãããš, well-known factãšã㊠$A,D,E,M$ ã¯åäžååšäžã«ãã, $MBC$ ãš $MDE$ ã¯çžäŒŒãªäºç蟺äžè§åœ¢, ããã§ã¯ç¹ã«æ£äžè§åœ¢ã§ãã.\\\r\nãäžè¬ã«ããããç·å $BC,DE$ äžã«ãã $BK:KC=EL:LD$ ãã¿ããç¹ã®çµ $(K,L)$ ãèãããš, $K$ ã $B$ ãã $C$ ãŸã§åããããšã, $\\angle BKL$ ã®å€§ããã¯ç矩å調æžå°ã, äžæ¹ã§ $\\angle DLK$ ã®å€§ããã¯ç矩å調å¢å ãã. ããªãã¡ $\\angle BKL=\\angle... | ã$AB\gt AC,\angle A=60^\circ$ ãªãéè§äžè§åœ¢ $ABC$ ã«ãããŠ, ãããã蟺 $AB,AC$ äžã®ç¹ $D,E$ ã $BD=CE$ ã, ããããç·å $BC,DE$ äžã®ç¹ $P,Q$ ã $BP:PC=EQ:QD$ ãã¿ãããŸã. ãã®ãšã,
$$\angle BPQ=\angle DQP,\quad (BC-DE):PQ=10:9$$
ãæç«ãããªãã°, $\displaystyle\frac{CP}{BP}=\frac{a-b\sqrt{c}}{d}$ ãšè¡šããŸã. ãã ã, $a,b,d$ ã¯æ倧å
¬çŽæ°ã $1$ ã§, $c$ ã¯å¹³æ¹å åããããªããã®ãšããŸã. $a+b+c+d$ ... |
OMC064 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc064/tasks/2717 | F | OMC064(F) | 700 | 8 | 47 | [
{
"content": "ãæäœã«ããã茪ã«çãŸã人æ°ã¯åžžã«å¶æ°äººã§ããïŒ ç¹ã«å
šå¡ã亀äºã« $2$ ã€ã®ã°ã«ãŒãã«åå²ãããšãïŒããããã®ã°ã«ãŒãããå¶æ°äººã茪ãã匟ãããããšã容æã«åããïŒãããã£ãŠ $1$ åã§æäœãçµããããšã¯ãªãïŒ\\\r\nãäžæ¹ã§ïŒ$2$ åã§æäœãçµããããããïŒãããæ°ãäžãããïŒ\r\n\r\n**解æ³1.**ãããã§ïŒä»¥äžã®ããã«ããïŒ\r\n$$S(m,n)=\\sum_{k=0}^m {n \\choose {2k}}{n \\choose {2m-2k}},\\quad T(m,n)=\\sum_{k=0}^{m-1} {n \\choose {2k+1}}{n \\c... | ãå圢ã®èŒªã«äžŠãã $2022$ 人ã®çåŸãããŸã.\
ãããã§, 以äžã®æäœãã¡ããã© $\ell$ åç¹°ãè¿ãããšãã, $\ell$ åç®ã®æäœã«ãã£ãŠå
šå¡ã茪ã®å€ã«åºãŸãã.
- å人ã茪ã«ãã£ãŠå·Šå³ã®çåŸã®äžæ¹ãäžæã«æåãã. å¶æ° ($0$ ãå«ã) 人ã«æåããã人ã¯äžæã«èŒªã®å€ã«åºã. ãã ã, 茪ã«äžŠãã çåŸã $1$ 人ã§ãããšã, èªèº«ã茪ã«ãã£ãŠå·Šå³ã«äœçœ®ãããšã¿ãªã.
ãã®ãšãïŒ$\ell$ ãæå°ãšãªãããã«æäœãè¡ãæ¹æ³ã¯ $N$ éãã§ã. $N$ ã $2021$ ã§å²ã£ãäœããæ±ããŠãã ãã.\
ããã ãïŒãã¹ãŠã®çåŸã¯åºå¥ãããã®ãšãïŒåãã® $2022$ 人ã®äžŠã³æ¹ã¯åºå®ã... |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/1874 | A | OMC063(A) | 100 | 264 | 266 | [
{
"content": "ãäºã $q$ å¹ãããšãããš, æ¡ä»¶ããã«ããã ã·ã¯ $q-37$ å¹, 鶎㯠$137-2q$ 矜ãã, 足ã®æ°ã«ã€ããŠ\r\n$$2(137-2q)+4q+6(q-37)=334$$\r\nããã解ãããšã§ $q=47$ ãåŸããã, æ±ããã¹ãå€ã¯ $43\\times 47=\\textbf{2021}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc063/editorial/1874"
}
] | ã鶎ãšäºãšã«ããã ã·ããã, ãããã®é ã®æ°ã¯åèšã§ $100$, 足ã®æ¬æ°ã¯åèšã§ $334$ ã§ã.\
ãããã«äºãã«ããã ã·ãã $37$ å€ããããšã, 鶎ã $p$ 矜, äºã $q$ å¹ãããšã㊠$pq$ ã解çããŠãã ãã. \
ããã ã, 鶎, äº, ã«ããã ã·ã«ã¯ãããã足ã $2$ æ¬, $4$ æ¬, $6$ æ¬ãããŸã. |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/2374 | B | OMC063(B) | 100 | 257 | 263 | [
{
"content": "ã$n\\geq 5$ ã®ãšã, $n!+5$ ã $5$ ãã倧ãã $5$ ã®åæ°ãšãªãããäžé©ã§ãã. $n\\leq 4$ ã®ç¯å²ã§èª¿ã¹ãã°,\r\n$$(n,p)=(2,7),(3,11),(4,29)$$\r\nã解ãšãªã. ãããã£ãŠ, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{163}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc063/editorial/2374"
}
] | ã$n!+5=p$ ãã¿ããæ£æŽæ° $n$ ããã³çŽ æ° $p$ ã®çµ $(n,p)$ ãã¹ãŠã«ã€ããŠ, $np$ ã®ç·åãæ±ããŠãã ãã. |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/1572 | C | OMC063(C) | 200 | 224 | 255 | [
{
"content": "ã$mn+m+n=(m+1)(n+1)-1$ ã«çæããã°, $2$ ä»¥äž $101$ 以äžã®åææ°ã®æ°ãæ±ããããšã«çãã, ãã㯠$\\textbf{74}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc063/editorial/1572"
}
] | ããã (çžç°ãªããšã¯éããªã) æ£æŽæ° $m,n$ ã«ãã£ãŠ $mn+m+n$ ã®åœ¢åŒã«è¡šãã, $100$ 以äžã®æ£æŽæ°ã¯ããã€ãããŸããïŒ |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/1321 | D | OMC063(D) | 200 | 148 | 195 | [
{
"content": "ãäžè§åœ¢ $OAD$ ã¯æ£äžè§åœ¢ã§ãã,\r\n$$\\angle DAE=\\angle OAE-\\angle OAD=\\dfrac{1}{2}(180^\\circ-30^\\circ)-60^\\circ=15^\\circ$$\r\nãæç«ããããšã«çæãããš, $AE=AO=AD$ ãã\r\n$$\\angle AEF=\\dfrac{1}{2}(180^\\circ-15^\\circ)=82.5^\\circ$$\r\nãã£ãŠåè§åœ¢ $OAEF$ ã«ãããå
è§ã®åãèããããšã§ $\\angle BFE=67.5^\\circ$ ã ãã, ç¹ã«è§£çãã¹ãå€ã¯ $\\te... | ã$O$ ãäžå¿ãšããååšäžã« $4$ ç¹ $A,C,D,B$ ããã®é ã«äžŠãã§ãã, 以äžã®æ¡ä»¶ãã¿ãããŸã.
