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 |
|---|---|---|---|---|---|---|---|---|---|
OMC059 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc059/tasks/1467 | F | OMC059(F) | 600 | 5 | 22 | [
{
"content": "ã$\\ell$ ã®äž¡ç«¯ã $A,B$ ãšã, $\\omega_1,\\omega_2$ ãš $\\ell$ ã®æ¥ç¹ããããã $U_1,U_2$ ãšãã.\\\r\nããã®ãšã, well-known factãšã㊠$S_1U_1$ ããã³ $S_2U_2$ ã®äº€ç¹ $M$ ã¯åŒ§ $AB$ ã®äžç¹ã«ããã, ããã« \r\n$$MS_1\\times MU_1=MA^2=MS_2\\times MU_2$$\r\nãã $4$ ç¹ $S_1,S_2,U_1,U_2$ ã¯å
±åã§ãã. ãã®åãš $\\omega_1,\\omega_2$ ã®æ ¹å¿ã¯ $M$ ã§ããããšãã, ç¹ $T$... | ãç¹ $O$ ãäžå¿ãšããååŸ $r$ ã®å $\Gamma$ ããã³ãã®åŒŠ $\ell$ ããããŸã. äºãã« $T$ ã§å€æ¥ããå $\omega_1$ ããã³ $\omega_2$ ã¯, äžå¿ããšãã« $\Gamma$ ã®å
éšã® $\ell$ ã«é¢ããŠåãåŽã«ãã, ã©ã¡ãã $\Gamma$ ããã³ $\ell$ ã®äž¡æ¹ãšæ¥ããŠããŸã. ãã®ãšã, $\omega_1,\omega_2$ ãš $\Gamma$ ã®æ¥ç¹ããããã $S_1,S_2$ ãšãããš, $\angle S_1OS_2=120^\circ$ ãæãç«ã¡ãŸãã.\
ãããã« $T$ ãš $\ell$ ã®è·é¢ $d$ ã«ã€ããŠ, $OT:d=7:10$ ã... |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/2245 | A | OMC058(A) | 100 | 156 | 173 | [
{
"content": "ãã©ã® $2$ æ¬ã端ç¹ãå
±æããªãããšãã, åŒãã察è§ç·ã¯é«ã
$[2021\\/2]=1010$ æ¬ã§ãã, ããã« $1010$ æ¬ãåŒãããšãããšèŸºãå«ãã§ããŸãããšã容æã«ããã. éã«, åé ç¹ãé ã« $A_1,A_2,\\cdots,A_{2021}$ ãšãããš,\r\n$$A_2A_{2021} ,\\quad A_3A_{2020} ,\\quad\\cdots, \\quad A_{1010}A_{1013}$$\r\nãšã㊠$1009$ æ¬ã®å¯Ÿè§ç·ãåŒãããšãã§ãããã, æ±ããæ倧å€ã¯ $\\textbf{1009}$ ã§ãã.",
"text": "å
¬... | ãæ£ $2021$ è§åœ¢ã«ãããŠ, ãã®å¯Ÿè§ç·ã $1$ æ¬ãã€åŒããŠãããŸã. ãã®ãšã, æ°ããåŒã察è§ç·ã¯, ãããŸã§ã«åŒããã©ã®å¯Ÿè§ç·ãšãå
±æç¹ïŒ**端ç¹ãå«ã**ïŒããã£ãŠã¯ãããŸãã. åŒãããšã®ã§ãã察è§ç·ã¯æ倧ã§äœæ¬ã§ããïŒ |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/1999 | B | OMC058(B) | 200 | 101 | 139 | [
{
"content": "$$S_k=\\sum_{i=4}^{k}\\dfrac{1}{2}i(i-3)=\\sum_{i=1}^{k-3}\\dfrac{1}{2}i(i+3)=\\dfrac{1}{6}(k-3)(k-2)(k+2)$$\r\n$2021=43\\times 47$ ãã $k=\\textbf{45}$ 㧠$S_k$ 㯠$2021$ ã®åæ°ãšãªã, æå°æ§ã容æã«åŸã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc058/editorial/1999"
}
] | ã$k\geq 4$ ãªãæŽæ° $k$ ã«å¯Ÿã, åž $k$ è§åœ¢ã®å¯Ÿè§ç·ã®æ¬æ°ã $a_k$ ãšã, ããã« $S_k$ ã以äžã§å®ããŸãïŒ
$$S_k=a_4+a_5+\cdots+a_k$$
ãã®ãšã, $S_k$ ã $2021$ ã®åæ°ãšãªãæå°ã® $k$ ãæ±ããŠãã ãã. |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/2230 | C | OMC058(C) | 300 | 66 | 84 | [
{
"content": "ãæ¹çšåŒã $2021$ åã®è² ã®æŽæ°è§£ãæã€ãã, å®æ°é
ã®çŽ å æ°å解ãèããããšã§ $f$ ã¯\r\n$$(x+1)^{2020}(x+2021),\\quad (x+1)^{2019}(x+43)(x+47)$$\r\nã®ããããã§ãã. ããããã«ã€ã㊠$S=f(1)-2022$ ãåèšãããš, \r\n$$T=(2^{2020}Ã2022-2022)+(2^{2019}Ã44Ã48-2022)=2^{2021}Ã3^4Ã19-4044$$\r\nãããã£ãŠ, 解çãã¹ãå€ã¯ $2+2021+3+4+19=\\bf2049$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
... | ãå®æ°ãä¿æ°ãšãã $x$ ã® $2021$ 次æ¹çšåŒ
$$x^{2021}+a_1x^{2020}+a_2x^{2019}++a_{2020}x+2021=0$$
ã, éè€ã蟌ã㊠$2021$ åã®è² ã®æŽæ°è§£ããã€ãããªçµ $(a_1, \cdots, a_{2020})$ ãã¹ãŠã«ã€ããŠ, $S=a_1+\cdots+a_{2020}$ ã®ç·åã $T$ ãšããŸã. ãã®ãšã, $T+4044$ 㯠$a,c,e$ ãçžç°ãªãçŽ æ°ãšããŠçŽ å æ°å解ã®åœ¢ã§
$$T+4044=a^b\times c^d\times e$$
ãšè¡šããã®ã§, $a+b+c+d+e$ ã®å€ã解çããŠãã ãã. |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/1605 | D | OMC058(D) | 300 | 55 | 70 | [
{
"content": "ãçåŒ $\\dbinom{k+1}{n+1}=\\dbinom{k}{n+1}+\\dbinom{k}{n}$ ãç¹°ãè¿ãçšããããšã«ãã,\r\n$$\\binom{k+1}{n+1}=\\binom{n+1}{n+1}+\\sum_{i=n+1}^{k}\\binom{i}{n}=\\sum_{i=n}^{k}\\binom{i}{n}$$\r\nãæãç«ã€ãã, æ±ãã $n$ ã«ã€ããŠä»¥äžãæç«ãã.\r\n$$\\binom{2021}{n}=\\binom{2021}{n+1}$$\r\nããªãã¡ $n+(n+1)=2021$ ã§ãããã, $n=\\textbf{1010}... | ã以äžã®çåŒãã¿ãããããª, $2020$ 以äžã®æ£æŽæ° $n$ ã®ç·åãæ±ããŠãã ãã.
$${}\_{2021}\mathrm{C}\_{n}=\sum_{i=n}^{2020}{}\_{i}\mathrm{C}\_{n}$$ |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/1777 | E | OMC058(E) | 300 | 47 | 63 | [
{
"content": "ãç¹ $P$ ããå¹³é¢ $BCDE,ABFD$ ã«ããããåç·ã®è¶³ããããã $G,H$ ãšãããš, British flag theoremãã (詳现ã¯[**OMC030(B)ã®è§£èª¬**](https:\\/\\/onlinemathcontest.com\\/contests\\/omc030\\/editorial\\/1340)ãåç
§ãã) 以äžãæç«ããïŒ\r\n$$BG^2+DG^2=CG^2+EG^2,\\quad AH^2+FH^2=BH^2+DH^2$$\r\nãã®ãšã, 第äžåŒã®äž¡èŸºã« $2PG^2$ ã, 第äºåŒã®äž¡èŸºã« $2PH^2$ ãå ããããšã§\r\n$$C... | ãæ£å
«é¢äœ $A-BCDE-F$ ããã³ãã®å€éšã«ããç¹ $P$ ã
$$BP=24,\quad CP=27,\quad DP=32$$
ãã¿ãããŠãããšã, $AP^2+EP^2+FP^2 $ã®å€ãæ±ããŠãã ãã. |
OMC058 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc058/tasks/1656 | F | OMC058(F) | 400 | 15 | 42 | [
{
"content": "ã芳芧è»ãæèšåãã§ãããšã, äžåšã®é·ãã $75$ ãšãã. ãŸãè¡å $A,B$ ããããã以äžã®ããã«å®çŸ©ããïŒ\r\n\r\n- $A$ïŒ$9$ æã®æ¹åããäžæã, $12$ æã®æ¹åã«éããããå
ã«èœäžããŠä¹ã移ã, åã³ $9$ æã®æ¹åã«éãã.\r\n- $B$ïŒ$9$ æã®æ¹åããäžæã㊠$12$ æã®æ¹åã«éããŠä»¥éã«èœäžããŠä¹ã移ã, $6$ æã®æ¹åã«éãã.\r\n\r\nè¡å $B$ ã¯é«ã
äžåã§ãã. ãã®ãšã $i\\ (0\\leq i\\leq 37)$ æ©å
ã®ãŽã³ãã©ã«ä¹ã移ã£ããšãã. ãã㧠$i=0$ ããã£ãŠè¡å $B$ ãè¡ããªãã£ãå Ž... | ã倧ããã®ç¡èŠã§ãã $75$ åºã®ãŽã³ãã©ãååšäžã«çééã«èšçœ®ãã, äžå®ã®é床ã§å転ãã芳芧è»ããããŸã. æäžç¹ããä¹ã蟌ã¿, $25$ åãããŠäžåšãããå¿
ãéããŸã. ããã, OMCåã¯ãã®èŠ³èŠ§è»ã«åºæ¥ãã ãé·ãä¹ã£ãŠãããã®ã§, 以äžã®è¡ãäœåŸããŸããïŒ
- ä¹ã£ãŠãããŽã³ãã©ã®çäžã«å¥ã®ãŽã³ãã©ããããšã, ãã®ãŽã³ãã©ãŸã§èœäžããŠä¹ã移ãããšãã§ãã. ãã ã, èœäžæéã¯ç¡èŠã§ãããã®ãšãã.
ãOMCåããŽã³ãã©ã«ä¹ã蟌ãã§ãã $2021$ ååŸã«åããŠåã³æäžç¹ã«ãããããªç§»åæ¹æ³ã®ç·æ°ã¯, éè² æŽæ° $a$ ã«ãã£ãŠ $2F_a$ ãšè¡šãããã®ã§, $a$ ã®å€ã解çããŠãã ãã. ããã§,... |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/1883 | A | OMC057(A) | 200 | 187 | 192 | [
{
"content": "ãäžåœå°äœå®çãã, $3$ ã€ã®äœãã®çµã¯ $(0,0,0)$ ãé€ããŠã¡ããã©äžã€ãã€çŸãããã, æ±ããç·åã¯\r\n\r\n$$\\sum_{i=0}^{6}\\sum_{j=0}^{10}\\sum_{k=0}^{12}ijk=\\sum_{i=0}^{6}i\\sum_{j=0}^{10}j\\sum_{k=0}^{12}k=\\dfrac{6\\times 7}{2}\\times\\dfrac{10\\times 11}{2}\\times\\dfrac{12\\times 13}{2}=\\textbf{90090}.$$",
"text": "å
¬åŒè§£èª¬",
... | ãæ£æŽæ° $n,m$ ã«å¯Ÿã, $n$ ã $m$ ã§å²ã£ãäœãã $n\ \mathrm{mod}\ m$ ã§è¡šããšã, 以äžã®ç·åãæ±ããŠãã ããïŒ
$$\sum_{n=1}^{1000}(n\ \mathrm{mod}\ 7)(n\ \mathrm{mod}\ 11)(n\ \mathrm{mod}\ 13)$$ |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/1831 | B | OMC057(B) | 300 | 111 | 156 | [
{
"content": "ãå
±ééšåã¯æ£å
è§éãäžäžã« $2$ ã€è²Œãåãããç«äœã§ãã. ããã¯å
ã®ç«æ¹äœããäžè§éã $6$ ã€åãé€ãããšè§£éã§ã, ãã®äœç©ã¯ $1-6\\times\\left(\\dfrac{1}{6}\\times\\dfrac{1}{2}\\times\\dfrac{1}{2}\\right)=\\dfrac{3}{4}$ ã§ãããã, ç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{7}$ ã§ãã.\r\n",
"text": "å
¬åŒè§£èª¬",
... | ãäžèŸºã®é·ãã $1$ ã®ç«æ¹äœã«ã€ããŠ, ããäœå¯Ÿè§ç·ã軞㫠$60^\circ$ å転ãããŠåŸãããç«æ¹äœãšã®å
±ééšåã®äœç©ãæ±ããŠãã ãã. ãã ã, æ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $x,y$ ã«ãã£ãŠ $\dfrac{x}{y}$ ãšè¡šããã®ã§, $x+y$ ã解çããŠãã ãã. \
ããªã, ç«æ¹äœã«ãããŠ**äœå¯Ÿè§ç·**ãšã¯, åãé¢äžã«ãªã $2$ é ç¹ãçµãã§åŸãããç·åã®ããšãæããŸã. |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/1314 | C | OMC057(C) | 500 | 64 | 136 | [
{
"content": "ã$f(n)$ ã¯ã$n$ ãäºé²æ°ã§è¡šèšãããšãã®åæ¡ã®ç·åãã§ããããšã容æã«ããã. ãã㧠$f(n)=k$ ãã€äºé²æ°ã§è¡šèšã㊠$i$ æ¡ç®ã $1$ ã§ãããã㪠$n$ ã¯, $i$ ã«ããã\r\n$$\\binom{N-1}{k-1}= \\frac{k}{N}\\binom{N}{k}$$\r\nåååšãããã,\r\n$$\\sum_{f(n)=k,n\\lt 2^N}nf(n)=\\sum_{i=1}^{N}k2^{i-1}\\binom{N}{k}\\frac{k}{N}=\\frac{k^2}{N}\\binom{N}{k}(2^N-1)$$\r\n----\... | ãéè² æŽæ°ã«å¯ŸããŠå®çŸ©ãããé¢æ° $f$ ã¯, $f(0)=0$ ããã³æ£æŽæ° $n$ ã«å¯ŸããŠ
$$f(n)=\begin{cases} f(n\/2) & (n\ \text{ãå¶æ°ã®ãšã}) \\\\ f((n-1)\/2)+1 & (n\ \text{ãå¥æ°ã®ãšã}) \end{cases}$$
ãã¿ãããŸã. ãã®ãšã, $N=2^{20}-1(=1048575)$ ã«å¯Ÿã以äžã®ç·å
$$M=\sum_{n=0}^{2^N-1}nf(n)$$
ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ãã. |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/2040 | D | OMC057(D) | 500 | 155 | 166 | [
{
"content": "ãäžè§åœ¢ã®å蟺ã $x+y,y+z,z+x$ ãšè¡šã,é¢ç©ã $S$, å€æ¥åã»å
æ¥åã®ååŸããããã $R,r$ ãšããã°,\r\n$$S=\\sqrt{xyz(x+y+z)},\\quad r=\\dfrac{S}{x+y+z}=\\sqrt{\\dfrac{xyz}{x+y+z}}$$\r\nãããã $Sr=xyz=1001$ ã§ãã, ããã« $x+y+z=\\dfrac{S^2}{xyz}=31$ ã§ãã.ãããã£ãŠ,\r\n$$(x+y+z)(xy+yz+zx)-xyz=(x+y)(y+z)(z+x)=4RS=8640$$\r\nãã $xy+yz+zx=311$ ãåŸããã... | ãé¢ç©ã $\sqrt{31031}$, å€æ¥åã®ååŸã $\dfrac{2160}{\sqrt{31031}}$, å
æ¥åã®ååŸã $\sqrt{\dfrac{1001}{31}}$ ã§ãããããªäžè§åœ¢ã«ãããŠ, $2$ çªç®ã«é·ã蟺ã®é·ããæ±ããŠãã ãã. ãã ã, æ±ããäžè§åœ¢ã«ãããŠãã¹ãŠã®èŸºã®é·ããç°ãªãããšãä¿èšŒãããŸã. |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/1709 | E | OMC057(E) | 600 | 62 | 99 | [
{
"content": "ã$\\displaystyle S_n=\\sum_{k=0}^{F_n-1}f(k)$ ãšå®ããã°, ããã¯æŒžååŒ $S_{n}=S_{n-1}+S_{n-2}+F_{n-2}$ ãã¿ãããã, ããã解ã㊠\r\n$$S_n=\\dfrac{1}{5}\\left((n-2)F_{n}+nF_{n-2}\\right)\\quad (n=2,3,\\cdots)$$\r\nãã㧠$2021=F_{17}+F_{14}+F_{9}+F_{7}$ ã§ãããã, 以äžã®ããã«åºéãåå²ããŠèšç®ããã°ããïŒ\r\n- $0\\leq k\\lt F_{17}$ ã«ã€ããŠ, $S_{17}... | ãæ°å $\\{F_{n}\\}$ ã¯ä»¥äžã®æ¡ä»¶ãã¿ãããŸãïŒ
$$F_0=0,\quad F_1=1,\quad F_{n}=F_{n-1}+F_{n-2}\ (n=2,3,\cdots)$$
éè² æŽæ° $k$ ã«å¯ŸããŠé¢æ° $f(k)$ ã $f(0)=0$ ããã³ä»¥äžã®èŠåã§å®ãããšã, $\displaystyle \sum_{k=0}^{2020}f(k)$ ãæ±ããŠãã ãã.
- $k\geq 1$ ã«ã€ããŠ, $F_n\leq k \lt F_{n+1}$ ã§ãããããªå¯äžã®éè² æŽæ°ã $n$ ãšããŠ, $f(k)=f(k-F_n)+1$. |
OMC057 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc057/tasks/2193 | F | OMC057(F) | 700 | 10 | 32 | [
{
"content": "ã$N=1$ ã®å¯äžã¯æããã§ãããã, ä»¥äž $N\\geq 2$ ã§ãããšãã. ãŸãåºå®ããã $\\sigma\\in S_N$ ã«ã€ã㊠$\\prod$ ã®äžèº«ãèãã. $3$ ãåºãšãã察æ°ããšãã°, çµå± $\\textrm{inv}(\\tau)$ ã®ç·åãèããã°ãã, ããã¯åçµ $1\\leq i\\lt j\\leq N$ ã®å¯äžãèããããšã§èšç®ã§ãã. å
·äœçã«ã¯, ãããããšãã«å«ã $\\tau$ 㯠$2^{N-2}$ åãã, $\\sigma(i)\\gt\\sigma(j)$ ã§ãããšãã®ã¿åå®ããããã, \r\n$$\\sum_{\\tau\... | ã以äžã§å®ãŸã $X$ ã«ã€ããŠ, ã$2$ ã§å²ãåããæ倧åæ°ããçŽ æ° $2017$ ã§å²ã£ãäœããæ±ããŠãã ãã.
$$X=\prod_{N=1}^{2^{2021}-1}\sum_{\sigma \in S_N}\prod_{\tau\subseteq \sigma}3^{\textrm{inv}(\tau)}$$
ãã ã, ããããã®èšå·ã®å®çŸ©ã¯ä»¥äžã®éãã§ã.