$$\angle AOC=\angle COD=\angle DOB=30^\circ$$
åçŽç· $AC$ äžã« $AO=AE$ ãªãç¹ $E$ ããšã, ç·å $OB$ ãšçŽç· $DE$ ã®äº€ç¹ã $F$ ãšãããšã, è§ $BFE$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã.\
ããã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ã«ãã£ãŠ $\dfrac{p}{q}$ 床ãšè¡šãããã®ã§, $p+q$ ã解çããŠãã ãã. |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/290 | E | OMC063(E) | 300 | 59 | 104 | [
{
"content": "ã察è§ç·ãäžæ¬ãã€æžã足ããŠããããšãèãããš, æ±ããé åã®æ°ã¯ $(1+$察è§ç·ã®æ¬æ°$+$å
éšã®å¯Ÿè§ç·ã®äº€ç¹æ°$)$ ã§è¡šãããããšã容æã«ããã. æ£ $n$ è§åœ¢ã®å¯Ÿè§ç·ã¯ $n(n-3)\\/2$ æ¬ã§, ãããã®å
éšã®äº€ç¹ã¯ $n$ é ç¹ãã $4$ ã€ãéžãã§ã§ããåè§åœ¢ãšäžå¯Ÿäžã«å¯Ÿå¿ãããã ${}\\_{n}\\mathrm{C}\\_{4}$ åã§ãã.\\\r\nããã£ãŠæ±ããé åæ°ã¯ä»¥äžã§äžããã, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{5796}$ ã§ãã.\r\n$$1+\\dfrac{n(n-3)}{2}+{}\\_{n}\\mathrm{C}\\_{4}... | ã$n$ ã $5$ 以äžã®å¥æ°ãšããŸã. æ£ $n$ è§åœ¢ã®å
éšã察è§ç·ã«ãã£ãŠåå²ãããé åã®æ°ã¯, æ£æŽæ° $a,b,c,d,e$ ãçšããŠ
$$\dfrac{1}{a}\left(n^4-bn^3+cn^2-dn+e\right)$$
ãšè¡šããŸã. $\dfrac{bcde}{a}$ ã解çããŠãã ãã. \
ããã ã, æ£ $n$ è§åœ¢ã®å¯Ÿè§ç· $3$ æ¬ã¯, ãã®éžã³æ¹ã«ãããå
éšã®äžç¹ã§äº€ãããªãããšã蚌æã§ããŸã. |
OMC063 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc063/tasks/1759 | F | OMC063(F) | 400 | 90 | 138 | [
{
"content": "ãäžãã $k$ æ¡ç®ã®æ°åã $a_k$ ãšãã. æããã« $a_k$ ãš $k$ ã®å¶å¥ã¯äžèŽã, ç¹ã« $a_5=5,\\ a_{10}=0$ ãçŽã¡ã«åŸã. ãŸã, $10a_3+a_4$ ããã³ $10a_7+a_8$ ã $4$ ã§å²ãåããããšãã, $a_4$ ããã³ $a_8$ 㯠$2$ ãŸã㯠$6$ ã§ãã. ããã«, $a_4+a_5+a_6$ ã $3$ ã§å²ãåããããšãã, å¶æ° $k$ ã«å¯Ÿãã $a_k$ ã®å®ãæ¹ã¯ $2$ éãã«çµããã. ããšã¯ $10a_7+a_8$ ã $4$ ã§å²ãåããããš, ããã³ $a_7+a_8+a_9$ ã $3$... | ã以äžã®æ¡ä»¶ããã¹ãŠã¿ãã, $10$ æ¡ã®æ£æŽæ° $N$ ãæ±ããŠãã ãã. ãã ã, ãã®ãã㪠$N$ ã¯äžæçã«ååšããŸã.
- $N$ ã®åæ¡ã«ã¯ $0$ ãã $9$ ãã¡ããã©äžåºŠãã€çŸãã. ãŸã, $N$ ã®æé«äœã¯ $0$ ã§ã¯ãªã.
- ä»»æã® $k=1,2,\cdots, 10$ ã«å¯Ÿã, $N$ ã®äžãã $k$ æ¡ãåãåºããŠåŸãããæŽæ°ã¯, $k$ ã§å²ãåãã.
ããã§, äŸãã° $12345$ ã®äžãã $3$ æ¡ãåãåºããŠåŸãããæŽæ°ã¯ $123$ ã§ã. |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/1725 | A | OMC062(A) | 200 | 145 | 171 | [
{
"content": "ã解ãšä¿æ°ã®é¢ä¿ãã, éè² æŽæ° $n$ ã«ã€ããŠ\r\n$$x^{n+2}-8x^{n+1}+x^n=0,\\quad y^{n+2}-8y^{n+1}+y^n=0$$\r\nãã£ãŠ $a_{n+2}=8a_{n+1}-a_{n}$ ãæç«ãã. ãããã, $b_0=2,b_1=1$ ããå§ããã°, $b$ 㯠$2,1,6,5,6,1$ ã®åšæãç¹°ãè¿ãããšãåãã. ãããã, æ±ããã¹ãç·å㯠$(2+1+6+5+6+1)\\times(2022\\/6)=\\textbf{7077}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https:/... | ãå®æ° $x\lt y$ 㯠$x+y=8,xy=1$ ãã¿ãããŸã. ãã®ãšã, æ£æŽæ° $n$ ã«å¯Ÿã
$$a_n=x^n+y^n$$
ã¯åžžã«æŽæ°å€ãšãªãã®ã§, ããã $7$ ã§å²ã£ãäœãã $b_n$ ãšãããŸã. 以äžã®ç·åãæ±ããŠãã ããïŒ
$$b_1+b_2+b_3+\cdots+b_{2021}+b_{2022}$$ |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/2551 | B | OMC062(B) | 300 | 135 | 147 | [
{
"content": "ãäžå¹³æ¹ã®å®çãã $CD^2=AD^2+BC^2-AB^2=40$ ã§ãã, ãã®ãšãPtolemyã®å®çãã $ACÃBD=40+14 \\sqrt{10}$ ã§ãã. ããŸæ±ããé¢ç©ã¯ $AC\\times BD\\/2$ ã§ãããã, 解çãã¹ãå€ã¯ $20+7+10=\\textbf{37}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc062/editorial/2551"
},
{
"content": "$BD$ ã®åçŽäºçåç·ã«ã€ã㊠$C... | ãåã«å
æ¥ããåè§åœ¢ $ABCD$ ã, 以äžã®æ¡ä»¶ãã¿ãããŠããŸãïŒ
$$AB=7,\quad BC=8,\quad AD=5,\quad AC \perp BD$$
ãã®ãšã, åè§åœ¢$ABCD$ã®é¢ç©ã¯, æ£æŽæ° $a,b,c$ ãçšã㊠$a+b \sqrt c$ ãšè¡šããã®ã§ (ãã ã $c$ ã¯å¹³æ¹å åããããªã), $a+b+c$ ãæ±ããŠãã ãã. |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/2616 | C | OMC062(C) | 300 | 112 | 143 | [
{
"content": "ã$a,b,c$ ã $7$ ã§å²ã£ãäœãã¯çžç°ãªãããïŒ$a,b,c$ ã®å€§å°ã®å¶çŽãåãé€ããŠèãïŒ$3!$ ã§å²ãã°ããïŒ\\\r\nã以äžãã $a+b+c$ 㯠$7$ ã®åæ°ã§ããããšã«çæããïŒ\r\n$$\\begin{aligned}\r\n4(a+b+c) &= a+b+c+(a+b)+(a+c)+(b+c)+(a+b+c) \\\\\\\\\r\n&\\equiv 0+1+2+\\cdots+6\\\\\\\\\r\n&\\equiv 0 \\pmod{7}\r\n\\end{aligned}$$\r\nããããïŒ$a,b,c$ ã $7$ ã§å²ã£ãäœãã¯ïŒ$\\\\... | ã$1\leq a\lt b\lt c\leq 500$ ãªãæŽæ°ã®çµ $(a,b,c)$ ã§ãã£ãŠïŒä»¥äžã®æ¡ä»¶ãã¿ãããã®ã¯ããã€ãããŸããïŒ
- éå $\\{a,b,c\\}$ ã®ç©ºã§ãªã $7$ ã€ã®éšåéåã«ã€ããŠïŒããããã®èŠçŽ ã®ç·åã $7$ ã§å²ã£ãäœãã¯çžç°ãªãïŒ |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/2033 | D | OMC062(D) | 400 | 74 | 96 | [
{
"content": "ã$4$ ç¹ $A,B,D,C$ ã®å
±åã¯å®¹æã«åãã. ããã§, 蟺 $BC$ äžã« $AQ=8$ ãªãç¹ $Q(\\neq C)$ ããšãã°, å
ã®äºå®ãšäœµããŠäžè§åœ¢ $ABD$ ãš $AQC$ ã¯çžäŒŒã§ãã, ç¹ã« $\\angle BAD=\\angle QAC$, åæã« $\\angle BAQ=\\angle PAC$ ã§ãã. ãããã£ãŠ,\r\n$$BQ:PC=(14 \\times8):(7 \\times8)=2:1,\\quad BP:QC=(14 \\times7):(8 \\times8)=49:32$$\r\nãã $BQ:QP:PC=34:15:17$ ãåŸ... | ã$AB=14,AC=8$ ãªãäžè§åœ¢ $ABC$ ãš (åãã蟌ããŠ) ååãªäžè§åœ¢ $ADE$ ã«ã€ããŠ, $C$ ã¯èŸº $DE$ äžã«ãã, 蟺 $AD$ ãšèŸº $BC$ ã®äº€ç¹ $P$ ã $AP=PD$ ãã¿ãããŸãã. ãã®ãšã, $BP$ 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. |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/2509 | E | OMC062(E) | 400 | 9 | 41 | [
{
"content": "ã以äžã§ã¯ $v_i,w_i$ ãçž®çŽããŠäžã€ã®é ç¹ãšã¿ãªããæ°ããªã°ã©ããèãã (人ã®æãçšããè¡šçŸãèãããšè¯ã).\\\r\nãé£çµæåã®ãµã€ãºã®çµã¿åãããèããã°, 以äžã® $4$ éããããåŸã.\r\n$$\\lbrace6\\rbrace,\\quad \\lbrace4,2\\rbrace,\\quad \\lbrace3,3\\rbrace,\\quad \\lbrace2,2,2\\rbrace$$\r\nãŸã, ããããã®é£çµæåã®ç¶æ
ãšããŠã¯, äžã€ã®ã«ãŒãã, äžã€ã®ãã¹ã®ããããã§ãã.\\\r\nã$n\\ (\\geq 2)$ é ç¹ã®çµã¿åãããåºå®ãã... | ã$12$ åã®é ç¹ $v_1,\ldots,v_6$ ããã³ $w_1\ldots,w_6$ ããããŸã. ãããã®ãã¡çžç°ãªã $2$ é ç¹ãçµã¶ç¡åã® (åãããããªã) 蟺ãäœæ¬ãåŒãæ¹æ³ã§ãã£ãŠ, 以äžã®æ¡ä»¶ãã¿ãããã®ã¯äœéããããŸããïŒ
- åã $2$ ç¹ãçµã¶èŸºã¯é«ã
$1$ æ¬ã§ãã.