- æ£ã®æŽæ° $N$ ã«å¯Ÿã, $S_N$ 㯠$1$ ãã $N$ ãŸã§ãã¡ããã©äžã€ãã€çŸããæ°å $N!$ åå
šäœã®éåã§ãã.
- $\tau\subseteq \sigma$ ã®ç·ç©ãšã¯, $\tau$ ã $\sigma$ ã®**é£ç¶ã... |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/230 | A | OMC056(A) | 100 | 205 | 206 | [
{
"content": "ã$P,Q$ ã® $x$ 座æšã $a$ ãšããã°, ç·å $PQ$ ã®é·ãã¯\r\n$$|(a^2+100)-(6a-700)|=|(a-3)^2+791|$$\r\nãã£ãŠ, æ±ããæå°å€ã¯ $\\textbf{791}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc056/editorial/230"
}
] | ã$xy$ å¹³é¢å
ã®ã°ã©ã $y=x^2+100$ äžã«ç¹ $P$ ã, ã°ã©ã $y=6x-700$ äžã«ç¹ $Q$ ããã, çŽç· $PQ$ ã $y$ 軞ã«å¹³è¡ãªãšã, ç·å $PQ$ ã®é·ããšããŠããåŸãæå°å€ãæ±ããŠãã ãã. |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/291 | B | OMC056(B) | 300 | 185 | 199 | [
{
"content": "ãæ¡ä»¶ã¯ $27a+3c=9b+d$ ãšåå€ã§ãã. ãã®ãšã $d=3d^\\prime\\ (d^\\prime=1,2)$ ãšãããŠ, $9a+c=3b+d^\\prime$ ã§, ç¹ã« $c=d^\\prime$ ãŸã㯠$c=d^\\prime+3$ ã§ãã. $c=d^\\prime$ ã®ãšã $(a,b)=(1,3),(2,6)$, $c=d^\\prime+3$ ã®ãšã $(a,b)=(1,4)$ ãšå®ãŸããã, æ±ãã確ç㯠$6\\/6^4=1\\/216$ ã§ãã, 解çãã¹ãå€ã¯ $\\textbf{217}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
... | ã$1$ ãã $6$ ã®ç®ãç確çã§åºããµã€ã³ãã $4$ åæ¯ã, ãã®åºç®ãé ã« $a,b,c,d$ ãšãããšã, å€é
åŒ $ax^3+bx^2+cx+d$ ã $x+3$ ã§å²ãåãã確çãæ±ããŠãã ãã. ãã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ã«ãã£ãŠ $\dfrac{p}{q}$ ãšè¡šããã®ã§, $p+q$ ã解çããŠãã ãã. |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/1239 | C | OMC056(C) | 300 | 123 | 171 | [
{
"content": "ãããããã®å¯Ÿè§ç·ã«ã€ããŠ, $101$ åã®æ°ã®ç·åã¯æäœã«ããã $515201$ ã§äžå®ã§ããããšã容æã«ããã. ãããã£ãŠ, $2$ æ¬ã®å¯Ÿè§ç·ãéè€ããäžå€®ã®ãã¹ã $10201$ ãšãªãå Žåãæå°ã§, ãããã¿ããæäœã¯æããã«ååšãããã, ãã®å€ã¯ $2\\times 515201-10201=\\textbf{1020201}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc056/editorial/1239"
}
] | ã$101$ è¡ $101$ åã®ãã¹ç®ã«, 次ã®ããã« $1$ ãã $10201$ ãŸã§ã®æŽæ°ãäžã€ãã€æžã蟌ã¿ãŸãïŒ
- $i$ è¡ç® $j$ åç®ã®ãã¹ã«ã¯ $101(i-1)+j$ ãæžã蟌ã.
äŸãã°, $1$ è¡ç®ã«ã¯ $1,2,\cdots,100,101$ ãå·Šããé ã«æžã蟌ãŸããŸã.\
ãããã«, 以äžã®äºçš®é¡ã®æäœã, ä»»æã®é åºã§ä»»æã®åæ° ($0$ åã§ããã) è¡ããŸã.
- ä»»æã«è¡ãäºã€éžã³, ããããè¡ããšãã¹ãŠå
¥ãæ¿ãã.
- ä»»æã«åãäºã€éžã³, ããããåããšãã¹ãŠå
¥ãæ¿ãã.
ãã®ãšã, æçµçã«åºæ¥äžãã£ããã¹ç®ã«ã€ããŠ, ãã®å¯Ÿè§ç·äžã®æ°ã®ç·åãšããŠèããã... |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/304 | D | OMC056(D) | 400 | 39 | 64 | [
{
"content": "ã$G,H$ ã¯ãããã $AD,ED$ ã $2:1$ ã«å
åããç¹ã§ãããã, äžè§åœ¢ $FGH$ ã®é¢ç©ã¯ $DGH$ ã®é¢ç©ã«çãã, ããã«ãã㯠$ADE$ ã®é¢ç©ã® $1\\/9$ ã«ããã. ãããã£ãŠ, äžè§åœ¢ $ADE$ ã®é¢ç©ãæ倧åããã°ãã, $AC$ ãçŽåŸãšããåãš $AD$ ã®äº€ç¹ã $D^\\prime$ ãšããã°, ãã®æ倧å€ã¯ $E$ ãåªåŒ§ $AD^\\prime$ ã®äžç¹ $M$ ã«äžèŽããå Žåã«éæããã.\\\r\nãããã§äžç·å®çãã $AD=2\\sqrt{7}$, äžå¹³æ¹ã®å®çãã $DD^\\prime=3\\/\\sqrt{7}$ ã§... | ã$AB=5,BC=6,CA=7$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ ã®äžç¹ã $D$ ãšããŸã. ããã« $AC$ ãçŽåŸãšããååšäžã®ç¹ $E$ ã«ã€ããŠ, $2\overrightarrow{AE}=\overrightarrow{DF}$ ãªãç¹ $F$ ããšã, äžè§åœ¢ $ABC,EBC$ ã®éå¿ããããã $G,H$ ãšãããŸã. ãã®ãšã, äžè§åœ¢ $FGH$ ã®é¢ç©ãšããŠããåŸãæ倧å€ã¯, æ£æŽæ° $a,b,c,d,e$ ã«ãã£ãŠ $\dfrac{a\sqrt{b}+c\sqrt{d}}{e}$ ãšè¡šããŸã. ãã ã, $b,d$ 㯠$1$ ãã倧ããå¹³æ¹æ°ã§å²ãåãã, $a,c,e$ ã¯äºãã«çŽ ... |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/268 | E | OMC056(E) | 500 | 95 | 167 | [
{
"content": "ã$x$ ã®äºæ¬¡æ¹çšåŒ $x^2-ax+b=0,x^2-bx+a=0$ ã®å€å¥åŒããããã $D_1,D_2$ ãšãã. ããªãã¡\r\n$$ D_1=a^2-4b,\\ \\ D_2=b^2-4a $$\r\næ¡ä»¶ãã, ãã®ãã¡å°ãªããšãäžæ¹ã¯éè² ã§ãã. ããã§äžæ¹ãæ£ã§äžæ¹ãè² ã®ãšã, ããªãã¡ä»¥äžã®ãããããæãç«ã€ãšã, æ¡ä»¶ã¯åžžã«æç«ãã.\r\n\r\n- $a^2\\gt 4b$ ã〠$b^2\\lt 4a$\r\n- $a^2\\lt 4b$ ã〠$b^2\\gt 4a$\r\n\r\nããããã£ãŠ, ä»¥äž $D_1,D_2$ ããšãã«éè² ã§ããå Žåã«ã€ããŠèãã... | ã$x$ ã®å次æ¹çšåŒ
$$(x^2 - ax + b)(x^2 - bx + a) = 0$$
ãçžç°ãªãå®æ°è§£ãã¡ããã© $2$ åãã€ãããª, $-6$ ä»¥äž $6$ 以äžã®**æŽæ°**ã®çµ $(a,b)$ ã¯ããã€ãããŸããïŒ |
OMC056 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc056/tasks/1735 | F | OMC056(F) | 500 | 37 | 56 | [
{
"content": "ã$4997-k$ ã $p$ ã§å²ã£ãåã $x$ ãšãã, $4997+k$ ã $p+2$ ã§å²ã£ãåãšã®å·®ã«ãã£ãŠå Žååããè¡ã.\r\n\r\n- å·®ã $0$ ã§ãããšã, $x=k$ ã§ããã, ãã®ãšã $4997=(p+1)k$ ããå¶å¥ãèããã°äžé©ã§ãã.\r\n- å·®ã $1$ ã§ãããšã, $px$ ããã³ $(p+2)(x\\pm 1)$ ã®å¶å¥ãäžèŽããããšãããã¯ãäžé©ã§ãã.\r\n\r\nã以äž, å·®ã $2$ ã§ããå Žåã«ã€ããŠèãã. ãŸã\r\n$$4997-k=px,\\quad 4997+k=(p+2)(x+2)$$\r\nãšæžãããšãã. ã... | ã$4997$ æªæºã®æ£æŽæ° $k$ ããã³çŽ æ° $p$ ã, 以äžã®æ¡ä»¶ãã¿ãããŸãïŒ
- $p+2$ ãçŽ æ°ã§ãã, $4997-k,4997+k$ ã¯ãããã $p,p+2$ ã§å²ãåãã.
ãã®ãããªçµã§ $\left\lvert \dfrac{4997-k}{p} - \dfrac{4997+k}{p+2} \right\rvert$ ãæå°å€ããšããã®ãã¹ãŠã«ã€ããŠ, $k+p$ ã®ç·åãæ±ããŠãã ãã.\
ããªã, [**ãã¡ã**](https:\/\/www.mathsisfun.com\/numbers\/prime-numbers-to-10k.html)ã®çŽ æ°è¡šãçšããŠãæ§ããŸãã. |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/300 | A | OMC055(A) | 100 | 209 | 211 | [
{
"content": "ã$3^{45}$ ã®äžã®äœãæ±ããã°ãã. ãã㧠$3^n$ ã®äžã®äœã¯ $3\\to 9\\to 7\\to 1$ ã®åšæãç¹°ãè¿ãããšã«çæããã°, æ±ããããŸã㯠$\\textbf{3}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc055/editorial/300"
}
] | ã$123^{45}$ ã $10$ ã§å²ã£ãäœãã¯ããã€ã§ããïŒ |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/199 | B | OMC055(B) | 200 | 154 | 197 | [
{
"content": "ã$ABCD$ ã«å¯Ÿãã $P,Q$ ã®äœçœ®é¢ä¿ã¯ $2$ éãããåŸãã, ãã®ãã¡ $BP$ ãš $CQ$ ã亀ããæ¹ãæå°å€ãå®çŸãã. äžèŸºã®é·ãã $k$ ãšã, $P$ ãã $BC$ ã«ããããåç·ã®è¶³ã $H$ ãšããã°, äžè§åœ¢ $BHP$ ã«ãããäžå¹³æ¹ã®å®çãã\r\n$$\\left(\\dfrac{k}{2}+1\\right)^2+\\left(\\dfrac{k}{2}\\right)^2=3^2$$\r\nããã解ããŠ$k\\gt 0$ ãã $k=\\sqrt{17}-1$ ãåŸããã, 解çãã¹ãå€ã¯ $\\textbf{16}$ ã§ãã.",
... | ãæ£æ¹åœ¢ $ABCD$ ã®å
éšã« $2$ ç¹ $P,Q$ ããã, 以äžã®æ¡ä»¶ãã¿ãããŸã.
$$AP=BP=CQ=DQ=3,\ \ PQ=2$$
ãã®ãšã, $ABCD$ ã®äžèŸºã®é·ããšããŠããåŸãæå°å€ãæ±ããŠäžãã.\
ããã ãçãã¯æŽæ° $p,q$ ã«ãã£ãŠ $p+\sqrt{q}$ ãšè¡šãããã®ã§, $p+q$ ã解çããŠãã ãã. |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/222 | C | OMC055(C) | 200 | 134 | 162 | [
{
"content": "ãåœé¡ã¯ã$1$ ä»¥äž $N^2$ æªæºã®æŽæ°ã®ç·å㯠$9N^3+9N^2$ ã§ããããšè¡šçŸã§ãããã, æ¹çšåŒ\r\n$$\\dfrac{1}{2}(N^2-1)N^2=9N^3+9N^2$$\r\nã解ã㊠$N=\\textbf{19}$ ãåŸã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc055/editorial/222"
}
] | ãã$1$ ä»¥äž $100$ **æªæº**ã®æŽæ°ã®ç·å㯠$9900$ ã§ããããšããåœé¡ã $N$ é²æ³è¡šèšã§è§£éãããšçã§ãããšã, $N$ ãšããŠããåŸã $10$ 以äžã®æŽæ°ã®ç·åãæ±ããŠãã ãã.\
ããã ã, èšæ°ã®å²ãåœãŠé åºã¯äžè¬çãªãã® ($0,1,\cdots,9,a,b,\cdots$) ã«åŸããŸã. |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/258 | D | OMC055(D) | 300 | 111 | 160 | [
{
"content": "ãäžè¬ã«æ£ã®é·ããå¥æ° $n$ ã§ããå Žåãèã, æ£ã®å·Šç«¯ã $O$, å³ç«¯ã $O^\\prime$ ãšãã. $OA=a,AB=b$ ãšããŠå·ŠåŽããç¯ç® $A,B$ ãéžæãããšã, ãããæ¡ä»¶ãã¿ããããšã¯ä»¥äžã®ããã«è¡šçŸã§ããïŒ\r\n$$a\\leq\\dfrac{n-1}{2},\\quad b\\leq\\dfrac{n-1}{2},\\quad a+b\\geq\\dfrac{n+1}{2}$$\r\nããã㧠$a$ ãåºå®ããã°, $b$ ãšããŠããåŸããã®ã¯ $\\dfrac{n+1}{2}-a$ ä»¥äž $\\dfrac{n-1}{2}$ ãŸã§ $a$ åã§ãããã... | ãé·ã $2021$ ã®æ£ããããŸã. ãã®æ£ã«ã¯ç«¯ããé·ã $1$ ã®ééã§ç¯ç®ãä»ããŠãã, ãããã§æ£ãèªç±ã«æãæ²ããããšãã§ããŸã. 次ã®æ¡ä»¶ãã¿ãããã㪠$2$ åã®ç¯ç®ã®éžã³æ¹ã¯ããã€ãããŸããïŒ
- æ¡ä»¶ïŒ$2$ åã®ç¯ç®ã§æ£ãæãæ²ãããšã, æ£ã®äž¡ç«¯ç¹ãåãããŠäžè§åœ¢ãäœãããšãã§ãã.
ããã ã, ç¯ç®ã¯ãã¹ãŠåºå¥ã§ã, ç¯ç®ã®éšåã®é·ãã¯ç¡èŠã§ããã»ã©çããã®ãšããŸã. |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/259 | E | OMC055(E) | 300 | 38 | 80 | [
{
"content": "ã$B$ ãéã $AC$ ã«å¹³è¡ãªçŽç· $\\ell$ ã«ã€ããŠ, $AP,AQ$ ãšã®äº€ç¹ããããã $D,E$ ãšã, $A$ ããããããåç·ã®è¶³ã $F$ ãšããã°, 以äžã®ããã«ããããã®é·ããèšç®ã§ããïŒ\r\n$$AF=BF=1,\\quad BD=DF-BF=\\sqrt{3}-1,\\quad DE=AD=2$$\r\nãããã, $BP:PC=(\\sqrt{3}-1):\\sqrt{3}$ ããã³ $BQ:QC=(\\sqrt{3}+1):\\sqrt{3}$ ã§ãããã\r\n$$BP:PQ:QC=(5-\\sqrt{3}):2\\sqrt{3}:(6-\\sqr... | ã$AB=\sqrt{2},AC=\sqrt{3},\angle A=45^\circ$ ãªãäžè§åœ¢ã«ãããŠ, 蟺 $BC$ äžã®ç¹ $P,Q$ã
$$\angle BAP = \angle PAQ = \angle QAC = 15^\circ $$
ãã¿ãããšã, äžè§åœ¢ $APQ$ ã®é¢ç©ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§, $a+b$ ãæ±ããŠãã ãã. |
OMC055 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc055/tasks/305 | F | OMC055(F) | 400 | 17 | 50 | [
{
"content": "ãåãã¹ã«å¯ŸããŠå¿
èŠãªæäœã®æå°åæ°ã¯, å·Šäžããé 次å®ãŸã, 以äžã®ãããªååž°çæ§é ã確èªã§ããïŒ\r\n$$\\begin{matrix}\r\n15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 & 15 \\\\\\\\\r\n15 & 14 & 14 & 15 & 15 & 14 & 14 & 15 & 15 & 14 & 14 & 15 & 15 & 14 & 14 & 15 \\\\\\\\\r\n15 & 14 & 13 & 13 & 13 & 13 & 14 & 15 & 15 ... | ãå³1ã«ç€ºã $3$ ãã¹ãããªãå³åœ¢ã**ã¿ã€ã«**ãšåŒã³ãŸã. ã㟠$2^{2021}\times2^{2021}$ ã®ãã¹ç®ããã, å·Šãã $i$ çªç®, äžãã $j$ çªç®ã®ãã¹ã $(i,j)$ ã§è¡šããŸã. Noyaåã¯, ãã®ã¿ã€ã«ãšãã¹ç®ãçšããŠæ¬¡ã®ãããªã²ãŒã ãè¡ããŸã.\
ããŸãNoyaåã¯æºåãšããŠ, ãã®ãã¹ç®ã«ã¿ã€ã«ãé
眮ããŸã. ããã§ã¿ã€ã«ã®é
眮ã¯, 以äžã§å®ãŸã $2$ èŠåã«ãã£ãŠååž°çã«å®çŸ©ãããŸã. ããããã®èŠåã¯éšåãã¹ç®ã«å¯ŸããŠé©çšãã, ãŸãåãã«ãã¹ç®å
šäœã«èŠå1ãé©çšããŸã.