- ããããã®é ç¹ã«æ¥ç¶ãã蟺ã¯é«ã
$1$ æ¬ã§ãã.
- $i=1,\ldots,6$ ã«ã€ããŠ, $v_i$ ãš $w_i$ ãçµã¶èŸºã¯ååšããªã.
- $i=1,\ldots,6$ ã«ã€ããŠ, $v_i$ ãš $w_i$ ã®å°ãªããšãäžæ¹ã«ã¯èŸºãæ¥ç¶ããŠãã.
(çŽæçãªè¡šçŸãšããŠã¯, $6$ 人ãåæã«... |
OMC062 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc062/tasks/2536 | F | OMC062(F) | 500 | 17 | 33 | [
{
"content": "ã$b_{ n }=a_{ n }+\\displaystyle \\frac{1}{3}Ã2^{n-1}$ ãšããã°, 以äžã®æŒžååŒãåŸã.\r\n$$b_{ 1 }=\\displaystyle \\frac{16}{3},\\quad b_{ 2 }=\\displaystyle \\frac{146}{3},\\quad b_{ n+2 }=16b_{ n+1 }-55b_{ n }$$\r\nããã解ã㊠$b_{ n }=\\displaystyle \\frac{1}{3}Ã(11^n+5^n)$ãåŸããã, $a_n$ ã®äžè¬é
ã¯\r\n$$a_{ n }=\\displa... | $$a_{ 1 }=5,\quad a_{ 2 }=48,\quad a_{ n+2 }=16a_{ n+1 }-55a_{ n }-9Ã2^{n-1}\ (n=1,2,\ldots)$$
ã§å®ãŸãæ°å $\lbrace a_{ n } \rbrace$ ã«ãããŠ, $a_{ n }$ ã $p$ ã§å²ãåããæ倧ã®åæ°ã $f_{ p }(n)$ ã§è¡šããŸã.\
ããã®ãšã, 以äžã®å€ãæ±ããŠãã ããïŒ
$$\displaystyle \sum_{n=1}^{1000} (f_{ 2 }(n)+f_{ 5 }(n))$$ |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/2410 | A | OMC061(A) | 100 | 222 | 228 | [
{
"content": "ã$AM=OB=OM=OA$ ããäžè§åœ¢ $OAM$ ã¯æ£äžè§åœ¢ã§ãã. ãããã $\\angle AOB=2\\angle AOM=120^\\circ$ ã§ãããã, æ±ããé¢ç©ã¯ $6^2\\times\\pi\\times(120\\/360)=\\bf{12} \\pi$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc061/editorial/2410"
}
] | ãäžå¿ã $O$ ãšããæ圢 $OAB$ ã«ã€ããŠ, 匧 $AB$ ã®äžç¹ã $M$ ãšãããš, ç·å $AM$ ããã³ $OB$ ã®é·ãã¯ãšãã« $6$ ã§ãã. ãã®ãšã, æ圢ã®é¢ç© $S$ ã«ã€ããŠ, $S\/\pi$ ãæ±ããŠãã ãã. |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/1610 | B | OMC061(B) | 100 | 200 | 224 | [
{
"content": "ã$l$ ãš $c$ ãé£ãåããããªäžŠã¹æ¹ã¯ $2\\times 5!=240$ éããã, $k$ ãš $r$ ãé£ãåãå Žåãåæ§ã§ãã. ãŸã, ããããåæã«é£ãåããããªäžŠã¹æ¹ã¯ $2\\times 2\\times 4!=96$ éãã§ãããã, æ±ããå Žåã®æ°ã¯ $6!-2\\times 240+96=\\textbf{336}$ éã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc061/editorial/1610"
}
] | ã$6$ æå $l,o,c,k,e,r$ ãäžåã«äžŠã¹ããšã, $l$ ãš $c$, $k$ ãš $r$ ãé£ãåããªããããªäžŠã¹æ¹ã¯äœéããããŸããïŒ |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/2439 | C | OMC061(C) | 200 | 164 | 203 | [
{
"content": "ã以äžã®ããã«èšç®ã§ãã.\r\n$$\\begin{aligned}\r\n1+1\\times 1!+2\\times 2!+ \\cdots + 2022\\times 2022! &= 2! + 2 \\times 2! + 3 \\times 3!+\\cdots + 2022 \\times 2022! \\\\\\\\\r\n&=3\\times 2!+3\\times 3! + \\cdots +2022\\times 2022!\\\\\\\\\r\n&= 3!+3\\times 3!+\\cdots + 2022\\times 2022! \\\\\\\\\r\n&= ... | ã$1+\displaystyle\sum_{k=1}^{2022} (k \times k!)$ 㯠$2$ ã§æ倧äœåå²ãåããŸããïŒ |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/2490 | D | OMC061(D) | 300 | 101 | 146 | [
{
"content": "ã$1$ ãã $10$ ãŸã§ã®æŽæ°ãé ç¹ãšã, $k$ ãã $a_k$ ã«èŸºã匵ã£ãæåã°ã©ããèãããš, ãã¹ãŠã®é ç¹ã«ã€ããŠå
¥æ¬¡æ°ã»åºæ¬¡æ°ããšãã« $1$ ã§ããããšãã, åé£çµæåããµã€ã¯ã«ãšãªãããšãåãã. ãã®ãšãæ¡ä»¶ã¯, ãã¹ãŠã®é£çµæåã®å€§ããã®æå°å
¬åæ°ã $30$ ãšãªãããšã§ãã, $10=5+3+2$ ãšåå²ããã»ããªã. ãããã£ãŠæ±ããçãã¯\r\n$${}\\_{10}\\mathrm{C}\\_{5}\\times {}\\_{5}\\mathrm{C}\\_{3} \\times4!\\times 2!\\times 1!=\\textbf{1209... | ãåºå¥ã§ãã $10$ æã®ã«ãŒããå·Šå³äžåã«äžŠãã§ããŸã. ã㟠$(1,2,\cdots,10)$ ã®äžŠã¹æ¿ã $(a_1, a_2, \cdots, a_{10})$ ã«ã€ããŠ, 以äžã®æäœãèããŸãïŒ
- $k=1,2,\cdots,10$ ã«å¯Ÿã, å·Šãã $k$ çªç®ã®ã«ãŒããå·Šãã $a_k$ çªç®ã«æ¥ãããã«äžŠã¹æ¿ãã.
ãã®æäœãç¹°ãè¿ãè¡ã£ããšã, $30$ åç®ã§ã¯ãããŠã«ãŒãã®äžŠã³ãå
ã«æ»ããŸãã. ãã®ãšã, $(a_1, a_2, \cdots, a_{10})$ ãšããŠããåŸããã®ã¯ããã€ãããŸãã. |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/2190 | E | OMC061(E) | 300 | 45 | 75 | [
{
"content": "ãæŽæ°ã®éå $X$ ã«ãããŠ, å°ããæ¹ãã $i$ çªç®ã $x_i$ ã§è¡šãã°, $3$ ã€ã®éå $A,B,C$ ã«ã€ããŠ, ã¹ã³ã¢ã®ç·åã¯\r\n$$\\sum_{k=1}^{674} (2k-675)(a_k+b_k+c_k)$$\r\n$A,B,C$ ã«çŸãã $2022$ ã®æ°ã®éå $D$ ãåºå®ããã°, ãããæ倧ãšãªãã®ã¯æããã« $k=1,2,\\ldots,674$ ã«å¯Ÿã\r\n$$\\\\{a_k,b_k,c_k\\\\}=\\\\{d_{3k-2},d_{3k-1},d_{3k}\\\\}$$\r\nãæãç«ã€ãšãã§ãã, ããã« $D$ ãåããã°æããã«... | ãæŽæ°ãããªãéåã«å¯Ÿã, $2$ å
ãéžã¶æ¹æ³ãã¹ãŠã«ã€ããŠ, ããããã®å·®ã®çµ¶å¯Ÿå€ã®ç·åã**ã¹ã³ã¢**ãšåŒã³ãŸã. äŸãã°,
$$|1-2|+|2-3|+|1-3|=4$$
ãã, éå $\\{1,2,3\\}$ ã®ã¹ã³ã¢ã¯ $4$ ã§ã.\
ãã㟠$10^{1000}$ 以äžã®æ£æŽæ°ããçžç°ãªã $2022$ åãéžã³, ãããããããã $674$ åãããªãéå $3$ ã€ãžããã«åå²ããŸã. ãã®ãšã, $3$ éåã®ã¹ã³ã¢ã®ç·åãæ倧ãšãªãæ¹æ³ã¯ $M$ éããããŸã. $M$ ã¯åé²æ³è¡šèšã§äœæ¡ã§ããïŒ\
ããã ã, $M$ ãžã®èšäžã«ããã£ãŠ, $3$ ã€ã®éåã®é åºã¯åºå¥ããªããã®ãšããŸã. ãŸã... |
OMC061 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc061/tasks/2247 | F | OMC061(F) | 400 | 15 | 104 | [
{
"content": "ã以äžã®ããã« $p,q,r,s$ ããšãã°ããããã¯çžç°ãªãæ£æŽæ°ã§ãã.\r\n$$p=\\frac{a+b+c+d}{a},\\quad q=\\frac{a+b+c+d}{b},\\quad r=\\frac{a+b+c+d}{c},\\quad s=\\frac{a+b+c+d}{d}$$\r\nããã«, $p,q,r,s$ ã¯æ¬¡ã®æ¡ä»¶ãã¿ããïŒ\r\n$$\\frac1p+\\frac1q+\\frac1r+\\frac1s=1$$\r\nãã®ãšã, ãã®ãã㪠$(p,q,r,s)$ ã®çµã¯\r\n$$(2,3,7,42), (2,3,8,24), (2,3,9,18), (... | 以äžã®æ¡ä»¶ãæºããçžç°ãªãæ£æŽæ°ã®çµ $(a,b,c,d)$ ãã¹ãŠã«ã€ããŠ, $a+b+c+d$ ã®ç·åã解çããŠãã ãã.
- $a,b,c,d$ ã¯äºãã«çŽ ã§ãã.
- $b+c+d$ 㯠$a$ ã®åæ°.
- $a+c+d$ 㯠$b$ ã®åæ°.
- $a+b+d$ 㯠$c$ ã®åæ°.