- èŠå1ïŒæ£æ¹åœ¢ã®é åã«é©çšããã. é åã $1\times 1$ ã®ãšã, äœãããªã.... |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/1479 | A | OMC054(A) | 200 | 135 | 184 | [
{
"content": "ã$(a-2021)(b-2021)=2021^2$ ãšäžåŒãå€åœ¢ããã°, $2021^2$ ã®çŽæ° (è² ãèš±ã) ã®åæ°ãæ±ããããšãšåé¡ã¯ç䟡ã§ãã, $2021^2=43^2\\times47^2$ ãšçŽ å æ°å解ã§ãããããã㯠$2\\times 3^2=18$ åã§ãã. ãã ã $(a,b)=(0,0)$ ãé€å€ããããšã«çæããã°, æ±ããã¹ãå€ã¯ $\\textbf{17}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc054/editorial/1479"... | ã$\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{2021}$ ãã¿ããé åºä»ãã®**æŽæ°**ã®çµ $(a,b)$ ã¯ããã€ãããŸããïŒ ãã ã $2021=43\times 47$ ã§ã. |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/2070 | B | OMC054(B) | 200 | 140 | 161 | [
{
"content": "ãäžããããç«äœãäžèŸºã®é·ãã $4$ ã®æ£åé¢äœã®å
éšã«é©åã«åã蟌ãããšãèããã°, 貌ãåããã«ãã£ãŠé£ãåã£ãé¢ã¯åäžå¹³é¢äžã«ãã. ãã£ãŠé¡æã®äžè§åœ¢ã¯èŸºã®é·ãã $2\\sqrt{3}, 2\\sqrt{3}, 2$ ã®äºç蟺äžè§åœ¢ã§, ãã®é¢ç©ã¯ $\\sqrt{\\textbf{11}}$.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc054/editorial/2070"
}
] | ãäžèŸºã®é·ãã $2$ ã®æ£åé¢äœãš, ãã¹ãŠã®èŸºã®é·ãã $2$ ã®æ£åè§éããããŸã. ããããããé©åœã«æ£äžè§åœ¢ã®é¢ãéžã³, ãããã«æ²¿ã£ãŠ $2$ ç«äœãå€åŽã« (é¢ãé€ããŠå
±ééšåããããªãããã«) 貌ãåããããšã, 貌ãåããã«çšããªãã£ã $3$ é ç¹ãããªãäžè§åœ¢ã®é¢ç©ã® $2$ ä¹ãæ±ããŠãã ãã. |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/1912 | C | OMC054(C) | 300 | 105 | 153 | [
{
"content": "ã$X=2^x3^y5^z7^w$ ãšè¡šã, $y+z+w\\leq 3$ ã§ãã. 以äž, $y+z+w$ ã®å€ã«å¿ããŠå Žååããè¡ã.\r\n\r\n(i) $y+z+w=0$ ã®ãšã, $s,b,l$ 㯠$1,2,4,8$ ã®ããããã§ãã, $x$ 㯠$0$ ä»¥äž $9$ 以äžã®æŽæ°å€ããšãåŸã.\r\n\r\n(ii) $y+z+w=1$ ã®ãšã, åºæ¬çã«äžãšåæ§ã« $x$ 㯠$0$ ä»¥äž $6$ 以äžã®æŽæ°å€ããšãåŸãã, $y=1$ ã®å Žåã«éã $3$ ã $6$ ã«çœ®ãæããããšã§ $x=7$ ãšã§ãã. ããªãã¡, $7\\times 3+1=22$ éãã§ãã... | ãããã¹ããŒãçš®ç®ã¯ $S,B,L$ ã® $3$ éšåãããªã, åéžæã¯ããããã¹ãŠã«åå ããŸã. $S,B,L$ ã«ãããããéžæã®é äœããããã $s,b,l$ ãšãããšã, ãã®éžæã®ç²åŸãããã€ã³ã $X$ 㯠$X=sbl$ ãšå®çŸ©ãããŸã. ãã®çš®ç®ã« $8$ 人ã®éžæãåå ãããšã, ããéžæãç²åŸãããã€ã³ã $X$ ãšããŠããåŸãæ£æŽæ°å€ã¯ããã€ãããŸããïŒ\
ããã ã, åéšåã®é äœã¯ $1$ äœãã $8$ äœãŸã§ã®æ£æŽæ°å€ãéè€ãªãä»ããã®ãšããŸã. |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/2009 | D | OMC054(D) | 400 | 64 | 78 | [
{
"content": "ã$T$ ã®åé
ã«ã€ããŠ,\r\n$$ \\frac{n^3}{n^4+4} =\\frac{1}{2}\\left(\\frac{n-1}{(n-1)^2+1}+\\frac{n+1}{(n+1)^2+1}\\right) $$\r\nãšæŽçã§ãããã,\r\n$$\\begin{aligned}\r\nS-T &=\\frac{1}{2}\\left(\\frac{1}{1^2+1}+\\frac{99}{99^2+1}\\right)-\\frac{1}{2}\\left(\\frac{0}{0^2+1}+\\frac{100}{100^2+1}\\right) \\\\\\\\\r... | ã以äžã® $2$ çš®é¡ã®åæ°ã®å $S,T$ ã«ã€ããŠ, ãã®å·®ã®çµ¶å¯Ÿå€ãæ±ããŠãã ããïŒ
$$ S = \frac{1}{2}+\frac{2}{5}+\frac{3}{10}+\cdots+\frac{n}{n^2+1}+\cdots+\frac{99}{99^2+1} $$
$$ T=\frac{1}{5}+\frac{8}{20}+\frac{27}{85}+\cdots+\frac{n^3}{n^4+4}+\cdots+\frac{99^3}{99^4+4} $$
ãã ã, æ±ããå·®ã®çµ¶å¯Ÿå€ã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ãçšã㊠$\dfrac{p}{q}$ ãšè¡šãããã®ã§, $p+q$ ã解çããŠãã ãã. |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/1942 | E | OMC054(E) | 500 | 92 | 130 | [
{
"content": "ã$3^n$ ã®äžã®äœã¯ $n$ ã $4$ ã§å²ã£ãŠ $2$ äœããšãã« $9$ ãšãªããã, $3^{4m+2}$ ã®æé«äœã $9$ ã§ãããããªæå°ã® $m$ ãæ±ããã°ãã. ãã®ããšã¯, $3^{4m}$ ãš $3^{4m+2}$ ã®æ¡æ°ãçããããš, ããªãã¡ $4m\\log_{10}3$ ãš $(4m+2)\\log_{10}3$ ã®æŽæ°éšåãçããããšãšåå€ã§ãã. ãã®æŽæ°éšåã $i$ ãšãããš, æ¡ä»¶ã¯ããã«æ¬¡ã®ããã«æžãããšãã§ããïŒ\r\n$$ \\frac{i}{2m}\\lt 2\\log_{10}3 \\lt \\frac{i+1}{2m+1} $$\r... | ã$10$ 以äžã®æŽæ° $x$ ã«ã€ããŠ, $f(x)$ 㧠$x$ ã® (åé²æ°è¡šèšã§ã®) æé«äœã®æ°åãšäžã®äœã®æ°åã®ç©ãè¡šããŸã. äŸãã° $f(2021)=2$ ã§ã. $n$ ã $3$ 以äžã®æŽæ°ãšãããšã, $f(3^n)$ ãæ倧å€ããšããã㪠$n$ ã®ãã¡, æå°ã®ãã®ãæ±ããŠãã ãã.\
ããã ãå¿
èŠã§ããã°, $\log_{10}3=0.4771212547\cdots$ ãå©çšããŠãæ§ããŸãã. |
OMC054 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc054/tasks/2305 | F | OMC054(F) | 500 | 19 | 44 | [
{
"content": "ã$N=54$ ãšãã, ããã§ã¯ã«ãŒãããã±ããã®çªå·ã $0$ ããå§ããŠæ°ãããã®ãšãã.\\\r\nããããã®äžãã $n$ æç®ã®ã«ãŒãã, äžåºŠã® $d$-ã·ã£ããã«ã§äžãããããã®äžãã $g(n)$ æç®ã«ç§»åãããšãã. $n$ ã® $d$ ã«ããå²ãç®ã $n=jd+k$ ãšè¡šããš, ååã®æé ã§ãã±ãã $k$ ã®äžãã $j$ æç®ã«ç§»ããã, \r\n$$g(n)= N-k\\times\\dfrac{N}{d}-j-1\\equiv -\\dfrac{N}{d}\\times n \\pmod{N-1} $$\r\nãã㧠$0$ æç®ãš $N-1$ æç®ã¯äžå... | ãçžç°ãªãå
š $54$ æã®ã«ãŒããéãããã®ã**ããã**ãšåŒã³, ããããããã€ãã«åå²ãããã®ã**ãã±ãã**ãšåŒã³ãŸã. ãŸã, ãããã«å¯ŸããŠä»¥äžã§å®çŸ©ãããäžé£ã®æäœã **$d$-ã·ã£ããã«**ãšå®çŸ©ããŸã.
- æäœåã«ãããã«éããããã«ãŒãã**äžãã**é ã« $1$ æç®, $2$ æç®, ... , $54$ æç®ãšãã.
- ãããããã±ãã $1$ ãããã±ãã $d$ ã«åå²ãã. ãã±ãã $i$ 㯠$i$ æç®, $i+d$ æç®, ... ãããªã.
- åãã±ããå
ã®äžäžããã¹ãŠå
¥ãæ¿ã, ãã±ãã $i$ ã®äžã«ãã±ãã $i+1$ ãç©ãèŠé ã§, ããããåæ§æãã.
... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/292 | A | OMC053(A) | 200 | 147 | 177 | [
{
"content": "ããŸã以äžã®ããã«å€æ°å€æãè¡ãïŒ\r\n$$a=2x,\\ \\ b=3y,\\ \\ c=5z,\\ \\ d=7p,\\ \\ e=11q,\\ \\ f=13r$$ \r\nãã®ãšã, äžæ¹çšåŒã®ä¿æ°ã«ã¯Pascalã®äžè§åœ¢ãçŸããïŒ\r\n$$\\begin{aligned}\r\n|a|+|b|+|c|+|d|+|e|+|f|&=32 \\\\\\\\\r\n1a+1b+1c+1d+1e+1f&=0 \\\\\\\\\r\n5a+4b+3c+2d+1e&=0 \\\\\\\\\r\n10a+6b+3c+1d&=0 \\\\\\\\\r\n10a+4b+1c&=0 \\\\\\\... | ã以äžã®é£ç«æ¹çšåŒ
$$\begin{aligned}
|2x|+|3y|+|5z|+|7p|+|11q|+|13r|&=32 \\\\
2x+3y+5z+7p+11q+13r&=0 \\\\
10x+12y+15z+14p+11q&=0 \\\\
20x+18y+15z+7p&=0 \\\\
20x+12y+5z&=0 \\\\
10x+3y&=0
\end{aligned}$$
ã®å®æ°è§£ $(x,y,z,p,q,r)$ ãã¹ãŠã«ã€ããŠ, 以äžã®å€ã®ç·åãæ±ããŠãã ãã.
$$|x|+|y|+|z|+|p|+|q|+|r|$$
ããã ã, çãã¯äºãã«çŽ ãªæ£æŽæ° $u,v$ ã«ãã£ãŠ $\dfrac{u}{... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/1641 | B | OMC053(B) | 500 | 56 | 119 | [
{
"content": "ãä»¥äž $A_{i+1024}=A_i$ ãšãã. ãŸã, $\\displaystyle \\sum_{i=1}^{1024} f(A_i,A_{i+1})$ ã®æå°å€ãæ±ããã°ãã. ãªããšãªãã°, æ·»åãé©åœã«å·¡åãã㊠$A_1$ ãå
šäœéåãšããã°, æ±ããæå°å€ã¯ãããã $55$ ãæžãããã®ãšããŠåŸãããããã§ãã.\\\r\nãããŸ, $A_i$ ã«å«ãŸããã $A_j$ ã«ã¯å«ãŸããªãã«ãŒãã«æžãããæ°ã®åèšã $g(A_i,A_j)$ ãšããã°,\r\n$$\\displaystyle \\sum_{i=1}^{1024} f(A_i,A_{i+1})=55 \\ti... | ã$1$ ãã $10$ ã®æŽæ°ãã€ããã«ãŒãããããã $1$ æãã€, èš $10$ æãããŸã. ãããã®ã«ãŒãããäœæããéžãã éå (空ãèš±ã) 㯠$1024$ åãããŸãã, ããããé©åœã«äžŠã¹æ¿ã㊠$A_1,A_2,\cdots,A_{1024}$ ã§è¡šããŸã. $A_i$ ããã³ $A_j$ ã®å°ãªããšãäžæ¹ã«å«ãŸããŠããã«ãŒãã«æžãããæ°ã®åèšã $f(A_i,A_j)$ ãšãããŸã. äŸãã°
$$f(\lbrace 1,4,5 \rbrace,\lbrace 3,4,5,9 \rbrace)=1+3+4+5+9=22$$
ã§ã. ãã®ãšã, $\displaystyle \sum_{i=1}^{1023... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/1259 | C | OMC053(C) | 500 | 24 | 53 | [
{
"content": "ãæäœã¯ä»¥äžã®ããã«èšãæããŠãç䟡ã§ãã.\r\n\r\n- $a_i=a_{i+1}$ ãŸã㯠$b_i=b_{i+1}$ ãªã $i$ ãéžæã, $a_i$ ãš $a_{i+1}$ ããã³ $b_i$ ãš $b_{i+1}$ ã®å€ããããã亀æãã. \r\n\r\nããã®ãšã $(a_i,b_i)$ ã¯åžžã«é£åããŠåããã, ãã¢ãšããŠèããã°ãã. ã㢠$(0,0)$ ãš $(1,1)$ ããã³ã㢠$(0,1)$ ãš $(1,0)$ ã¯äº€æã§ããªãã, ãã以å€ã¯äº€æã§ããããšã«çæããã°, 以äžã®æ§è³ªãåŸã.\r\n\r\n- $a_i=b_i$ ãªããã¹ãŠã®ãã¢ã®çžå¯Ÿç... | ãããã§ã¯**ãã€ããªå**ã§åé
ã $0$ ãŸã㯠$1$ ã§ãããããªæéåãæããã®ãšããŸãïŒ\
ãé·ãã $5183(=71 \times 73)$ ã§ããäºã€ã®ãã€ããªå $a,b$ ãããïŒåãã¯ãããã以äžã®ããã«å®çŸ©ãããŸãïŒ
- $a$ ã¯ã'$0$' ãé£ç¶ã㊠$26$ åç¶ããããš '$1$' ãé£ç¶ã㊠$47$ åç¶ãåäœãã $71$ åç¹°ãè¿ããåã§ããïŒ
- $b$ ã¯ã'$0$' ãé£ç¶ã㊠$29$ åç¶ããããš '$1$' ãé£ç¶ã㊠$42$ åç¶ãåäœãã $73$ åç¹°ãè¿ããåã§ããïŒ
ãsiosioåã¯ãããã®ãã€ããªåã«ïŒä»¥äžã® $2$ æäœã奜ããªé åºã§ $0$ ... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/1419 | D | OMC053(D) | 700 | 16 | 42 | [
{
"content": "ãäžåŒãé©åœã«å€åœ¢ããããšã§\r\n$$(f(y)-(f^2(x)+1))^2+(x-z)((f(x)-x)-(f(z)-z))\\geq 1$$\r\nç¹ã«ããã« $z=x$ ã代å
¥ããã°\r\n$$(f(y)-(f^2(x)+1))^2\\geq 1$$\r\n$y=f(x)$ ã代å
¥ãããš,\r\n$$(x-z)((f(x)-x)-(f(z)-z))\\geq 0$$\r\néã«ããããæç«ããã°äžåŒãæç«ãããã, æ¡ä»¶ã¯ããã $2$ åŒã«åé¢ããã. ããã«,\r\n\r\n- $(f(y)-(f^2(x)+1))^2\\geq 1$ 㯠$f^2(x)+1 \\neq f(y... | ãæ£æŽæ°å
šäœã§å®çŸ©ãã, æ£æŽæ°å€ãåãé¢æ° $f$ ã¯, ä»»æã®æ£ã®æŽæ° $x,y,z$ ã«ã€ããŠ
$$\begin{aligned}
&f^2(x)^2+2f^2(x)+f(y)^2+xf(x)+zf(z) \\\\
\geq ~ &2f^2(x)f(y)+2f(y)+xf(z)+zf(x)+(x-z)^2
\end{aligned}$$
ãã¿ãããŸã. ãã®ãšã $f^{16}(2)$ ãšããŠããåŸãæå°å€ãæ±ããŠäžãã. ãã ã, æ£æŽæ° $n$ ã«å¯Ÿã $f^{n}$ 㯠$f$ ã® $n$ ååæã§ã. ããªãã¡ $f^1(x)=f(x),\\, f^{n}(x)=f(f^{n-1}(x))$ ã§ã. |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/1414 | E | OMC053(E) | 700 | 4 | 11 | [
{
"content": "ãäžè§åœ¢ $ABC$ ã®å
å¿ã $Z$, å
æ¥åã $O^{\\prime}$ ãšãã. æåäºå®ãšã㊠$D$ ã¯äžè§åœ¢ $ABC$ ã®è§ $A$ å
ã®åæ¥åã $BC$ ãšæ¥ããç¹ã§ãã.ãŸã, $E$ ãã $BC$ ã«ããããåç·ã®è¶³ã $P$ ãšãããš, $AB-AC=BP-CP$ ãã $P$ 㯠$O^{\\prime}$ ã $BC$ ãšæ¥ããç¹ã§ãã, æåäºå®ãšã㊠$E$ ã¯å
æ¥åã«ãã㊠$P$ ã®å¯Ÿè¹ ç¹ãšãªã. ç¹ã« $O^{\\prime}$ 㯠$FG$ ãšæ¥ãããã, å $O$ äžã« $RFG$ ã®å
æ¥åã $O^{\\prime}$ ãšãªããããªç¹ $R... | ãå€æ¥åã $O$ ãšããéè§äžè§åœ¢ $ABC$ ã®èŸº $BC$ äžã«ç¹ $D$ ã, ç·å $AD$ äžã«ç¹ $E$ ããã, $E$ ãéã $BC$ ã«å¹³è¡ãªçŽç·ã $m$ ãšããŸã. $m$ ãš $O$ ã®å£åŒ§ $AB,AC$ ã®äº€ç¹ããããã $F,G$ ãšã, å£åŒ§ $BC$ äžã«ç¹ $H$ ããšããš, $FGH$ ã¯éè§äžè§åœ¢ãšãªããŸãã. ããã« $AB,BC$ ãš $FH$ ã®äº€ç¹ããããã $I,J$, $AC,BC$ ãš $GH$ ã®äº€ç¹ããããã $K,L$ ãšãããš, äžè§åœ¢ $HJL$ ã®åšé·ã¯ $494$ ã§, äžè§åœ¢ $BIJ$ ã®åšé·ã¯äžè§åœ¢ $CKL$ ã®åšé·ãã $154$ é·ã, å ããŠ
$... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/1446 | F | OMC053(F) | 700 | 0 | 4 | [
{
"content": "ãäžåã§ç§»åã§ããæ£æŽæ°ã®çµã«æå蟺ã匵ããš, 以äžã®ãããªã°ã©ããåŸã (èªå·±èŸºãå€é蟺ã¯é©åœã«é€å»).\r\n\r\n\r\nã$50000005000000$ ã¯äžè§æ°ã§ãããã, äžãã $10^7$ 段ã®ã¿èããã°ãã. 蟺ã®åŸã㯠$3$ çš®é¡ååšããã, æççµè·¯ãšããŠé©ãããã®ã§ã¯é«ã
$2$ çš®é¡ããçšããªã. ãã®ãã¡,察称æ§ãã暪åãã®èŸºã䜿ãã, äžããäžãžåãããã®ãæ°ããã°ãã. ãã®ãšã, äžãã $n$ 段ç®ã®ããåºå®ãããæ°ãã... | ãæ°çŽç·äžã®æ£æŽæ° $a$ ã®äœçœ®ã«ããsimasimaåã¯, æ£æŽæ° $b$ ã®äœçœ®ã«ããPCTåã«äŒãããã§ã.\
ãsimasimaåãäœçœ® $x$ ã«ãããšã, 以äžã® $6$ æ¡ä»¶ã®ãã¡ $1$ ã€ä»¥äžãã¿ãã**æ£æŽæ°**ã®äœçœ®ã«äžåã§ç§»åã§ããŸãïŒ
- $x+1$
- $x-1$
- $x$ **æªæº**ã§æ倧ã®äžè§æ° $y$ ã«ã€ããŠ, $2y-x+2$ã($x=1$ ã§ã¯èããªã)
- $x$ **以äž**ã§æ倧ã®äžè§æ° $z$ ã«ã€ããŠ, $2z-x+1$
- $x$ **以äž**ã§æå°ã®äžè§æ° $s$ ã«ã€ããŠ, $2s-x+1$
- äžãšåã $s$ ã«ã€ããŠ, $2s-x+2$
ãã... |
OMC053 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc053/tasks/216 | G | OMC053(G) | 900 | 3 | 10 | [
{
"content": "ã$k\\equiv 2\\pmod 4$ ãã $x^k+y^k$ ã $x^2+y^2$ ã§å²ãåããããšã«çæããã°, LTEã®è£é¡ãã\r\n$$n=v_{p}(x^k+y^k)=v_{p}(x^2+y^2)+v_{p}(k\\/2)$$\r\nãããã£ãŠç¹ã« $x^k+y^k\\leq k\\/2\\times (x^2+y^2)$ ã§ãã, ãã®ãšã $k=2$ ãå¿
èŠã§ããããšã容æã«ããã.\\\r\nãããã§well-known factãšã㊠$p\\equiv 1 \\pmod 4$ ã§ãã, ããã« $x^2+y^2=p^n$ ãã¿ããæ£æŽæ° $x\\leq y$ ... | ãæ£ã®æŽæ°ã®çµ $(k,n,p,x,y)$ ã¯ä»¥äžã®æ¡ä»¶ãã¿ãããŸã.