- $a+b+c$ 㯠$d$ ã®åæ°. |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/243 | A | OMC060(A) | 200 | 158 | 169 | [
{
"content": "ã$X=\\log x$ ãªã©ãšããã°, åºã®å€æå
¬åŒããæå°åãã¹ãå€ã¯\r\n$$81\\dfrac{Y}{X}+72\\dfrac{Z}{Y}+64\\dfrac{X}{Z}$$\r\nçžå ã»çžä¹å¹³åã®é¢ä¿ãããã㯠$3\\sqrt[3]{81\\times72\\times64}=216$ 以äžã§ãã, $8X=9Y=9Z$ ã§çå·ãæç«ãããã, æ±ããæå°å€ã¯ $\\textbf{216}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc060/editoria... | ã$1$ ãã倧ããå®æ° $x,y,z$ ã«å¯Ÿã, $81\log_{x}y+72\log_{y}{z}+64\log_{z}{x}$ ã®ãšãåŸãæå°å€ãæ±ããŠãã ãã. |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/244 | B | OMC060(B) | 400 | 102 | 160 | [
{
"content": "ãããããã®çåŸããã³ä¿ã $12$ åã®é ç¹ãšã, çåŸãžã®ä¿ã®å²ãåœãŠã«åŸã£ãŠèŸºã貌ã£ãäºéš(ç¡å)ã°ã©ããèãããš, ãã¹ãŠã®é ç¹ã®æ¬¡æ°ã $2$ ã§ããããšãã, ãã®èŸºã¯ããã€ãã®ãµã€ã¯ã«ã«å解ã§ãã. ããã§, 3ã€ç®ã®æ¡ä»¶ãããã®ãµã€ã¯ã«ã®é·ã㯠$4$ ã§ã¯ãªã, ããã«äºéšã°ã©ãã§ããããšãããµã€ã¯ã«ã®é·ãã¯å¥æ°ã§ã¯ãªããã, ããããã $6$ 以äžã§ãã. ãããã, ã°ã©ãã¯é·ã $6$ ã®ãµã€ã¯ã«äºã€ã«å解ãããã, é·ã $12$ ã®ãµã€ã¯ã«äžã€ããæ§æããããããªã. åè
ã®å Žåã®æ°ã¯ä»¥äžã§äžãããã.\r\n$$\\dfrac{1}{2}({}\\_{6}\... | ã以äžã®æ¡ä»¶ãã¿ããããã«, $6$ 人ã®çåŸã« $6$ ã€ã®ä¿ãå²ãåœãŠãæ¹æ³ã¯äœéããããŸããïŒ
- å人ã«ã¯ã¡ããã© $2$ ã€ã®çžç°ãªãä¿ãå²ãåœãŠã.
- ããããã®ä¿ã¯ã¡ããã© $2$ 人ã®çåŸãåãæã€.
- ã©ã® $2$ ã€ã®ä¿ã«ã€ããŠã, ãã®äž¡æ¹ãå²ãåœãŠãããçåŸã¯é«ã
$1$ 人ã§ãã.
ãã ã, çåŸããã³ä¿ã¯ãã¹ãŠåºå¥ãããã®ãšããŸã. |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/274 | C | OMC060(C) | 400 | 109 | 125 | [
{
"content": "ã$M$ 㯠$DF$ ã®äžç¹ã§ãããããšãã, $AB$ ãš $CF$ ã¯å¹³è¡ã§ãã,\r\n$$\\angle DAF=\\angle BAF=\\angle BCF=\\angle MCF$$\r\nããäžè§åœ¢ $ADF$ ãš $CFM$ ã¯çžäŒŒ, ããªãã¡ $AD=2CF$ ã§ãã. äžæ¹ã§æ¹ã¹ãã®å®çãã\r\n$$AD\\times CF=AD\\times BD=DF\\times DG=72$$\r\nã§ãããã, $AD=12,CF=6$ ãåŸã, äžå¹³æ¹ã®å®çãã $AC=6\\sqrt{10}$ ã§ãã.\\\r\nãããã§, $AC$ ãš $DM$ ã®äº€ç¹ã $X$... | ãå $\Gamma$ ã«å
æ¥ããäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ ã®äžç¹ã $M$ ãšã, $M$ ãã $AB,AC$ ã«ããããåç·ã®è¶³ããããã $D,E$ ãšããŸã. ãŸã, $M$ ãã $AB$ ã«ããããåç·ãš $\Gamma$ ã®äº€ç¹ã®ãã¡, $BC$ ã«é¢ã㊠$A$ ãšå察åŽã«ãããã®ã $F$, ããäžæ¹ã $G$ ãšããŸã. $DG=4,DM=MF=9$ ãæç«ãããšã, $DE$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\sqrt{\dfrac{a}{b}}$ ãšè¡šãããŸã. $a+b$ ã解çããŠãã ãã. |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/273 | D | OMC060(D) | 600 | 33 | 80 | [
{
"content": "ãå·Šãã $k$ çªç®ã®ç®±ã $B_k$ ã§è¡šã. ãŸã, \r\n$$\\begin{aligned}\r\nc_k=\\begin{cases}\r\nk&(k\\leq n)\\\\\\\\\r\n2n+1-k&(k\\gt n)\r\n\\end{cases}\r\n\\end{aligned}$$\r\nãšå®ãã. $B_k$ ã«å
¥ã£ãŠããçã« $\\sum^{k-1}\\_{i=1}c_i+1$ ä»¥äž $\\sum^{k}\\_{i=1}c_i$ 以äžã®æŽæ°ãäžã€ãã€æžã蟌ã. ãã®ãšã, $n(n+1)$ 以äžã®æ£ã®æŽæ°ã $1$ ã€ãã€æžã蟌ãŸããããšã«ãªã. ããã§, $... | ã$2n$ åã®ç®±ãå·Šå³äžåã«äžŠãã§ãã, åãå·Šãã $k$ çªç®ã®ç®±ã«ã¯, $k\leq n$ ã®ãšã $k$ å, $k\gt n$ ã®ãšã $2n+1-k$ åã®çãå
¥ã£ãŠããŸã ($k=1,2,\ldots,2n$). ãããã«å¯Ÿã,
- äžçªå³ä»¥å€ã®ç®±ã«å
¥ã£ãçãäžã€éžã³, ãããç®±ããåãåºã, ãã®ç®±ã®äžã€å³ã®ç®±ã«ç§»ã.