- $k$ 㯠$4$ ã§å²ã£ãŠ $2$ äœã
- $n\leq 2^{1000}$
- $p$ 㯠$3\leq p\leq 19$ ãªãçŽ æ°
- $x\leq y$
- $x^k+y^k=p^n$
ãã®ãããªçµãã¹ãŠã«ã€ã㊠$xy$ ã®**ç·ç©**ãèã, ããã $2$ ã§å²ãåããåæ°ã $Q$ ãšããŸã.\
ã$Q$ ã $2^{998}-1$ ã§å²ã£ãäœãã $R$ ãšãããšã, $(2^{998}-1)-R$ ãæ±ããŠãã ãã. |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/1726 | A | OMC052(A) | 100 | 177 | 185 | [
{
"content": "ã$\\angle{BCP}=108^\\circ-67^{\\circ}=\\angle PAB$ ãã, äžè§åœ¢ $ABP$ ãš $CBP$ ã¯ååã§ããããšãããã. ãã£ãŠæ±ããè§åºŠã¯ $108^\\circ\\/2=\\textbf{54}^\\circ$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc052/editorial/1726"
}
] | ãæ£äºè§åœ¢ $ABCDE$ ã®å
éšã«ç¹ $P$ ããã, 床æ°æ³ã§ $\angle{PAB}=41^{\circ}$ ããã³ $\angle{PCD}=67^{\circ}$ ãã¿ãããŸã. ãã®ãšã, $\angle PBC$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã. |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/2066 | B | OMC052(B) | 200 | 183 | 188 | [
{
"content": "ãé©åœã«ãã¿ã³ãæŒãããšã§æ°ãã«è¿œå ã§ããããããã€ãã®æ倧éã¯, ãã®æç¹ã§å®¹åšã«å
¥ã£ãŠããããããã€ãã®éã«å¯ŸããŠå調ã«å¢å ãã. ãããã£ãŠ, ããæç¹ã§å®¹åšã«å
¥ã£ãããããã€ãã®éã $a\\\\,\\textrm{ml}$ ãšãããš, $a+40\\gt 1.1a$ ããªãã¡ $a\\lt 400$ ã®ãšãã¯èµ€ããã¿ã³ã, ããã§ãªããšãã¯ç·ã®ãã¿ã³ãæŒãã®ãæåã®æŠç¥ãšãªã\\\r\nã ã€ãŸã, åãã«èµ€ããã¿ã³ã $10$ åç¶ããŠæŒããŠããç·ã®ãã¿ã³ã $5$ åç¶ããŠæŒããç¶æ³ãèããã°ãã,\r\n$$\\displaystyle [N]=\\left[(40\\ti... | ãMasaåã¯**ããããã€ããµãŒããŒ**ã䜿ã£ãŠ $1\\,\textrm{L}$ ã®ç©ºã®å®¹åšã«ããããã€ããå
¥ããããšããŠããŸã. ããããã€ããµãŒããŒã«ã¯ $40\textrm{ml}$ ã®ããããã€ãã容åšã«è¿œå ããèµ€ããã¿ã³ãš, 容åšã«æ¢ã«å
¥ã£ãŠããéã® $0.1$ åã®éã®ããããã€ããè¿œå ããç·ã®ãã¿ã³ããããŸã. Masaåãåèšã§ $15$ åãã¿ã³ãæŒããšã, 容åšã«å
¥ããããããããã€ãã®æ倧å€ã¯å®æ° $N$ ã«ãã£ãŠ $N\textrm{ml}$ ãšè¡šããã®ã§, $N$ 以äžã®æ倧ã®æŽæ°ãæ±ããŠãã ãã. |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/1837 | C | OMC052(C) | 300 | 117 | 162 | [
{
"content": "ãæ£æ¹åœ¢ã®é¢ã« $8$ 以äžãæžã蟌ãŸããããšã¯ç¡ããã, æ£æ¹åœ¢ã®é¢ã«ã¯ã¡ããã© $2$ çš®é¡ã®æ°ãæžã蟌ãŸããŠãã. ãã㧠$5$ ããã³ $7$ ãæžã蟌ãŸããé¢ãåãæ£æ¹åœ¢ã®é¢ã«é£æ¥ããŠãããšã, æããã«æ£æ¹åœ¢ã®é¢ã«ã¯ $3$ çš®é¡ä»¥äžã®æ°ãæžã蟌ãŸããããšã«çæããã°, é©åœã«äžè§åœ¢ã®åé¢ã«æåãå²ãæ¯ãããšã§, æ¡ä»¶ã¯ä»¥äžã®ããã«è¡šçŸã§ããïŒ\r\n$$X=abc=ade=cdf,\\quad Y=abe=bcf=def$$\r\nãã㧠$X,Y$ 㯠$\\textrm{lcm}(1,2,3,4,6,8)=24$ ã®åæ°ã§ãã, $XY=abcdef=2\\times ... | ã**ç«æ¹å
«é¢äœ**ãšã¯, ç«æ¹äœã®åé ç¹ã«ã€ããŠä»¥äžã®æäœãè¡ãããšã§åŸãããç«äœã§ãïŒ
- åé ç¹ã«ã€ããŠ, ããã端ç¹ãšãã $3$ 蟺ã®äžç¹ãéãå¹³é¢ãèã, ããã«æ²¿ã£ãŠé ç¹åŽãåãèœãšã.
ãMasaåã¯ç«æ¹å
«é¢äœã®ãµã€ã³ãã«æ°ãæžã蟌ã¿ããã§ã. 圌ã¯ãŸãæ£äžè§åœ¢ã®é¢ã« $1$ ãã $8$ ãŸã§ã®æŽæ°ãäžã€ãã€æžã蟌ã¿, ããããã®æ£æ¹åœ¢ã®é¢ã«é£æ¥ããæ£äžè§åœ¢ã®é¢ã«æžã蟌ãŸãã $4$ æ°ã®ç©ãæžã蟌ã¿ãŸãã. ãããš, ãµã€ã³ãã«ã¯ã¡ããã© $10$ çš®é¡ã®æŽæ°ãæžã蟌ãŸããŠããŸãã. æžã蟌ãŸããæŽæ°ãšããŠããåŸãæ倧å€ãæ±ããŠãã ãã. |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/1901 | D | OMC052(D) | 300 | 45 | 86 | [
{
"content": "ã$X={}\\_{2021}\\mathrm{C}\\_{5}$ ãšãã, ${}\\_{2021}\\mathrm{C}\\_{6}=336X$ ã«çæãã. æ¡ä»¶ã¯ $3$ ã€ã®æåãªäºè±¡ã«åé¡ãããïŒ\r\n\r\n- ãŸã, åæèšåãã« $6$ ç¹ $ABCDEF$ ããšã, 察è§ç· $AD,BE,CF$ ãåŒãæ¹æ³ã¯ $336X$ éãã§ãã.\r\n- 次ã«, åæèšåãã« $6$ ç¹ $ABCDEF$ ããšã, 察è§ç· $AD,BF,CE$ ãåŒãæ¹æ³ã¯ $1008X$ éãã§ãã.\r\n- æåŸã«, åæèšåãã« $5$ ç¹ $ABCDE$ ããšã, 察è§ç· $AC,... | ãæ£ $2021$ è§åœ¢ã $6$ å以äžã®é åã«åå²ãã察è§ç· $3$ æ¬ã®åŒãæ¹ã¯ $N$ éããããŸã. $\dfrac{N}{{}\_{2021}\mathrm{C}\_{5}}$ ãæ±ããŠãã ãã.\
ããã ã, æ£ $2021$ è§åœ¢ã®é ç¹ã¯ãã¹ãŠåºå¥ããŠèãããã®ãšããŸã. |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/1902 | E | OMC052(E) | 400 | 22 | 45 | [
{
"content": "ã$x=2^j$ ãšãããš $f(\\log_2x)$ 㯠$x$ ã® $99$ 次åŒãšãªã, æ¡ä»¶ãããããã®æ ¹ã¯ $2^0,2^1,\\cdots,2^{98}$ ã§ãã. ããªãã¡, \r\n$$f(j)=a_{99}(2^j-2^0)(2^j-2^1)\\cdots(2^j-2^{98})$$\r\nãããã, $f(100)$ ã«ã€ããŠä»¥äžã®ããã«å€åœ¢ã§ããïŒ\r\n$$\\frac{f(100)}{f(99)}=2^{98}\\times\\frac{2^{100}-2^0}{2^{99}-2^{98}}=2^{100}-1$$\r\nãã£ãŠ, $f(100)=99(2^{10... | ã$a_0,a_1,\cdots,a_{99}$ ãå®æ°ãšã, éè² æŽæ° $j$ ã«å¯ŸããŠå®çŸ©ãããé¢æ° $f$ ã
$$f(j)=\sum_{i=0}^{99} a_i2^{ij}$$
ã§å®ãããš, 以äžãæç«ããŸãã. ãã®ãšã, $\log_2{f(100)}$ ã®æŽæ°éšåã解çããŠãã ãã.
$$f(0)=f(1)=\cdots=f(98)=0,\quad f(99)=99$$ |
OMC052 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc052/tasks/2067 | F | OMC052(F) | 400 | 28 | 59 | [
{
"content": "ã$B$ ã«é¢ã㊠$E$ ãšå¯Ÿç§°ãªç¹ã $F$ ãšããã°, $D,E,F$ 㯠$A$ ãäžå¿ãšããåäžååšäžã«ãã, $CD$ ã¯ããã«æ¥ãã. ãããã£ãŠ, æ£æŽæ° $a,b$ ãçšã㊠$BE=a, CE=b$ ãšè¡šãã°, æ¹ã¹ãã®å®çãã\r\n$$b(2a+b)=CE\\times CF=CD^2=2^{20}\\times 3^2$$\r\nãããã£ãŠ, $b$ ãšããŠããåŸããã®ã¯ $2^{20}\\times 3^{2}$ ã®çŽæ°ã§ãããã㪠$2^{10}\\times 3$ æªæºã®å¶æ°ã§ãã,\r\n$$2^{11}\\lt 2^{10}\\times 3\\lt 2... | ã$\angle B,\angle D$ ããšãã«çŽè§ã§ããåžåè§åœ¢ $ABCD$ ã¯, $AB\lt AD$ ããã³ $CD=3\times 2^{10}$ ãã¿ãããŸã. ããã§èŸº $BC$ äžã« $AD=AE$ ãªãç¹ $E$ ããšããš, $BE,CE$ ã®é·ãã¯ãšãã«æ£æŽæ°å€ãšãªããŸãã.\
ããã®ãšã, $CE$ ã®é·ããšããŠããåŸãå€ã®ç·åãæ±ããŠãã ãã. |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/307 | A | OMC051(A) | 100 | 189 | 194 | [
{
"content": "ãããã«ãã£ãå«ããæçµçãªæéã¯ãããã $60+6n$ å, $75.5+3n$ å, $98+n$ åãšè¡šãã. ãããã£ãŠ\r\n$$60+6n\\geq 75.5+3n,\\ \\ 98+n\\geq 75.5+3n$$\r\nã解ãã°ãã, ãããã $n=6,7,8,9,10,11$ ãåŸã. ç¹ã«ãããã®ç·å㯠$\\textbf{51}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc051/editorial/307"
}
] | ãããOMCäžã®ã³ã³ãã¹ãããã®ã³ã³ãã¹ããšåãã«ãŒã«ã§éå¬ãããŸãã. $A$ å, $B$ å, $C$ åã® $3$ 人ãSolverãšããŠåå ã, æçµçã«å
šå¡ãæéå
ã«å
šåã§CAãåºããŸãã. ããã«, WAãåºããåæ°ã¯ãããã $6$ å, $3$ å, $1$ åã§, æåŸã«CAãåºããŸã§ã®çµéæéã¯ãããã $60$ åã¡ããã©, $75$ å $30$ ç§, $98$ åã¡ããã©ã§ãã.\
ã$B$ åã¯ç«¶æã·ã¹ãã ã®Hackã«æåããã®ã§, ãã®ã³ã³ãã¹ãã®ããã«ãã£ãä»»æã®æ£æŽæ° $n$ ã«ã€ã㊠$n$ åã«èšå®ã§ããããã«ãªããŸãã. ãã®ãšã, $B$ åã $3$ 人ã®äžã§ããã(åçã§ãè¯ã) ã«... |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/308 | B | OMC051(B) | 300 | 128 | 167 | [
{
"content": "ã$s=x+y, t=x-y$ ãšããã°, æäœã«ãã£ãŠ $s,t$ ã¯ããããç¬ç«ã«ç確çã« $\\pm1$ ããããã®ãšæãã. ãã®ãšã, $st=2021$ ãšãªã確çãæ±ããã°ãã. 察称æ§ãã $(s,t)=(43,47)$ ã«å°éããå Žåã®ã¿èããã°ãã. ãã®ãšã $s$ 㯠$57$ åã®ãã¡ $50$ å㧠$+1$, $t$ 㯠$57$ åã®ãã¡ $52$ å㧠$+1$ ãããå¿
èŠããããã, ãã®ãããªç¢ºçã¯ä»¥äžã§äžãããã. ç¹ã«, å
šäœã§æ±ãã確çã¯ããã® $4$ åã§ãã.\r\n$$\\frac{{}\\_{57}\\mathrm{C}\\_{50}}... | ãèç«ãªã¬ãŒã®æçµã©ã³ããŒã§ããããåã¯, ããŸåº§æšå¹³é¢ã®åç¹ã«ãã, 以äžã®è¡åã $57$ åç¶ããŠè¡ããŸãïŒ
- ãããã $U,D,L,R$ ãšæžããã $4$ æã®ã«ãŒããã $1$ æãåŒã, åŒããã«ãŒãã«æžãããæåã $U$ ãªãã° $y$ 座æšã $+1$ , $D$ ãªãã° $y$ 座æšã $-1$ , $R$ ãªãã° $x$ 座æšã $+1$ , $L$ ãªãã° $x$ 座æšã $-1$ ããå Žæã«ç§»åãã.