ãšããæäœãç¹°ãè¿ããŸã. ã©ã®ç®±ã«ãçã $n$ ã®åæ°åå
¥ã£ãŠããããã«ããããã«å¿
èŠãªæå°ã®æäœã®åæ°ã $f(n)$ ãšããŸã. ãã ã, ãã®æäœãæéåç¹°ãè¿ããŠãã®ãããªç¶æ
ã«ã§ããããšãä¿èšŒãããŸã.\
ããã®ãšã, $f(10^6-1)+f(10^6)$ ãæ±ããŠãã ... |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/1484 | E | OMC060(E) | 700 | 9 | 35 | [
{
"content": "ã$k = \\dfrac{2021}2$ ãšãããšïŒ$\\ell_t$ ã $x$ åçãæã€ããšãš $t \\ne k$ ã¯åå€ã§ããïŒ$\\\\{x_n\\\\}$ ã«ã€ããŠ\r\n$$x_{n+1} = \\frac{x_n^2 - 4k^2}{2 \\left(x_n - k\\right)}. $$\r\nãã®ãšãïŒ$\\tan \\theta_n = \\dfrac{\\sqrt3\\\\, k}{x_n - k}$ ãšããã° $\\tan \\theta_{n+1} = \\tan 2\\theta_n$ ãåŸãããïŒ$\\tan \\alpha\\pi = \\dfrac... | ãé¢æ° $f(x) =x^2 - 2021x + 2021^2$ ããã³ä»»æã®å®æ° $t$ ã«ã€ããŠïŒ$xy$ 座æšå¹³é¢äžã§æ²ç· $y=f(x)$ ã®ç¹ $(t,f(t))$ ã«ãããæ¥ç·ã $\ell_t$ ãšããŸãïŒãŸãå®æ°å $\\{x_n\\}_{n=1,2,\ldots}$ ãïŒä»¥äžã®ããã«å®çŸ©ããŸãïŒ
* $a$ ãé©åœãªå®æ°ãšããŠïŒ$x_1=a$ ãšããïŒ
* $x_n$ ãå®çŸ©ãã㊠$â_{x_n}$ ã® $x$ åçãååšãããªãã°ïŒãã® $x$ åçã®å€ã $x_{n+1}$ ãšããïŒ
* $x_n$ ãå®çŸ©ãããã $â_{x_n}$ ã® $x$ åçã¯ååšããªããšãïŒä»»æã®æŽæ° $m\gt n$ ... |
OMC060 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc060/tasks/289 | F | OMC060(F) | 1000 | 2 | 20 | [
{
"content": "ã$\\varphi$ ã§Eulerã®totienté¢æ°ãè¡šããã®ãšãã. äžè¬ã« $N\\geq 2$ ãéè² æŽæ° $l_2,l_3,l_5,l_{17},l_{257}$ ãçšããŠ\r\n$$2^{l_{2}}\\times 3^{l_{3}}\\times 5^{l_{5}}\\times 17^{l_{17}}\\times 257^{l_{257}}$$\r\nãšè¡šããããšãã (ãã®ãã㪠$N$ ã**è¯ãæŽæ°**ãšãã¶ããšãšãã).\r\nãŸã, æ£ã®æŽæ° $n$ ãšéè² æŽæ° $k$ ã«å¯Ÿã\r\n$$a(n,0)=1,\\quad a(n,k+1)=n^{a(n,k)}-... | ã$N=2^5\times 3^4\times 5^3\times 17^2\times 257$ ã«ã€ããŠ, 以äžãã¿ãã $0$ ä»¥äž $N$ æªæºã®æŽæ° $m$ ã®åæ°ãæ±ããŠãã ãã.
- ããæ£ã®æŽæ° $n$ ãšæ£ã®æŽæ° $K$ ãååšã, æŽæ°å $a_0,a_1,\dots$ ã
$$a_0=1,\quad a_{k+1}=n^{a_k}-1\quad(k=0,1,\dots)$$
ã§å®ãããš, $a_K\equiv a_{K+1}\equiv\cdots\equiv m\pmod N$ ãšãªã. |
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/1637 | A | OMC059(A) | 100 | 185 | 194 | [
{
"content": "ãè¡ãç®ç·ã§äžãå, äžãå, å¹³åŠãªéãåèšã§ãããã $a\\\\,\\text{m},b\\\\,\\text{m},c\\\\,\\text{m}$ ãã£ããšãããš, 以äžã®äžåŒãæãç«ã€ïŒ\r\n$$\\frac{a}{40}+\\frac{b}{60}+\\frac{c}{50}=210,\\quad\\frac{a}{60}+\\frac{b}{40}+\\frac{c}{50}=220,\\quad a+b+c =10560$$\r\nãã®äžå
äžæ¬¡é£ç«æ¹çšåŒã解ã㊠$(a,b,c)=(1680,2880,6000)$ ãåŸããã, å¹³åŠãªéã¯åèšã§ $\\textbf{6... | ãnatuåã¯äžãåãåé $40\\,\text{m}$, äžãåãåé $60\\,\text{m}$, å¹³åŠãªéãåé $50\\,\text{m}$ ã§æ©ããŸã. natuåã $10560\\,\text{m}$ ã®éã®ãã $210$ åãããŠè¡ã, $220$ åãããŠæ»ã£ãŠãããšã, ãã®éã®ãã®ãã¡å¹³åŠãªéã¯åèšã§äœ $\text{m}$ ãããŸããïŒ |
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/1248 | B | OMC059(B) | 200 | 155 | 175 | [
{