$57$ åã®è¡åã®åŸ, æ²ç· $x^2-y^2=2021$ äžã«ããã°ããåã¯èç«å°ãžç¹ç«ããããšãã§ããŸã.\
ãã©ã®ã«ãŒããåŒããã確çãåæ§ã«ç¢ºãããããã®ãšãããšã, ããåãç¡... |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/310 | C | OMC051(C) | 400 | 81 | 140 | [
{
"content": "ã$CF=EF, \\angle CFE=108^\\circ$ ãã, æ£äºè§åœ¢ $CFEGH$ ããšãã. ãã®ãšã, $CE=EH=FH$ ãã $B$ 㯠$H$ ã«äžèŽããããšãããã. ããã« $\\angle AEG=60^\\circ,AE=GE$ ãã $\\triangle AEG$ ã¯æ£äžè§åœ¢ã§ãã. ãã£ãŠ, $G$ ã¯äžè§åœ¢ $ABE$ ã®å€å¿ã§ãã, $\\angle ABE=\\angle AGE\\/2=30^\\circ$ ãåŸããã, $\\angle EBC=72^\\circ$ ãšäœµã㊠$\\angle ABC=\\textbf{102}^\\ci... | ãåžåè§åœ¢ $ABCD$ ã«ãããŠ, 蟺 $BC$ ã®åçŽäºçåç·ãšèŸº $AD$ ã®äº€ç¹ã $E$, 蟺 $CE$ ã®åçŽäºçåç·ãšèŸº $CD$ ã®äº€ç¹ã $F$ ãšããã°, 以äžã®æ¡ä»¶ãæãç«ã¡ãŸããïŒ
$$AE=BC,\ \ BE=BF,\ \ \angle CDE=96^\circ,\ \ \angle DEF=12^\circ$$
ãã®ãšã, è§ $ABC$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ãã. |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/309 | D | OMC051(D) | 500 | 65 | 95 | [
{
"content": "ã$\\Delta$ ã¯è€æ°ã®éåã«å¯ŸããŠã¡ããã©å¥æ°åã«å±ããå
ãããªãéåãè¿ããã, ç¹ã«çµåçã«ã§ããããšã«çæãã.\\\r\nã以äž, $S_n$ ã¯ã$n$ ã®çŽæ° $x$ ã§ãã£ãŠ $n\\/x$ ãå¹³æ¹å åããããªããã®ãå
šäœãããªãéåã§ããããšãåž°çŽæ³ã§ç€ºã. ããæ£æŽæ° $n\\geq 2$ ã«ã€ããŠ, $n$ æªæºã§æç«ãä»®å®ã, $n$ ã§ã®æç«ã瀺ãã°ãã. $n$ èªèº«ããã³ $n$ ã®çŽæ°ã§ãªãæ°ã«ã€ããŠã¯æããã§ãã. $n$ ã®çŽæ° $x\\lt n$ ãäžã®æ¡ä»¶ãã¿ãããšã, ä»»æã® $x$ ã§å²ãåãã $n$ ã®çŽæ° $d\\lt n$ ã«ã€ããŠ... | ãäºã€ã®éå $A, B$ ã«å¯Ÿã $A, B$ ã®ã¡ããã©äžæ¹ã®ã¿ã«å«ãŸããèŠçŽ å
šäœã®éåã $A \Delta B$ ã§è¡šããŸã. å³å¯ã«ã¯
$$A\Delta B=(A \cap \overline B) \cup (\overline A \cap B)$$
ã§ã. ãã®ãšã, ããããæ£æŽæ°ãããªãæééå $S_1,S_2,\cdots$ ã, ä»»æã®æŽæ° $i\geq 2$ ã«å¯Ÿã以äžãã¿ãããŸããïŒ
- $i$ ã®ãã¹ãŠã®æ£ã®çŽæ°ã $d_1\lt d_2\lt \cdots\lt d_n$ ãšããã°, $(\cdots((S_{d_1} \Delta S_{d_2}) \Delta S_{d_3})\cd... |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/311 | E | OMC051(E) | 600 | 21 | 41 | [
{
"content": "**解æ³1.**ãäžåŒãå©çšããŠé©åœã«å æžãè¡ãããšã§, å€é
åŒ\r\n$$P(x)=x(x-1)(x-2)(x-333)(x-335)\\cdots (x-2021)$$\r\nã«ã€ããŠä»¥äžãæç«ãã.\r\n$$ S=\\sum_{k=0}^{2021} P(k)a_k = -(334!)\\times(2021-334)!\\times a_{334}$$\r\nãããã£ãŠ, æ±ãã $m$ ã¯çµå± $334!\\times(2021-334)!$ ã $5$ ã§å²ãåããåæ° $\\textbf{500}$ ã§ãã.\r\n\r\n**解æ³2.** ãäžåŒãå©çšããŠé©åœã«å æžãè¡ã... | ãããæ°å $A=\\{a_0,a_1,\cdots,a_{2021}\\}$ ã«ã€ããŠ, 以äžã®æ¡ä»¶ãæç«ããŸãã. ãã ã, $0^0=1$ ãšããŸã.
$$ \sum_{k=0}^{2021} k^{i}a_{k}= \begin{cases}
0 & (i=0,1,\cdots,2020) \\\\
S\neq 0 & (i=2021)
\end{cases}$$
ç¹ã« $S$ ãæŽæ° $m,n$ ãçšã㊠$S=5^m \times n\times a_{334}$ ãšè¡šãããšã, $m$ ãšããŠããåŸãæ倧ã®å€ãæ±ããŠäžãã. |
OMC051 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc051/tasks/312 | F | OMC051(F) | 700 | 1 | 28 | [
{
"content": "ãã¹ã³ã¢ãæ倧åãã $X$ ã«ã€ããŠèããïŒããæŽæ° $a$ ãåé²æ³è¡šèšãããšã $a = \\overline{a_1a_2 \\cdots a_{2021}}$ ãšè¡šãããšãã. ããã« $b_i = 9 - a_i$ ãšããã°ïŒ$a \\in X$ ãšãªãããã®å¿
èŠååæ¡ä»¶ã¯ $\\oplus$ ãæä»çè«çåãšããŠ\r\n$$b_1 \\oplus b_2 \\oplus \\cdots \\oplus b_{2021} = 0 \\tag{1}$$\r\nãšãªãããšã瀺ãïŒã¹ã³ã¢ã®å®çŸ©ããïŒ$S$ ã®å
ã倧ããæ¹ããé ã«èŠãŠïŒ$X$ ã®å
ãšãªãåŸããã®ã貪欲ã«éžæããŠããã®ã... | ã$S = \\{0, 1, 2, \ldots,10^{2021}-1\\}$ ãšãïŒ$S$ ã®éšåéå $\\{a_1, a_2, \ldots, a_n\\}$ ã«å¯Ÿããã®**ã¹ã³ã¢**ã以äžã§å®ããŸãïŒ
$$2^{a_1} + 2^{a_2} + \cdots + 2^{a_n}$$
ã以äžã®æ¡ä»¶ãã¿ãã $S$ ã®éšåéå $X$ ã§ãã£ãŠïŒãã®ã¹ã³ã¢ãæ倧ã«ãªããã®ã¯äžæã«ååšããããšãä¿èšŒãããŸãïŒãã®å
ã®åæ°ãçŽ æ° $1009$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ
- ä»»æã® $a,b \in X$ ã«å¯ŸããŠïŒ$a,b$ ã®åé²æ³è¡šèšã§åæ¡ãæ¯èŒãããšïŒã¡ããã© $2020$ ãµæãäžèŽããããšã¯ãªãïŒ
... |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2127 | A | OMCå¬é«æ¯2(A) | 200 | 37 | 45 | [
{
"content": "ã$2$ ç¹ $(x_s,y_s,z_s), (x_t,y_t,z_t)$ ã«å¯Ÿã, ãããã $33:4$ ã«å
éšããç¹\r\n$$\\left(\\dfrac{4x_s+33x_t}{37},\\dfrac{4y_s+33y_t}{37},\\dfrac{4z_s+33z_t}{37}\\right)\\$$\r\nãæ Œåç¹ãšãªãæ¡ä»¶ã¯, $2$ ç¹ã®åº§æšã $37$ ãæ³ãšããŠäžèŽããããšã ãã, æ±ããæå°å€ã¯ $37^3+1=\\textbf{50654}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcont... | ã$xyz$ 空éå
ã«çžç°ãªã $k$ å ($k\geq2$) ã®æ Œåç¹ $P_1,P_2,\cdots,P_k$ ããšããŸã. $1\leq i \lt j \leq k$ ãªãæŽæ°ã®çµ $(i,j)$ ãã¹ãŠã«å¯ŸããŠ, ç·å $P_iP_j$ ã $33:4$ ã«å
åããç¹ããšã, ãããã®ç¹ã®éåã $N$ ãšããŸã. $P_1,P_2,\cdots,P_k$ ã®ãšãæ¹ã«ããã, $N$ ã«æ Œåç¹ãå«ãŸãããã㪠$k$ ã®æå°å€ãæ±ããŠãã ãã. |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2126 | B | OMCå¬é«æ¯2(B) | 300 | 34 | 38 | [
{
"content": "$$\\begin{aligned}\r\n\\sum_{k=1}^{2032}a_{4k-3}=a_{1}+a_{5}+\\cdots+a_{8125}=A\\\\\\\\\r\n\\sum_{k=1}^{2032}a_{4k-2}=a_{2}+a_{6}+\\cdots+a_{8126}=B\\\\\\\\\r\n\\sum_{k=1}^{2032}a_{4k-1}=a_{3}+a_{7}+\\cdots+a_{8127}=C\\\\\\\\\r\n\\sum_{k=1}^{2032}a_{4k}=a_{4}+a_{8}+\\cdots+a_{8128}=D\\\\\\\\\r\n\\... | ãå®æ° $a_1,a_2,\cdots,a_{8128}$ 㯠$1\leq n\leq 8126$ ãªãä»»æã®æŽæ° $n$ ã«å¯Ÿã㊠$a_{n+2}=a_{n+1}+a_n$ ãã¿ãããŸã.
$$\begin{aligned} x&=a_1-a_3+\cdots +a_{8125}-a_{8127}\\\\ y&=a_2-a_4+\cdots+a_{8126}-a_{8128}\end{aligned}$$
ãšãããšã, $a_1+a_2+\cdots+a_{8128}$ ã¯æŽæ° $a,b$ ãçšã㊠$ax+by$ ãšåžžã«è¡šããã®ã§, $|a+b|$ ã解çããŠãã ãã. |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2201 | C | OMCå¬é«æ¯2(C) | 400 | 38 | 42 | [
{
"content": "ã$n$ ãçŽ å æ°å解ã㊠$n={p_1}^{\\alpha_1}{p_2}^{\\alpha_2}\\cdots{p_m}^{\\alpha_m}$ ãšããã°, äžåŒã¯\r\n$$\\dfrac{n}{{(T(n^2)})^2}=\\dfrac{{p_1}^{\\alpha_1}{p_2}^{\\alpha_2}\\cdots{p_m}^{\\alpha_m}}\r\n{\\left\\\\{\r\n(2{\\alpha_1}+1)(2{\\alpha_2}+1)\\cdots(2{\\alpha_m}+1)\r\n\\right\\\\}^2\r\n}$$\r\nãã㧠${f_p}... | ãæ£æŽæ° $n$ ã«å¯Ÿã, $T(n)$ 㧠$n$ ã®ãã€æ£ã®çŽæ°ã®åæ°ãè¡šããŸã. ãã®ãšã,
$$\dfrac{n}{(T(n^2))^2}$$
ã®ãšãããæå°å€ã¯, äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2196 | D | OMCå¬é«æ¯2(D) | 500 | 12 | 16 | [
{
"content": "ãæ£æŽæ° $k$ ã«å¯ŸããŠ, 以äžã $n$ ã«é¢ããæçåŒã«ãªããããªå®æ° $b_1,\\cdots,b_k$ ãèãã.\r\n$$n^k={b_1}\\cdot{\\_n}\\mathrm{C}\\_1+{b_2}\\cdot{\\_n}\\mathrm{C}\\_2+\\cdots+{b_{k-1}}\\cdot{\\_n}\\mathrm{C}\\_{k-1}+{b_k}\\cdot{\\_n}\\mathrm{C}\\_k\r\n$$\r\nãã®ãã¡æ·»ãåã®å¶å¥ã $k$ ãšäžèŽãããã®ã®åã $C(k)$ ãšã, ããã§ãªããã®ã®åã $D(k)$ ãšãããš,\r\n$$\r... | ãä»»æã®æ£æŽæ° $k$ ã«å¯ŸããŠ,
$$kn^k={a_1}\cdot{\_n}\mathrm{C}\_1+{a_2}\cdot{\_n}\mathrm{C}\_2+\cdots+{a_{k-1}}\cdot{\_n}\mathrm{C}\_{k-1}+{a_k}\cdot{\_n}\mathrm{C}\_k
$$
ãæ£æŽæ° $n$ ã«é¢ããæçåŒãšãªããããªå®æ° $a_1,\cdots,a_k$ ãäžæã«å®ãŸããŸã. ãã®ãšã, $a_1,\cdots,a_k$ ã®ãã¡æ·»ãåã®å¶å¥ã $k$ ãšäžèŽãããã®ã®åã $A(k)$ ãšã, ããã§ãªããã®ã®åã $B(k)$ ãšããŠ, 以äžã®å€ãæ±ããŠãã ãã.
$$\sum_... |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2205 | E | OMCå¬é«æ¯2(E) | 700 | 5 | 16 | [
{
"content": "ã$n=2^{2020}$ ãšãã. ç·åãå¥æ°ã®æ¡ä»¶ãç¡èŠããã° ${\\_{4n}}\\mathrm{C}\\_{2n}$ éãã§ãã. ãã®ãã¡, ãã¹ãŠãå¶æ°ã〠$a_{2i-1}=a_{2i}$ ãã¿ãããã® ${\\_{2n}}\\mathrm{C}\\_{n}$ éããé€ã, ç·åãå¶æ°ã®ãã®ãšå¥æ°ã®ãã®ãäžå¯Ÿäžã«å¯Ÿå¿ããããšã確èªããã. å®éã«, æ¡ä»¶ã$a_{2i-1}=a_{2i}$ ãã€ããããå¶æ°ã§ããããã¿ãããªãæå°ã® $i$ ããšãã°, 以äžã®ããã«å¯Ÿå¿ãåŸããã.\r\n\r\n- $a_{2i-1}$ ãå¥æ°ã®ãšã, $a_{2i-1} \\longm... | ãç·åãå¥æ°ã§ããæŽæ°ã®çµ $0\leq a_1\leq a_2\leq\cdots\leq a_{2^{2021}}\leq 2^{2021}$ 㯠$M$ éããããŸã. $M$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ãã. |
OMCå¬é«æ¯2 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou2/tasks/2202 | F | OMCå¬é«æ¯2(F) | 900 | 0 | 3 | [
{
"content": "ã以äžã®äž¡è£é¡ãæåäºå®ãšããŠèªããïŒ\r\n\r\n----\r\n**è£é¡1.**ã å
å¿ã $I$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, $AB,AC$ ã«æ¥ãåæã« $\\Gamma$ ã«å
æ¥ããåã $C_A$ ãšã, $C_A$ ãš $AB,AC$ ã®æ¥ç¹ããããã $X,Y$ ãšãã.ãã®ãšã, $3$ ç¹ $X,I,Y$ ã¯åäžçŽç·äžã«ãã.\r\n\r\n----\r\n**è£é¡2.**ãè§ $A$ å
ã®åå¿ã $I_A$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, $AB,AC$ ã«æ¥ãåæã« $\\Gamma$ ã«å€æ¥ããåã $D_A$ ãšã, $D_A$ ãš $AB,A... | ãå
æ¥åã $\omega$, è§ $A$ ã®åæ¥åã $\omega_A$, å€æ¥åã $\Gamma$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, $\omega$, $\omega_A$, $\Gamma$ ãã¹ãŠã«æ¥ããå $\gamma$ ãèããŸã. ãã ã $\omega$ ã $\gamma$ ã«å
æ¥ãããã®ãšããŸã. $\gamma$ ã®ååŸã $20$ ã§ãã, $AC-AB=21$ ãæãç«ã£ãŠãããšã, $\omega$, $\omega_A$ ã®ååŸããããã $r$,$r_A$ ãšããŠ, $\dfrac{r_A}{r}$ ã®ãšãããæå°å€ãæ±ããŠãã ãã. ãã ã, 解çãã¹ãå€ã¯æ£æŽæ° $a,b,c,d$ ... |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/1802 | A | OMC050(A) | 200 | 183 | 191 | [
{
"content": "ã解ãšä¿æ°ã®é¢ä¿ãã,\r\n$$x+y+z=-111,\\quad xy+yz+zx=222,\\quad xyz=-333$$\r\nã§ããããšã«çæããã°, æ±ããå€ã¯\r\n$$\\begin{aligned}\r\n\\frac{y+z}{x}+\\frac{z+x}{y}+\\frac{x+y}{z}&=\\left(\\frac{x+y+z}{x}-1\\right)+\\left(\\frac{x+y+z}{y}-1\\right)+\\left(\\frac{x+y+z}{z}-1\\right)\\\\\\\\\r\n&=(x+y+z)\\left(\\frac{1}{... | ã$t$ ã«ã€ããŠã®äžæ¬¡æ¹çšåŒ
$$t^3+111t^2+222t+333=0$$
ã® $3$ ã€ã®è€çŽ æ°è§£ã $t=x,y,z$ ãšãããšã, 以äžã®åŒã®å€ãæ±ããŠãã ããïŒ
$$\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}$$ |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/251 | B | OMC050(B) | 300 | 117 | 152 | [
{
"content": "ãäžç¹é£çµå®çãã $PQRS$ ã¯å¹³è¡å蟺圢ã§ãã, æ¡ä»¶ã®å
±ç¹ã $T$ ãšããã°ããã¯ãã®äžå¿ã§ãã. $DT$ ãš $PQ$ ãå¹³è¡ã§ããããšãã $D$ 㯠$QR$ ã®äžç¹ã§ãã, ããªãã¡ $BC$ ã®äžç¹ã§ãã. ãã®ãšã, äžè§åœ¢ $ABC$ ã«ãããäžç·å®çãã $AD^2=31\\/4$ ã§ãããã, äžè§åœ¢ $TQR$ ã«ãããäžç·å®çãã\r\n$$PR^2+QS^2=4(QT^2+RT^2)=8(DT^2+DQ^2)=\\dfrac{1}{2}(AD^2+BC^2)=\\dfrac{707}{8}$$\r\nç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{715}$ ã§... | ãäžè§åœ¢ $ABC$ ããã³èŸº $BC$ äžã®ç¹ $D$ ã«ã€ããŠ, ç·å $AB,BD,DC,CA$ ã®äžç¹ããããã $P,Q,R,S$ ãšããã°, $AD,PR,QS$ ã¯äžç¹ã§äº€ãããŸãã. $AB=6,BC=13,CA=8$ ã§ãããšã, $PR^2+QS^2$ ãæ±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. |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/1943 | C | OMC050(C) | 300 | 102 | 129 | [
{
"content": "ãæ£å
è§åœ¢ã®é ç¹ã«æãè¿ãæ£äžè§åœ¢ (å³ã®é»è²) ã®åãæ¹ãèããã°é 次åšãããåãŸã, 䞡端㮠$2$ éãã®ã¿ãé©ãã. ããã«ãã£ãŠ, äžèŸºã®é·ãã $2$ å°ããå Žåã«åž°çããããã, æ±ããã¹ãå Žåã®æ°ã¯ $2^{20\\/2}=\\bm{1024}$ éãã§ãã.\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc050/editorial/... | ãäžèŸºã®é·ãã $20$ ã®æ£å
è§åœ¢ã, 以äžã®èŠé ã§äžèŸºã®é·ãã $1$ ã®æ£äžè§åœ¢ã«åå²ãããŠããŸã. ãŸã, äžèŸºã®é·ãã $1$ ã®æ£äžè§åœ¢ $4$ ã€ã以äžã®ããã«çµã¿åããã**ã¿ã€ã«**ãç¡æ°ã«ãããŸã. æ£å
è§åœ¢ãééãéè€ãªãã¿ã€ã«ã§æ·ãè©°ããæ¹æ³ã¯äœéããããŸããïŒãã ã, å転ãããè£è¿ãããããŠäžèŽãããã®ãç°ãªããã®ãšããŠæ°ããŸã.
 |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/282 | D | OMC050(D) | 400 | 92 | 150 | [
{
"content": "ã$f$ ã¯æŽæ°å€ã®ã¿ããšãããšãã, 以äžã®äžçåŒãš $f(10^6)=111222$ ã䜵ããã°ãã®æ倧å€ã¯ $111222$ ã§ãã.\r\n$$f(n)\\leq \\dfrac{111222444888}{n}\\times\\dfrac{n}{10^6}=111222.444888$$\r\nãããã£ãŠ, ä»¥äž $f(n)=111222$ ãªã $n$ ã«ã€ããŠèããã°ãã. $111222=2\\times3^2\\times37\\times 167$ ã«çæãã.\\\r\nããã®ãšã $m=[n\\/10^6](\\leq 111222)$ ã«ã€ããŠ, $m$ 㯠$1... | ãæ£æŽæ°ã«å¯ŸããŠå®çŸ©ãããé¢æ° $f$ ã¯, ä»»æã®æ£æŽæ° $n$ ã«å¯ŸããŠä»¥äžãã¿ãããŸãïŒ
$$f(n)=\biggl[\dfrac{111222444888}{n}\biggr]\biggl[\dfrac{n}{10^6}\biggr]$$
ãã®ãšã, $f(n)$ ãæ倧å€ããšããããªæ£æŽæ° $n$ ã¯ããã€ãããŸããïŒ\
ããã ã, å®æ° $x$ ã«å¯Ÿã $[x]$ 㧠$x$ ãè¶
ããªãæ倧ã®æŽæ°ãè¡šããŸã. |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/1583 | E | OMC050(E) | 500 | 39 | 61 | [
{
"content": "ã$ABD$ ããã³ $BCD$ ã«ããããäœåŒŠå®çãé©çšã㊠$BD^2$ ã $2$ éãã«è¡šçŸããããšã§ $AD=7$ ãåŸã. ãã®ãšã $XAD$ ãš $XCB$ ã¯çžäŒŒæ¯ $1:2$ ã®é¢ä¿ã«ããããšãã, $XA=15,XD=13$ ã容æã«åŸã.\\\r\nããã㧠$\\Gamma,\\Omega_{B},\\Omega_{C}$ ã®äžå¿ããããã $O,O_B,O_C$, $XB$ ã®äžç¹ã $M$, $\\Gamma$ ã«ããã $B$ ã®å¯Ÿè¹ ç¹ã $B^\\prime$ ãšããã°, $O_B$ 㯠$BX$ ã®åçŽäºçåç·ãš $BO$ ã®äº€ç¹ã§ãã, $\\angl... | ã$AB=11,BC=14,CD=17,\angle C=60^\circ$ ãªãåè§åœ¢ $ABCD$ ãå $\Gamma$ ã«å
æ¥ããŠããŸã. $2$ çŽç· $AB,CD$ ã®äº€ç¹ $X$ ãéã, ããããç¹ $B,C$ 㧠$\Gamma$ ã«æ¥ããåã $\Omega_{B},\Omega_{C}$ ãšããã°, ãããã®ååŸæ¯ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $a:b$ ãšè¡šãããŸã. $a+b$ ã解çããŠãã ãã. |
OMC050 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc050/tasks/1752 | F | OMC050(F) | 500 | 24 | 78 | [
{
"content": "ã$\\alpha=\\cos\\theta_0 +2$ ãªãå®æ° $2\\pi\\/3\\leq\\theta_0\\leq\\pi$ ãäžæã«ååšããããšã«çæãã. ããã« $x_n=\\cos\\theta+2$ ã«ã€ããŠ,\r\n$$\\begin{aligned}\r\nx_{n+1}=2(\\cos\\theta+2)^2-8(\\cos\\theta+2)+9=2\\cos^2\\theta+1=\\cos2\\theta+2\r\n\\end{aligned}$$\r\nãããã£ãŠ, $\\cos2^{10}\\theta_0=\\cos\\theta_0$ ãªã $\\... | ã$1$ ä»¥äž $3\/2$ 以äžã®å®æ° $\alpha$ ã§ãã£ãŠ, 以äžã§å®ãŸãå®æ°å $\\{x_n\\}$ ã $x_{10}=x_0$ ãã¿ãããã®ã¯ããã€ãããŸããïŒ
$$\begin{aligned}
x_0=\alpha,\quad x_{n+1}=2x_n^2-8x_n+9\quad (n=0,1,\cdots)
\end{aligned}$$ |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/2125 | A | OMCå¬é«æ¯1(A) | 100 | 94 | 96 | [
{
"content": "ã$AB=3a$ ãšããã°, å³1ããã³å³3ã«ã€ããŠæç·éšã®åšé·ã¯ãããã $30a,50a$ ã§ãã, 解çãã¹ãå€ã¯ $\\textbf{8}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omckoukou1/editorial/2125"
}
] | ãéé«çã®Birdyåã¯, 次ã«ç€ºãæ¹æ³ã§å³3ã®ãããªé岡é«æ ¡ã®æ ¡ç« ãæãããšã«ããŸãã.
- æ£äºè§åœ¢ $ACEGI$ ããå³1ã®åè§åœ¢ $ABCDEFGHIJ$ ãäœã.
- 蟺 $AB, BC, CD, DE, EF, FG, GH, HI, IJ, JA$ ããããã $3$ çåããç¹ããšã.
- å³2ã®ããã«, ãããã®ç¹ãéãæ£äºè§åœ¢ã $2$ ã€äœã.
ãã®ãšã, å³1ã®æç·éšåã®åšã®é·ããšå³3ã®æç·éšåã®åšã®é·ãã®æ¯ã¯, äºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$a:b$ ãšè¡šããã®ã§, $a+b$ ã解çããŠäžãã.
 | 300 | 62 | 79 | [
{
"content": "ã$(A_s,A_t,A_u)$ ãåºå®ã, ãã®ç·åãžã®å¯äžãèããããšã§, 以äžã®æç«ã容æã«ããã.\r\n$$M=\\binom{2021}{3}\\times\\binom{2021}{3}\\times 2018!=\\dfrac{1}{6^2}\\times 2019\\times 2020\\times 2021\\times 2021!$$\r\nLegendreã®å®çãã, ããã $2$ ã§å²ãåããåæ°ã¯ $\\textbf{2013}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest... | ã$(1,2,\cdots,2021)$ ã®çœ®æãšããŠåŸãããçµ $(A_1,A_2,\cdots,A_{2021})$ ãã¹ãŠã«å¯Ÿã, 以äžã®å€ã®ç·åã $M$ ãšããŸã.
- $s\lt t\lt u$ ã〠$A_s\gt A_t\gt A_u$ ãªãçµ $(s,t,u)$ ã®åæ°
ãã®ãšã, $M$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ãã. |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/1365 | C | OMCå¬é«æ¯1(C) | 300 | 60 | 72 | [
{
"content": "ã$x+y+z=p,xy+yz+zx=q,xyz=r$ ãšããã°, æ¡ä»¶ãæŽçããããšã§ä»¥äžãåŸãïŒ\r\n$$ r-2q+3p=4,\\quad r-3q+8p=20,\\quad r-4q+15p=54 $$\r\nããã解ããš $p=9,q=29,r=35$ ãåŸã. åæ§ã« $f(4)$ ã«ãããŠãåæ¯ãæãã°\r\n$$f(4)(r-4q+16p-64)=4(q-8p+48)$$\r\nãããã $|f(4)|=|-20|=\\textbf{20}$ ãåŸã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.... | ããããã $1,2,3,4$ ã§ãªãè€çŽ æ° $x,y,z$ ã«ã€ããŠ,
$$f(k)=\dfrac{k}{x-k}+\dfrac{k}{y-k}+\dfrac{k}{z-k}$$
ãšãããš, $f(1)=1,f(2)=2,f(3)=3$ ãæç«ããŸãã. ãã®ãšã, $|f(4)|$ ãæ±ããŠãã ãã. |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/2194 | D | OMCå¬é«æ¯1(D) | 400 | 56 | 61 | [
{
"content": "ã${\\dfrac{1}{s_i}+\\dfrac{1}{s_{2021-i}}=1}$ ãå€åœ¢ã㊠$(s_i-1)(s_{2021-i}-1)=1$ ãåŸããã,\r\n\r\n$$\r\n\\begin{aligned}\r\n\\sum_{i=1}^{2021}\\dfrac{1}{{s_i}^3-3{s_i}^2+3{s_i}}&=\\sum_{i=1}^{2021}\\dfrac{1}{(s_i-1)^3+1}\\\\\\\\\r\n&=1+\\sum_{i=1}^{1010}\\left(\\frac{1}{(s_i-1)^3+1}+\\frac{1}{(s_{2021-i... | ã$s_i=\dfrac{2021}{i}$ ã«ã€ããŠ, 以äžã®ç·åãæ±ããŠãã ããïŒ
$$\displaystyle\sum_{i=1}^{2021}\frac{1}{{s_i}^3-3{s_i}^2+3{s_i}}$$ |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/2142 | E | OMCå¬é«æ¯1(E) | 400 | 32 | 50 | [
{
"content": "ãä»»æã®äººã«ã€ããŠ, çµã¿æãã«ãããå±ããã°ã«ãŒãã®å¶å¥ã¯äžå®ã§ãããã, æ¡ä»¶ã¯ã¯ããããªããå¥æ°åã®ã°ã«ãŒãã«å±ããããšã§ãã. ããªãã¡, ${\\_{2021}}\\mathrm{C}\\_{m}$ ãå¥æ°ãšãªã $0\\leq m\\leq2021$ ã®åæ°ãæ°ããã°ãã. ããã¯Lucasã®å®çãã $m$ ãš $2021$ ã®è«çç©ã $m$ ã«äžèŽããããšãšåå€ã§, $2021_{(10)}=11111100101_{(2)}$ ãã $2^8=\\textbf{256}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://o... | ã$n$ ã $1$ ä»¥äž $2022$ 以äžã®æŽæ°ãšããŸã. $2021$ 人ã®äººãšããªãããã, äœäººããããªã**ã°ã«ãŒã**ãããã€ãååšããŸã. ã¯ãã, ã°ã«ãŒãã¯ãã¹ãŠ $n$ 人ãããªã, éã«ä»»æã® $n$ 人ããæ§æãããã°ã«ãŒããã¡ããã©äžã€ååšããŸã.\
ãããŸ, ãããã®ã°ã«ãŒãã«ã€ããŠ, ã°ã«ãŒããäžã€ã«ãªããŸã§ä»¥äžã®èŠé ã§çµã¿æããè¡ããŸã.
- é©åœãªã°ã«ãŒã $A\neq B$ ãéžã³, ããããã解æ¶ãã.
- $A$ ãš $B$ ã®ã¡ããã©äžæ¹ã«å±ããŠãã人ãããªãã°ã«ãŒããæ°ããäœã.
çµã¿æãã®æ¹æ³ã«ããã, æåŸã«æ®ã£ãã°ã«ãŒãã«ããªããå«ãŸããŠãããã㪠$n$ ã¯ã... |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/2130 | F | OMCå¬é«æ¯1(F) | 700 | 4 | 16 | [
{
"content": "ã$\\triangle{ABC}$ ã®åšé·ã $S$ ãšãããš, 以äžã容æã«åãããã, $A$ ãš $BC$ ã®è·é¢ãæ±ããã°ãã.\r\n$$S_BS_C=\\dfrac{S}{2}-AD$$\r\n蟺 $AB,AC$ ã®äžç¹ã $M,N$ ãšã, $F_B,F_C$ ã®äžå¿ããããã $O_B,O_C$ ãšãã. çŽç· $MN$ ãšçŽç· $O_BB,O_CC$ ã®äº€ç¹ããããã $Q,R$ ãšã, $B$ ãã $AO_B$ ã«äžãããåç·ã®è¶³ã $H$ ãšãã.\\\r\nã$4$ ç¹ $B,O_B,S_B,H$ 㯠$BO_B$ ãçŽåŸãšããååšäžã«ãããã, ç°¡åãªè§åºŠèšç®... | ã$AB=4,AC=5$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ äžã«ç¹ $D$ ããšã, äžè§åœ¢ $ABD$ ã®è§ $BAD$ å
ã®åæ¥åã $F_B$, äžè§åœ¢ $ACD$ ã®è§ $CAD$ å
ã®åæ¥åã $F_C$ ãšããŸã. ãŸã, $F_B$ ãš $BC,AD$ ã®æ¥ç¹ããããã $S_B,T_B$, åæ§ã« $F_C$ ãš $BC,AD$ ã®æ¥ç¹ããããã $S_C,T_C$ ãšããŠ, $S_BT_B$ ãš $S_CT_C$ ã®äº€ç¹ã $P$ ãšããŸã. \
ãããŸ, $D$ ã蟺 $BC$ äžãåãããšã, $P$ ã¯ããååŸ $4$ ã®ååšäžãåããŸãã. ãã®ãšã, ç·å $S_BS_C$ ã®é·ãããš... |
OMCå¬é«æ¯1 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omckoukou1/tasks/2150 | G | OMCå¬é«æ¯1(G) | 800 | 1 | 12 | [
{
"content": "ãå¹³é¢äžã®ããã€ãã®ç¹ã®éåã«ã€ããŠ, ã©ã® $3$ ç¹ãåäžçŽç·äžã«ãªãã©ã® $4$ ç¹ãåäžååšäžã«ãªããšã, ããã**æŽã£ã**ãšåŒã¶ããšãšãã. 以äž, åã«**å**ã**äžè§åœ¢**ãªã©ãšããã°, ãã®é ç¹ã¯æŽã£ãç¹ã®éåããéžã°ãã $3$ ç¹ã§ãããšãã.\r\n\r\n----\r\n**è£é¡1.**ãæŽã£ã $s$ åã®ç¹ã«å¯ŸããŠ, ä»ã®ãã¹ãŠã®ç¹ãå
éšã«å«ãåãååšãã. ã€ãŸã $M=s-3$ ã§ãã.\\\r\n**蚌æ.**ã$s$ åã®ç¹ã®åžå
ã $P_1\\cdots P_k$ ãšã, $\\angle{P_1P_3P_2},\\cdots,\\angle{... | ãå¹³é¢äžã« $1000$ åã®ç¹ããã, ã©ã® $3$ ç¹ãåäžçŽç·äžã«ãªã, ã©ã® $4$ ç¹ãåäžååšäžã«ãããŸãã. ãããã $3$ ç¹ãéžã¶æ¹æ³ã§ãã£ãŠ, ããããéãåã®å
éš (åšäžãå«ãŸãªã) ã«ã¡ããã© $k$ åã®ç¹ãå«ããããªãã®ã®åæ°ã $f(k)$ ã§è¡šããŸã. ããŸ, $3$ ç¹ãéãåãå
éšã«å«ãããšã®ã§ããç¹ã®åæ°ã®æ倧å€ã $M$ ãšãããš, $f(M)=500$ ãæç«ããŸãã. ãã®ãšã, 以äžã®ãšãããæ倧å€ãæ±ããŠãã ãã.
$$f(1)+2f(2)+\cdots+(M-1)f(M-1)+Mf(M)$$ |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/1544 | A | OMC049(A) | 100 | 225 | 227 | [
{
"content": "ãåžå€è§åœ¢ã®å€è§ã®å㯠$360$ 床ã§ãããã, æ£ãã㯠$5$ 以äžã«ãªãåŸãªã.\\\r\nãéã«é·æ¹åœ¢ã®æ£ãã㯠$\\textbf{4}$ ã§ãããã, ãããæ±ããæ倧å€ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc049/editorial/1544"
}
] | ãåžå€è§åœ¢ã«å¯ŸããŠ, ãã®å
è§ã®ãã¡çŽè§ã§ãããã®ã®æ°ããã®**æ£ãã**ãšå®çŸ©ããŸã.\
ãæ£ããã®ãšãåŸãæ倧ã®å€ã¯ããã€ã§ããïŒ |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/220 | B | OMC049(B) | 200 | 191 | 216 | [
{
"content": "ãäžåŒã¯ $(ab-1)(c+d-1)=10$ ãšå€åœ¢ããã.\r\n\r\n- $(ab,c+d)=(2,11)$ ã®ãšã, $2\\times10=20$ éã.\r\n- $(ab,c+d)=(3,6)$ ã®ãšã, $2\\times5=10$ éã.\r\n- $(ab,c+d)=(6,3)$ ã®ãšã, $4\\times2=8$ éã.\r\n- $(ab,c+d)=(11,2)$ ã®ãšã, $2\\times1=2$ éã.\r\n\r\n以äžããæ±ããçµã®æ°ã¯ $\\textbf{40}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "http... | ã以äžãã¿ããæ£æŽæ°ã®é åºä»ããçµ $(a,b,c,d)$ ã¯ããã€ãããŸããïŒ
$$abc+abd=ab+c+d+9$$ |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/221 | C | OMC049(C) | 200 | 125 | 182 | [
{
"content": "ã$a+b+c+d=19$ ãªãæŽæ° $0\\leq a,b,c,d\\leq 9$ ã®çµãæ°ãäžããã°ãã.\r\n\r\n**解æ³1.**ã$(a+b,c+d)=(1,18)$ ãªããã®ã¯ $2\\times1=2$ éã, $(a+b,c+d)=(2,17)$ ãªããã®ã¯ $3\\times2=6$ éããã, åæ§ã«ããã $(18,1)$ ãŸã§èããããšã§æ±ããå Žåã®æ°ã¯\r\n$$2\\times 1+3\\times2+\\cdots+10\\times 9+9\\times 10+\\cdots+1\\times 2=\\textbf{660}$$\r\n\r\n**解æ³2... | ãåé²æ³ã§åäœã®åã $19$ ã§ãããã㪠$9999$ 以äžã®æ£æŽæ°ã¯ããã€ãããŸããïŒ |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/1506 | D | OMC049(D) | 300 | 101 | 136 | [
{
"content": "ãå®çŸ©ãã $\\\\{3x^2\\\\}$ ã¯åžžã« $0$ ä»¥äž $1$ æªæºã§ãããã,\r\n$$\\frac{3}{2}\\lt \\left\\lfloor\\frac{1}{2}x\\right\\rfloor\\le\\frac{5}{2} \\implies \\left\\lfloor\\frac{1}{2}x\\right\\rfloor=2$$\r\nããªãã¡ $4\\le x\\lt 6$ ã§ãã. ãã㧠$\\\\{3x^2\\\\}=3x^2-m$ ($m=48,\\cdots,107$) ãšããã°,\r\n$$ \\\\{3x^2\\\\}+\\left\... | ã次ã®æ¹çšåŒã®å®æ°è§£ã®**å¹³æ¹ã®ç·å**ãæ±ããŠãã ããïŒ
$$ \\{3x^2\\}+\left\lfloor\frac{1}{2}x\right\rfloor=\frac{5}{2}$$
ããã ãå®æ° $a$ ã«å¯ŸããŠ, $\lfloor a\rfloor$ 㧠$a$ ãè¶
ããªãæ倧ã®æŽæ°ãè¡šãã$\\{a\\}$ 㧠$a-\lfloor a\rfloor$ ãè¡šããã®ãšããŸã. |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/1269 | E | OMC049(E) | 300 | 123 | 147 | [
{
"content": "ã$ACR$ ãæ£äžè§åœ¢ãšãªããããªç¹ $R(\\neq B)$ ã«ã€ããŠ, $AQ$ ãš $CR$ ã®äº€ç¹ã $S$ ãšããã°, $ABP$ ãš $ACS$ ã¯ååã§ãã, ã〠$AR$ ãš $CQ$ ã¯å¹³è¡ã§ãããã, $AR:CQ=RS:SC=10:11$ ãåŸã. ç¹ã« $CQ=\\dfrac{231}{10}$ ã§ãããã, 解çãã¹ãå€ã¯ $\\textbf{241}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc049/editorial/1269"
... | ãäžèŸº $21$ ã®æ£äžè§åœ¢ $ABC$ ã«ãããŠ, 蟺 $BC$ äžã«ç¹ $P$ ã, 蟺 $BC$ ã® $C$ åŽãžã®å»¶é·ç·äžã«ç¹ $Q$ ããã, $BP=11$ ããã³ $\angle BAP=\angle CAQ$ ãã¿ãããŠããŸã. ãã®ãšã, $CQ$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§, $a+b$ ã解çããŠãã ãã. |
OMC049 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc049/tasks/1437 | F | OMC049(F) | 400 | 83 | 122 | [
{
"content": "ãçµè·¯ã«ãã£ãŠåå²ããã $2$ é åã®ãã¡, äžåŽã«å«ãŸãããã¹ã®åæ°ã $x$ ãšããã°, $x$ ãšããŠããåŸãå€ã¯\r\n$$x=0,2,5,8,17,20,23,25$$\r\nããšã¯å $x$ ã«ã€ããŠ, 以äžã®æ¡ä»¶ãã¿ããçµ $(a,b,c,d,e)$ ã®åæ° $p(x)$ ãæ±ããã°ãã.\r\n\r\n- ãã¹ãŠ $0$ ä»¥äž $5$ 以äžã®æŽæ°ã§ãã.\r\n- $a\\leq b\\leq c\\leq d\\leq e$\r\n- $a+b+c+d+e=x$\r\n\r\nãã㧠$a,b,c,d,e$ ã¯åè¡ã«ã€ããŠäžåŽã®é åã«å±ãããã¹ã®åæ°ã«å¯Ÿå¿ãã.\\\r... | ã$5\times 5$ ã®ãã¹ç®ã«ãããŠ, å·Šäžã®ç¹ããå³äžã®ç¹ãŸã§èŸºäžãæçã§ç§»åãã (ããªãã¡ $10$ æ¬ã®èŸºãéã) ãããªçµè·¯ã®ãã¡, 以äžã®æ¡ä»¶ãã¿ãããã®ã¯ããã€ãããŸããïŒ
- çµè·¯ã«ãã£ãŠãã¹ç®ãåå²ãããŠã§ãã $2$ é åã«ã€ããŠ, ããããå«ãŸãããã¹ã®åæ°ã®å·® (ã®çµ¶å¯Ÿå€) ãåææ°ã§ãã.