"content": "ã$5$ åã®ããŒã«ã«å¯ŸããŠ, 以äžã§å®ãŸãæäœ $k\\ \\ (k=1,2,3,4,5)$ ãèãã (æäœ $0$ ã§ã¯äœãè¡ããªã)ïŒ\r\n\r\n- æäœ $k$ïŒæäœ $k-1$ ãŸã§ã«éžã°ããŠããªãããŒã«ãã, $a_k$ åãéžã㧠$k$ ãæžã蟌ã.\r\n\r\nãããš, æäœ $k$ ã®æœãæ¹ã¯ $\\_{5-a_1-...-a_{k-1}}\\mathrm{C}\\_{a_k}$ éãã§ãããã, æ±ããç·åã¯æäœ $5$ åãéããŠã®æœãæ¹ã®å Žåã®æ°ã«çããããšãããã. äžæ¹ã§, ãã®æäœ $5$ åã¯åããŒã«ã« $1,2,3,4,5$ ã®ãããããå²ãæ¯ã... | ã$a_1+a_2+a_3+a_4+a_5=5$ ãªãéè² æŽæ°ã®çµ $(a_1,a_2,a_3,a_4,a_5)$ ãã¹ãŠã«ã€ããŠ, 以äžã®å€
$$\_{5}\mathrm{C}\_{a_1}\times{}\_{5-a_1}\mathrm{C}\_{a_2}\times{}\_{5-a_1-a_2}\mathrm{C}\_{a_3}\times{}\_{5-a_1-a_2-a_3}\mathrm{C}\_{a_4}\times{}\_{5-a_1-a_2-a_3-a_4}\mathrm{C}\_{a_5}$$
ã足ãåããããã®ã解çããŠãã ããïŒ |
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/1488 | C | OMC059(C) | 300 | 104 | 165 | [
{
"content": "ã$10^a$ ã®æ£ã®çŽæ°ã§, ã〠$10^b$ ã®åæ°ã§ãããã®ã®åæ°ã¯, $10^{a-b}$ ã®æ£ã®çŽæ°ã®åæ°ã«çãããã $(a-b+1)^2$ ã§ãã. äžæ¹ã§ $10^a$ ã®æ£ã®çŽæ°ã¯ $(a+1)^2$ åã§ãããã, æ¡ä»¶ã¯\r\n$$10^{100}=(a+1)^2-(a-b+1)^2=(2a-b+2)b$$\r\nãã㧠$2a-b+2$ ãš $b$ ã®å¶å¥ã¯äžèŽãããã, ãããã¯ãšãã«å¶æ°ã§ãã. ããã« $a\\geq b$ ãã\r\n$$2a-b+2\\gt b$$\r\nç¹ã«æ±ããã¹ããã®ã¯ $AB=2^{98}\\times5^{100}$ ãªãæ£æŽæ°... | ã次ã®æ¡ä»¶ãã¿ããæ£æŽæ°ã®çµ $a\geq b$ ã®åæ°ã¯ããã€ã§ããïŒ
- æ¡ä»¶ïŒ$10^a$ ã®æ£ã®çŽæ°ã§ãã£ãŠ, $10^b$ ã®åæ°ã§ãªããã®ã¯ã¡ããã© $10^{100}$ åããïŒ |
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/250 | D | OMC059(D) | 400 | 52 | 99 | [
{
"content": "ã$\\Gamma$ ã®é¢ç©ã $S$ ãšãã. $AB,CD$ ã®äžç¹ã $M,N$ ãšããã°, $AP^2+BP^2$ ã®ãšãåŸãå€ã®ç¯å²ã¯\r\n$$\\dfrac{2S}{\\pi}=AB^2\\/2\\leq2(AM^2+MP^2)=AP^2+BP^2\\leq AB^2=\\dfrac{4S}{\\pi}$$\r\nãããã£ãŠ,\r\n$$NQ^2=\\dfrac{1}{2}(CQ^2+DQ^2)-CN^2=\\dfrac{1}{2}(AP^2+BP^2)-36$$\r\nãã, $Q$ ã®ééãåŸãé åã¯ååŸ $\\sqrt{2S\\/\\pi-36}$ ã®åç€ããååŸ $... | ãå¹³é¢äžã«å®ç¹ $A,B,C,D$ ããã³ $AB$ ãçŽåŸãšããåç€ $\Gamma$ ããã, $CD$ ã®é·ã㯠$12$ ã§ã. ç¹ $P$ ã $\Gamma$ å
ãåããšã, 以äžã®æ¡ä»¶
$$AP=CQ,\ \ BP=DQ$$
ãã¿ããç¹ $Q$ ãééãåŸãé åã®é¢ç©ã¯ $30$ ã§ãã. ãã®ãšã, $\Gamma$ ã®é¢ç©ãæ±ããŠãã ãã.\
ããã ã, çãã¯æŽæ° $a,b$ ãçšã㊠$a+b\pi$ ãšè¡šããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/264 | E | OMC059(E) | 500 | 15 | 45 | [
{
"content": "ã$a_i=x_{i+1}\\/x_{i}$ ãšããã°, æ¡ä»¶ã¯ $a_0\\lt a_1\\lt\\cdots\\lt a_6$ ã§ãã, æ±ãããã®ã¯\r\n$$\\displaystyle\\frac{a_0a_1+1}{a_1-a_0}+\\frac{a_1a_2+1}{a_2-a_1}+...+\\frac{a_{5}a_{6}+1}{a_6-a_{5}}\\gt c$$\r\nãåžžã«ã¿ããå®æ° $c$ ã®æ倧å€ã§ãã. äžè¬ã« $6$ ã $n$ ãšãã, æ±ããæ倧å€ã $A_n$ ãšãã.\\\r\n ãå $i=0,1,...,n$ ã«å¯Ÿã, $a_i=\\tan \\t... | ã$8$ åã®æ£ã®å®æ° $x_0,x_1,\ldots,x_7$ ã¯, $n=1,\ldots,6$ ã«å¯Ÿã $x_{n-1}x_{n+1}\gt x_{n}^2$ ãã¿ãããŸã. ãã®ãšã, 以äžã®äžçåŒ
$$\frac{x_1(x_0+x_2)}{x_0x_2-x_1^2}+\frac{x_2(x_1+x_3)}{x_1x_3-x_2^2}+\cdots+\frac{x_6(x_5+x_7)}{x_5x_7-x_6^2}\gt c$$
ãåžžã«ã¿ããå®æ° $c$ ã®æ倧å€ãæ±ããŠãã ãã.\
ããã ã, æ±ããå€ã¯æ£æŽæ° $s,t$ ã«ãã£ãŠ $s+\sqrt{t}$ ãšè¡šããã®ã§, $s+t$ ã解çããŠãã ãã. |
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