ããã ã, ãã¹ç®ã $0$ ãã¹ããã³ $25$ ãã¹ãžåå²ãããå Žåã,ã$2$ é åã«åå²ãããããšã¿ãªããã®ãšããŸã.\
ãããã§, åææ°ãšã¯ $2$ 以äžã®çŽ æ°ã§ãªãæŽæ°ãæããŸã. |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/1613 | A | OMC048(A) | 300 | 148 | 183 | [
{
"content": "**解æ³1.**ãäžæ¹çšåŒã¯ $x=0$ ã解ã«æããªããã, $20x^{2020}=1-\\dfrac{21}{x}$ ãšããã°, 解ã®éæ°åãæ±ããã°ãã. ããã§,\r\n$$21y^{2021}-y^{2020}+20=y^{2021}\\left(20\\left(\\frac{1}{y}\\right)^{2021}-\\dfrac{1}{y}+21\\right)=0$$\r\nã¯, äžæ¹çšåŒã®ããããã®è§£ $x=X$ ã«å¯Ÿã㊠$y=1\\/X$ ã解ã«ãã€æ¹çšåŒã§ãããã, 解ãšä¿æ°ã®é¢ä¿ããäžæ¹çšåŒã®è§£ã®éæ°å㯠$1\\/21$ ã§ãã, ç¹ã«æ±ããç·å㯠$(20... | ã$x$ ã®æ¹çšåŒ $20x^{2021}-x+21=0$ ã®ïŒéè€ã蟌ããŠïŒ$2021$ åã®è€çŽ æ°è§£ãã¹ãŠã«ã€ããŠïŒããããã® $2020$ ä¹ã®ç·åãæ±ããŠãã ããïŒ |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/1615 | B | OMC048(B) | 400 | 93 | 157 | [
{
"content": "ã$105$ 以äžã®æ£æŽæ°ã¯, $3,5,7$ ã§å²ã£ãäœããããããå®ããããšã§äžæã«å®ãŸãããšã«çæãã.\\\r\nã$S$ ã«ãããŠ, $3,5,7$ ã®åæ°ã¯ããããé«ã
äžã€ã§ãããã, ãããã®åæ°ã§ããªãå
ã«ã€ããŠèãã. ãããã«ã€ããŠ, $3,5,7$ ã§å²ã£ãäœãã¯é«ã
$1,2,3$ éãã§ãããã, $N\\leq 3+1\\times 2\\times 3=9$ ãåŸã.\\\r\nã以äž, $N=9$ ãªãè¯ãéåãæ°ãäžãã. äŸãã° $7$ ã§å²ã£ãäœãã«ã€ããŠ, $(1,6),(2,5),(3,4)$ ã®åãã¢ããäžã€ãã€ãéžæããããšã«ãªã. ãŸã, äŸ... | ãéå $\\{1,2,3,\cdots,105\\}$ ã® $2$ å
以äžãããªãéšåéåã«ã€ããŠ, ã©ã®çžç°ãªã $2$ å
ã«ã€ããŠããã®åã $105$ ãšäºãã«çŽ ã§ãããšã, ããã**è¯ã**éåãšåŒã³ãŸã. è¯ãéåã®èŠçŽ æ°ãšããŠããåŸãæ倧å€ã $N$ ãšãããšã, $N$ å
ãããªãè¯ãéåã¯ããã€ãããŸããïŒ |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/288 | C | OMC048(C) | 500 | 72 | 125 | [
{
"content": "ã$39305$ ç§åŸãŸã§ã« $X$ ã $k$ å移åãã確ç㯠${}\\_{39305}\\textrm{C}\\_{k}\\/2^{39305}$ ã§ãããã, æ±ãã確ç $P$ ã¯\r\n$$P=\\dfrac{1}{2^{39305}}\\left({}\\_{39305}\\textrm{C}\\_{0}+{}\\_{39305}\\textrm{C}\\_{4}+\\cdots+{}\\_{39305}\\textrm{C}\\_{39304}\\right)$$\r\nãšããã§, äºé
å®çãã以äžã®ååŒãããããæç«ããïŒ\r\n$$\\begin{alignedat}... | ãæ£æ¹åœ¢ã®ããé ç¹ã«äžã€ã®ç²å $X$ ããã, ã¡ããã© $1$ ç§çµã€ããšã« $50\\%$ ã®ç¢ºçã§æèšåãã«é£ã®é ç¹ã«ç§»åã, æ®ãã® $50\\%$ ã®ç¢ºçã§ç§»åããªãæ§è³ªããã¡ãŸã. $X$ ã $39305.5$ ç§åŸã«å
ãšåãé ç¹ã«ãã確çã¯, äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããŸã. $a$ ã®ãã€æå°ã®çŽ å æ°ãš, $a$ ãããã§å²ãåããæ倧ã®åæ°ã®**ç©**ãæ±ããŠãã ãã. \
ããªã, ç²å㯠(é£ç¶çã§ã¯ãªã) ç¬éçã«ç§»åãããã®ãšããŸã. |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/1611 | D | OMC048(D) | 600 | 69 | 81 | [
{
"content": "ã挞ååŒãã以äžãåŸããã, $\\alpha^2+\\beta^2=1$ ã§ãã.\r\n$$a_{n+1}^2+b_{n+1}^2=(\\alpha^2+\\beta^2)(a_n^2+b_n^2)$$\r\nããªãã¡, ãã $\\theta$ ã«ãã£ãŠ $(\\alpha,\\beta)=(\\cos\\theta,\\sin\\theta)$ ãšãã, ãã $\\theta_n$ ã«ãã£ãŠ\r\n$$(a_n,b_n)=5\\sqrt{26}(\\cos\\theta_n,\\sin\\theta_n)$$\r\nãšè¡šãã° $\\theta_{n+1}\\equiv\\thet... | ãç§ natu_math ã®èªçæ¥ã¯ $2004$ 幎 $5$ æ $25$ æ¥ã§ã. ãšããã§, å®æ° $\alpha,\beta$ ã«å¯Ÿã,
$$(a_0,b_0)=(5,25),\quad a_{n+1}=\alpha a_{n}-\beta b_{n},\quad b_{n+1}=\beta a_{n}+\alpha b_{n}\quad (n=0,1,\cdots)$$
ã«ãã£ãŠæ°å $\\{a_n\\},\\{b_n\\}$ ãå®ãããš, $(a_x,b_x)=(a_0,b_0)$ ãªãæ£æŽæ° $x$ ãååšã, ãã®æå°å€ã¯ $2004$ ã§ãã.\
ããã®ã³ã³ãã¹ã㯠$2021$ 幎 $9$ æ $30$ æ¥... |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/1933 | E | OMC048(E) | 700 | 41 | 66 | [
{
"content": "ã$Q(x)=(x-1)P(x)$ ãšãããš, åé¡æã®æ¡ä»¶ãã以äžã®ããã«è¡šããïŒ\r\n$$Q(x)=\\sum^t_{i=1}x^{A_i}-\\sum^s_{i=1}x^{B_i}-8$$\r\nãã ã, $A_i$ ããã³ $B_i$ ã«éè€ã¯ãªã, $A_1=17$ ãšãã. ãŸã $Q(1)=0$ ãã $t=s+8$ ã§ãã, ãã®ãšã $Q$ ãšåé¡ã®ååã®æ¡ä»¶ãã¿ãã $P$ ã¯äžå¯Ÿäžã«å¯Ÿå¿ãã. ããã«, ç°¡åãªè©äŸ¡ã«ãã£ãŠ $P$ 㯠$1$ ãæ ¹ã«ãããªããã, ä»¥äž $1$ 以å€ã®æçæ°æ ¹ãã〠$Q$ ã«ã€ããŠèããã°ãã. æçæ°æ ¹ $q$ ã®åè£ã¯ $-1... | ãæŽæ°ä¿æ°ã® $x$ ã® $16$ 次å€é
åŒ
$$P(x)=a_{16}x^{16}+a_{15}x^{15}+\cdots+a_1x+a_0$$
ã«ã€ããŠïŒ$a_{16} =1,~ a_0=8$ ã§ããïŒ$i=1,2,\ldots,16$ ã«å¯Ÿã㊠$|a_i - a_{i-1}|\leq 1$ ãæç«ããŠããŸãïŒãã®ãã㪠$P$ ã§ãã£ãŠïŒ$x$ ã®æ¹çšåŒ $P(x)=0$ ãæçæ°è§£ããã€ãã®ã¯ããã€ãããŸããïŒ |
OMC048 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc048/tasks/1609 | F | OMC048(F) | 800 | 11 | 32 | [
{
"content": "ãäžè¬ã« $BC=a,AC=b,AB=c,ID=r\\\\,(b\\neq c)$ ãšããŠèãã. ãã ã, ç¹ $I$ 㯠$ABC$ ã®å
å¿ã§ãã. \\\r\nãç°¡åãªè§åºŠèšç®ã«ãã£ãŠ $P$ 㯠$IBC$ ã®åå¿ã§ããããšãåãããã, $PD$ 㯠$\\omega$ ã®çŽåŸã§ãã. ãããã, çŽç· $AP$ ãš $BC$ ã®äº€ç¹ã $S_A$ ãªã©ãšããã°, well-known factãšã㊠$BS_A=CD$ ã§ãã. äžæ¹ã§ $BS_C=CS_B=a$ ã§ãããã, ãããããCevaã®å®çãçšããŠç«åŒããã°, $3a=b+c$ ãæç«ããããšã確èªã§ãã.\\\r... | ã$AB=18416,AC=17296$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠ, å
æ¥åã $\omega$ ãšã, $\omega$ äžã®ç¹ $P$ ã以äžãã¿ãããŸãïŒ
$$\angle PBC=90^{\circ}-\frac{1}{2}\angle C,\quad \angle PCB=90^{\circ}-\frac{1}{2}\angle B$$
$\omega$ ãš $BC$ ã®æ¥ç¹ã $D$ ãšã, çŽç· $AD$ ã $\omega$ ãšåã³äº€ããç¹ã $K$, çŽç· $AD$ ãäžè§åœ¢ $BCK$ ã®å€æ¥åãšåã³äº€ããç¹ã $L$ ãšããŸã. $LD=LX,\angle PXL=90^\circ$ ãã¿ããç¹ $... |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/1995 | A | OMC047(A) | 100 | 242 | 250 | [
{
"content": "ãå $4$ æ°ã®éžã³æ¹ã«å¯Ÿã, æ¡ä»¶ãã¿ãã $4$ æ¡ã®æŽæ°ã $1$ ã€ãã€å¯Ÿå¿ãããã, æ±ããå€ã«ã€ã㊠${}_9 \\mathrm{ C }_4=\\bf{ 126 }$ éã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc047/editorial/1995"
}
] | ã$1$ ä»¥äž $9$ 以äžã®æŽæ°ãã, çžç°ãªã $4$ ã€ã䞊ã¹ãŠ $4$ æ¡ã®æ£æŽæ°ãäœããšã, 次ã®æ¡ä»¶ãã¿ãããã®ã¯ããã€ãããŸããïŒ
- åã®äœ, çŸã®äœ, åã®äœ, äžã®äœã®æ°ããããã $a,b,c,d$ ãšãããšã, $a \lt b \lt c \lt d$ ãæç«ãã. |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/1836 | B | OMC047(B) | 200 | 229 | 238 | [
{
"content": "ãæ¡ä»¶ãã $(x+3)(y+1)=2025$ ã§ãã, çžå ã»çžä¹å¹³åã®é¢ä¿ãã\r\n$$x+y+4\\geq 2\\sqrt{(x+3)(y+1)}=90$$\r\nãåŸã. éã« $(x,y)=(42,44)$ ã§çå·ãæç«ãããã, æ±ããæå°å€ã¯ $\\textbf{86}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc047/editorial/1836"
}
] | ã$xy+x+3y=2022$ ãªãæ£ã®å®æ° $x,y$ ã«ã€ããŠ, $x+y$ ã®ãšãåŸãæå°å€ãæ±ããŠãã ãã. |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/1921 | C | OMC047(C) | 300 | 179 | 216 | [
{
"content": "ãäžè¬ã« $2n$ è§åœ¢ã§èãã. çŽè§äžè§åœ¢ã®éžæã«ããã£ãŠ, æ蟺ãšãªãçŽåŸãåºå®ããããšã§, æ±ããã¹ã³ã¢ã®ç·åã¯\r\n$$\\begin{aligned}\r\nM&= \\displaystyle \\sum_{t=1}^n \\left(\\displaystyle \\sum_{i=1}^{2n}(i+2t+n) - (3t+n) - (3t+2n) \\right ) \\\\\\\\\r\n&=6n^3-3n^2-3n\r\n\\end{aligned}$$\r\nãã ãåŸãã® $2$ é
ã¯, çŽåŸã®äž¡ç«¯ã $3$ ç¹ç®ãšããŠéžæããå¯èœæ§ãæžããŠãã. \r\n\r\... | ãããæ£ $100$ è§åœ¢ $P_1P_2\cdots P_{100}$ ã«ãããŠ, $3$ é ç¹ãéžãã§ã§ããäžè§åœ¢ã®**ã¹ã³ã¢**ãããããé ç¹ã®æ·»åçªå·ã®ç·åã§å®ããŸã. ãã®ãšã, **çŽè§äžè§åœ¢**ãã¹ãŠã«ã€ããŠã¹ã³ã¢ã®ç·åãæ±ããŠãã ãã. \
ããã ã, åãäžè§åœ¢ã«ã€ããŠ, é ç¹çªå·ã®é åºãç°ãªããã®ã¯åºå¥ããŠæ°ããªããã®ãšããŸã. |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/2018 | D | OMC047(D) | 300 | 138 | 204 | [
{
"content": "ã以äžã®èŠé ã§, $2$ å以äžã® $2$ ãšæ®ããã¹ãŠ $3$ ãšããç¶æ³ ($\\textrm{mod}\\ 3$ ãèããã°ããã¯äžæ) ã«åž°çã§ããïŒ\r\n\r\n- $1$ ãå«ãå Žåã¯, é©åœãªãã®ã« $1$ å ç®ãã.\r\n- $2$ ã $3$ åå«ãå Žåã¯, ããã $2$ åã® $3$ ã«çœ®ãæãã.\r\n- $4$ ãå«ãå Žåã¯, ããã $2$ åã® $2$ ã«çœ®ãæãã.\r\n- $5$ 以äžã®æŽæ° $a$ ãå«ãå Žåã¯ããã $2,a-2$ ã«çœ®ãæãã.\r\n\r\nãããã£ãŠ, $2021=2+3\\times 673$ ãã $M=2\\time... | ãç·åã $2021$ ã§ãã**çžç°ãªããšã¯éããªã**ããã€ãã®æ£æŽæ°ã«å¯ŸããŠïŒãã®ç·ç©ãšããŠããåŸãæ倧å€ã $M$ ãšããŸãïŒ$M$ ãçŽ å æ°å解ãããšãïŒææ°ã®ç·åãæ±ããŠãã ããïŒäŸãã° $2\times5^{20}\times 7^{21}$ ãªãã° $42$ ã解çããŠãã ããïŒ |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/1960 | E | OMC047(E) | 400 | 47 | 126 | [
{
"content": "ã$\\angle CDP$ ã®äºçåç·ãš $CP$ ã®äº€ç¹ã $Q$ ãšãããš, $CQ:QP=10:7$ ã§ãã. ãŸã\r\n$$\\angle QDC=\\angle BDC\\/2=\\angle BAC\\/2=\\angle ACB=\\angle ADB$$\r\nããã³ $\\angle QCD=\\angle ACD=\\angle ABD$ ããäžè§åœ¢ $ABD$ ãš $QCD$ ã¯çžäŒŒã§ãããã $AD:DQ=7:5$ ã§ãã. ãã㯠$AP:PQ$ ã«çãããã, 以äžãã以äžã®æ¯ãåŸã.\r\n$$AP:PQ:QC=49:35:50$$\r\nãã£ãŠæ¹ã¹ãã®å®... | ãåã«å
æ¥ããåè§åœ¢ $ABCD$ ã«ãããŠ, 察è§ç·ã®äº€ç¹ã $P$ ãšãããšãã, 以äžã®æ¡ä»¶ãæç«ããŸããïŒ
$$BP=PD=7,\quad CD=10,\quad \angle BAC=2\angle ACB$$
ãã®ãšã, $CP$ ã®é·ãã® $2$ ä¹ãæ±ããŠãã ãã. |
OMC047 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc047/tasks/2015 | F | OMC047(D2) | 500 | 75 | 138 | [
{
"content": "ãããã€ãã®æ£æŽæ°ãå°ããé ã« $a_1,a_2,\\cdots,a_n$ ãšãã. ãŸã, 以äžã®èªæãªäžçåŒã«æ³šæããïŒ\r\n$$m\\gt n+1 \\implies (m-1)(n+1)\\gt mn$$\r\nãããç¹°ãè¿ãå©çšããã° $a_1$ ä»¥äž $a_n$ 以äžã®æ£æŽæ°ã«ãããŠ, ç»å Žããªããã®ã¯é«ã
$1$ ã€ãšããŠããããšãããã. ãŸã, å
åãšåæ§ã«ã㊠(ãã ã倧å°é¢ä¿ã厩ããªããã泚æãã) $a_1=2$ ãŸã㯠$a_1=3$ ã§ãããšããŠãã. \\\r\nããªã, $a_1=4$ ã«ã€ããŠã¯ $(4,a_2)\\to(2,2,a_2)\\to(2,... | ãç·åã $2021$ ã§ãã**çžç°ãªã**ããã€ãã®æ£æŽæ°ã«å¯ŸããŠïŒãã®ç·ç©ãšããŠããåŸãæ倧å€ã $M$ ãšããŸãïŒ$M$ 㯠$2$ ã§æ倧äœåå²ãåããŸããïŒ |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1815 | A | OMC046(A) | 100 | 240 | 241 | [
{
"content": "ã以äžã«ç¹ã®äœçœ®é¢ä¿ã瀺ã. ç·å $EF$ ã§åå²ããŠèãããšïŒåè§åœ¢ $AEFD$ ããã³ $BEFC$ ã¯å¹³è¡å蟺圢ã§ãã, äžè§åœ¢ $EFG,EFH$ ã¯ããããã® $1\\/4$ ã«ããã. ãã£ãŠ, æ±ãã $ABCD$ ã®é¢ç©ã¯ $7\\times4=\\textbf{28}$ ã§ããïŒ \r\n\r\n",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests... | ãå¹³è¡å蟺圢 $ABCD$ ã«ãããŠ, 蟺 $AB,CD$ ã®äžç¹ããããã $E,F$ ãšã, ç·å $AF$ ãš $DE$ ã®äº€ç¹ã $G$, $BF$ ãš $CE$ ã®äº€ç¹ã $H$ãšãããš, åè§åœ¢ $EGFH$ ã®é¢ç©ã¯ $7$ ã§ãã. ãã®ãšã, $ABCD$ ã®é¢ç©ãæ±ããŠãã ãã. |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1370 | B | OMC046(B) | 300 | 150 | 229 | [
{
"content": "ã$n=500$ ãšã, äžè¬ã« $2n\\times 2n$ ã®ãã¹ç®ã« $2n^2-1$ åã®ç³ãæ¡ä»¶ãã¿ããããã«çœ®ãããšãèãã. åè¡ã«ã¯é«ã
$n$ åããç³ã眮ããªãããšãã, ãã $k$ ã«ã€ããŠäžãã $k$ è¡ç®ã« $n-1$ å, æ®ãã® $2n-1$ è¡ã« $n$ åã®ç³ã眮ããããªã. 察称æ§ãã $k\\leq n$ ã®å Žåãèã㊠$2$ åããã°ãã.\\\r\nãããã§, ãã¹ç®å
šäœãçœé»ã®åžæŸæš¡æ§ã«å¡ãåããããšãèããã. ãã®ãšã $k$ è¡ç®ä»¥äžã®ç³ã¯ãã¹ãŠåãè²ã®ãã¹ã«çœ®ãããŠãã. äžè¬æ§ã倱ãããããé»è²ã§ããå Žåã®ã¿ãèã㊠$2$ å... | ã$1000\times 1000$ ã®ãã¹ç®äžã«, 以äžã®æ¡ä»¶ãã¿ããããã« $499999$ åã®ç³ã眮ãæ¹æ³ã¯äœéããããŸããïŒ
- åãã¹ã«çœ®ãããç³ã¯é«ã
$1$ åã§ãã.
- äºãã«é£ãåãä»»æã® $2$ ãã¹ã®çµã«ã€ããŠ, ãã®äž¡æ¹ã«ç³ã眮ãããŠããããšã¯ãªã. |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1816 | C | OMC046(C) | 300 | 120 | 164 | [
{
"content": "ã$xa$ å¹³é¢ã§èãããš, æŸç©ç· $x^2-2x-a-3=0$ ãšé å $x^2-4x+a^2-5\\leq0$ ãå
±æç¹ãã〠$a$ ã®ç¯å²ãæ±ããã°è¯ãïŒ2åŒãé£ç«ãããŠè§£ããš, å¢çã®äº€ç¹ãšããŠ\r\n$$(x,a)=(-1,0),(2,-3),\\left(\\dfrac{3\\pm\\sqrt{17}}{2},\\dfrac{1\\pm\\sqrt{17}}{2}\\right)$$\r\n(埩å·åé ) ãåŸããããã, ã°ã©ããæžããšä»¥äžã®ããã«ãªãããšããããïŒ\r\n\r\n ãšè¡šããã®ã§, $s+t+u$ ã解çããŠãã ãã. |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1817 | D | OMC046(D) | 400 | 142 | 192 | [
{
"content": "ãäžè¬ã« $N=500$ ãšã, ãŸãç¹ $(0,0,0)$ ãã移åãç¹°ãè¿ããŠç¹ $(2N,2N,2N)$ ãŸã§å°éããæ¹æ³ãèãã. $y$ 座æšã®å€åã«çç®ãããšïŒç§»å $A$ ãšç§»å $B$ ãåãããŠã¡ããã© $2N$ åè¡ãå¿
èŠããã. åæ§ã«ããŠ, çµå±ã¯ããããã®ç§»åã $N$ åãã€è¡ãå¿
èŠãããããšãããã. ãããã£ãŠ, ãã®åé¡ã¯æ¬¡ã®ããã«èšãæããããšãã§ãã.\r\n\r\n- èšå· $A,B,C$ ããããã $N$ åãã€ãã. ããããäžåã«äžŠã¹ãæ¹æ³ã®ãã¡, ç°ãªãèšå·å士ãé£ãåã£ãŠããç®æãã¡ããã© $4$ ç®æã§ãããããªãã®ã¯äœéããïŒ\r\n... | ã$xyz$ 空éã«ãããŠ, ç¹ $(0,0,0)$ ãã次ã®ããããã®ç§»åãç¹°ãè¿ããŠç¹ $(1000,1000,1000)$ ãŸã§å°éãã移åçµè·¯ãèããŸã. ãã®ãããªç§»åçµè·¯ã®ãã¡, çŽåãšç°ãªã移åæ¹æ³ãçšããåæ°ãã¡ããã© $4$ åãšãªããããªãã®ã¯äœéããããŸããïŒ
- 移å $A$ : $(x,y,z)$ ãã $(x+1,y+1,z)$ ã«ç§»åãã.
- 移å $B$ : $(x,y,z)$ ãã $(x,y+1,z+1)$ ã«ç§»åãã.
- 移å $C$ : $(x,y,z)$ ãã $(x+1,y,z+1)$ ã«ç§»åãã.
äŸãã°, $AAABBCAABBB$ ãšç§»åãããšã, çŽåãšç°ãªã... |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1819 | E | OMC046(E) | 500 | 70 | 98 | [
{
"content": "ã$x,y,z$ ã¯ãããã $0$ ã§ãªãããšã容æã«åãã. äžåŒãå€åœ¢ããããšã§ä»¥äžãåŸã.\r\n$$x^2y^2z^2-2xyz(x+y+z)+x^2+y^2+z^2+2(xy+yz+zx)=102^2$$\r\näžæ¹ã§, äžåŒãå€åœ¢ããããšã§ä»¥äžãåŸã.\r\n$$x^2y^2+y^2z^2+z^2x^2+2xyz(x+y+z)-2(xy+yz+zx)+1=91^2$$\r\nãããã蟺ã
足ãåãããããšã§, 以äžãåŸã.\r\n$$(x^2+1)(y^2+1)(z^2+1)=5\\times 37\\times 101=(2^2+1)(6^2+1)(10^2+1)$$\r\n... | ã以äžãã¿ããæŽæ°ã®é åºä»ããçµ $(x,y,z)$ ãã¹ãŠã«ã€ããŠ, $x+y+z$ ã®ç·åãæ±ããŠãã ãã.
$$\begin{cases}xyz=x+y+z+102\\\\xy+yz+zx=92\end{cases}$$ |
OMC046 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc046/tasks/1407 | F | OMC046(F) | 600 | 12 | 55 | [
{
"content": "ã$90^\\circ+\\angle BAC\\/2=\\angle BIC=180^\\circ-\\angle BAC$ ãã $\\angle BAC=60^\\circ$ ã§ãã. ãã®ãšãæ£åŒŠå®çãã $BC=18$ ã§ãã, $AB=x,AC=y$ ãšããã°äœåŒŠå®çãã $x^2-xy+y^2=18^2$ ã§ãã. ãŸã $\\angle DAI=30^\\circ$ ã§ãããã, äžè§åœ¢ $ADI$ ã«å¯Ÿããæ£åŒŠå®çãã $DI=2\\sqrt{3}$ ã§ãã.\\\r\nãããã§, è§ã®äºçåç·å®çãã $AE:EC=x:18$ ããã³ $DI:IC=x:(y+18)$ ... | ãå
å¿ã $I$ ãšããäžè§åœ¢ $ABC$ ã«ãããŠ, ãã®å€æ¥åã®ååŸã¯ $6\sqrt{3}$ ã§ã. ãŸã, $AB$ ãš $CI$ ã®äº€ç¹ã $D$, $AC$ ãš $BI$ ã®äº€ç¹ã $E$ ãšãããš, $4$ ç¹ $A,D,E,I$ ã¯ååŸ $2\sqrt{3}$ ã®ååšäžã«ãããŸãã. ãã®ãšã, æ£æŽæ° $p,q$ ã«ãã£ãŠ $AB+AC=p+\sqrt{q}$ ãšè¡šããã®ã§, $p+q$ ã®å€ã解çããŠãã ãã.\
ããã ã, $XY$ ã§ç·å $XY$ ã®é·ããè¡šããŸã. |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/239 | A | OMC045(A) | 100 | 186 | 252 | [
{
"content": "ãäºã€ã®è§£ã $pi$ ããã³ $qi$ ãšããã°, $p,q$ 㯠$x$ ã®äºæ¬¡æ¹çšåŒ\r\n$$(x-4)^2=16-n$$\r\nã® $2$ 解ã§ãã. ãããã£ãŠ, ãããçžç°ãªãäºã€ã®å®æ°è§£ããã€ããšãå¿
èŠååæ¡ä»¶ã§, ãã㯠$n\\lt 16$ ãšåå€ã§ãã. 以äžãã, æ±ããç·å㯠$1+2+\\cdots+15=\\textbf{120}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc045/editorial/239"
}
] | ã$n$ ãæ£ã®æŽæ°ïŒ$i$ ãèæ°åäœãšããŸãïŒ$z$ ã«ã€ããŠã®äºæ¬¡æ¹çšåŒ
$$z^2-8iz=n$$
ã $2$ ã€ã®çžç°ãªãçŽèæ°ã®è§£ãæã€ãšãïŒ$n$ ãšããŠããããå€ã®ç·åãæ±ããŠãã ããïŒãã ãïŒããã§ã¯**çŽèæ°**ãšã¯ïŒ$0$ ã§ãªãå®æ° $a$ ã«ãã£ãŠ $ai$ ãšè¡šãããæ°ããããã®ãšããŸãïŒ |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/1371 | B | OMC045(B) | 200 | 192 | 211 | [
{
"content": "ã $\\ell$ ãåºå®ããŠ, $f$ ãäžããå
æºã®äœçœ®ã $A_M$, $g$ ãäžããå
æºã®äœçœ®ã $A_m$, æ£ã®äžç«¯ã $B$, $R$ ããéçŽæ¹åã«é«ã $30$ ã¡ãŒãã«ã®ç¹ïŒå£ã®äžç«¯ïŒã $C$, ç¹ $A_m$ ã«å
æºã眮ãããšãã« $W$ ã«æ ãç¹ $B$ ã®åœ±ã®äœçœ®ã $K$ ãšãã. ç¹ $A_M$ ã«å
æºã眮ãããšãã« $W$ ã«æ ãç¹ $B$ ã®åœ±ã®äœçœ®ã¯ $R$ ã§ãããã, æ£ã®é·ãã«ãããäžè§åœ¢ $A_MA_mB$ ãš $RKB$ ã¯çžäŒŒã§ãã, ãããã以äžã®ããã«è©äŸ¡ã§ãã. éã«æããã«çå·ã¯å®çŸå¯èœã§, ãããæ±ããæ倧å€ã§ãã.\r\n$$... | ãæ°Žå¹³ãªå°é¢äžã« $3$ ç¹ $P,Q,R$ ãåäžçŽç·äžã«ãã®é ã«äžŠãã§ãã, $PQ,QR$ éã®è·é¢ã¯ãããã $16$ ã¡ãŒãã«, $24$ ã¡ãŒãã«ã§ã. ãŸã, ç¹ $R$ ãéãçŽç· $PQ$ ã«åçŽãª, é«ã $30$ ã¡ãŒãã«ã»é·ãç¡éã®å£ $W$ ãç«ã£ãŠããŸã.\
ãããŸ, ç¹ $P$ ããéçŽæ¹åã«é«ã $x$ ã¡ãŒãã« ($x\geq0$) ã®å°ç¹ã«ç¹å
æºã眮ã, ç¹ $Q$ ã«éçŽãªé·ã $\ell$ ã¡ãŒãã« ($\ell\gt 0$) ã®æ£ãç«ãŠãŸã. ãã®æ£ã®äžç«¯éšåã®ç¹å
æºã«ãã圱ã, $W$ ã«æ ããã㪠$x$ ã®æ倧å€ã $f(\ell)$, æå°å€ã $g(\ell)$ ãšãããšã, $... |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/241 | C | OMC045(C) | 200 | 153 | 212 | [
{
"content": "ãæ±ããç·åã $S$ ãšããã°, 以äžã®çåŒãæç«ããããšã容æã«ããã.\r\n$$\\left(\\sum_{n=1}^{10}n\\right)^3=6S+3\\left(\\sum_{n=1}^{10}n\\right)\\left(\\sum_{n=1}^{10}n^2\\right)-2\\sum_{n=1}^{10}n^3$$\r\nãããã, $S=\\textbf{18150}$ ãšèšç®ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc045/editoria... | ã$1$ ä»¥äž $10$ 以äžã®æŽæ°ããçžç°ãªã $3$ ã€ãéžã¶ããšãèããŸã. ãã®ãããªéžã³æ¹ãã¹ãŠã«ã€ããŠ, $3$ æ°ã®ç©ã®ç·åãæ±ããŠãã ãã. |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/255 | D | OMC045(D) | 300 | 138 | 191 | [
{
"content": "ã$a\\leq c\\leq a$ ãããããã, ç¹ã«çå·ãæç«ãã. ããªãã¡æ¡ä»¶ã¯ \r\n$$b\\leq d\\leq 1000-a$$\r\nãã㧠$a$ ãåºå®ã, $k=1000-a$ ãšãããªããã°, çµ $(b,d)$ ãšããŠããåŸããã®ã¯ $\\dfrac{k^2+k}{2}$ åã§ããããšã容æã«ããããã, æ±ããå Žåã®æ°ã¯\r\n$$ \\sum_{k=1}^{999}\\dfrac{k^2+k}{2}=\\dfrac{999\\times1000\\times1001}{6}=\\textbf{166666500}$$",
"text": "å
¬åŒè§£... | ã以äžã®äžçåŒãæºããæ£æŽæ°ã®çµ $(a,b,c,d)$ ã¯ããã€ãããŸããïŒ
$$a+b\leq b+c\leq c+d\leq d+a\leq1000$$ |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/1694 | E | OMC045(E) | 400 | 45 | 88 | [
{
"content": "ã$(1+x)^n(1+x)^n(1+x)^n=(1+x)^{3n}$ ã«ã€ããŠ, 䞡蟺㧠$x^n$ ã®ä¿æ°ãæ¯èŒããããšã§ä»¥äžãåŸãïŒ\r\n$$\\sum_{p+q+r=n}\\binom{n}{p}\\binom{n}{q}\\binom{n}{r}=\\binom{3n}{n}$$\r\näžæ¹ã§, 以äžã®çåŒãæç«ããããšããããïŒ\r\n$$\\frac{1}{p!q!r!(p+q)!(q+r)!(r+p)!}=\\frac{1}{(10!)^3}\\binom{10}{p}\\binom{10}{q}\\binom{10}{r}$$\r\nãããã£ãŠ, $S$ ã«ã€ããŠä»¥äžã®... | ã$p+q+r=10$ ãªãéè² æŽæ°ã®çµ $(p,q,r)$ ãã¹ãŠã«ã€ããŠ, 以äžã®å€ã®ç·åã $S$ ãšããŸã.
$$\frac{1}{p!q!r!(p+q)!(q+r)!(r+p)!}$$
$S$ ãæ¢çŽåæ°ã«è¡šçŸãããšã, ãã®ååãšãªãæ£æŽæ°ã解çããŠãã ãã. |
OMC045 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc045/tasks/295 | F | OMC045(F) | 400 | 85 | 96 | [
{
"content": "ã$\\angle AXC=90^\\circ$ ããç¹ $X$ ã¯æ£æ¹åœ¢ $ABCD$ ã®å€æ¥åã®å£åŒ§ $AB$ äžãåã. ãããã\r\n$$\\angle AXD=\\angle ABD=45^\\circ=\\angle AXZ$$\r\nã§ãããã $D,Z,X$ ã¯åäžçŽç·äžã«ãã, ããã« $A$ ãš $Y$ ã¯ãã®çŽç·ã«é¢ããŠå¯Ÿç§°ã§ãã. ãããã£ãŠ,\r\n\r\n- $DA=DY$ ãã, ç¹ $Y$ 㯠$D$ ãäžå¿ãšã $A,C$ ãéãåã®å£åŒ§ $AC$ äž (æ²ç· $K$ ãšãã) ãåã.\r\n- $\\angle AZD=135^\\circ$ ãã... | ãé¢ç© $2021$ ã®æ£æ¹åœ¢ $ABCD$ ããã, $4$ ç¹ $P,X,Y,Z$ ã以äžã®æ¡ä»¶ãã¿ãããŸã.
- $P$ ã¯èŸº $AB$ äžã«ãã.
- $X$ ãš $Y$ ã¯çŽç· $CP$ äžã«ãã, $3$ ç¹ $C,Y,X$ ã¯ãã®é ã«äžŠã¶.
- åè§åœ¢ $AXYZ$ ã¯æ£æ¹åœ¢ã§ãã.
ã$P$ ã蟺 $AB$ äžãåããšã, ç·å $XZ$ ã®ééããé åãšç·å $YZ$ ã®ééããé åã®å
±ééšåã®é¢ç©ãæ±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. |
OMC044 (Wolfram Cup) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc044/tasks/1 | A | OMC044(A) | 200 | 219 | 230 | [
{
"content": "ãããç¹ã«ã€ããŠ, è·é¢ã $1$ 以äžã®ç¹ãå°ãªããšã $3$ ã€ååšãããš, æ¡ä»¶ãããããã® $3$ ç¹ã®éã®è·é¢ã¯ãããã $1$ æªæºã§ããã, ãã®ãšãããã $3$ ç¹ãããªãäžè§åœ¢ã¯æ¡ä»¶ãã¿ãããªã. ãããã£ãŠ, åç¹ã«ã€ããŠè·é¢ã $1$ 以äžã®ç¹ã¯é«ã
$2$ ã€ã§ãã, åæ§ã«ããŠåç¹ã«ã€ããŠè·é¢ã $1$ 以äžã®ç¹ãé«ã
$2$ ã€ã§ãããã, ãã㯠$n\\leq 5$ ãè¡šã. éã«æ£äºè§åœ¢ãèããããšã§ $n=5$ ãé©ããããšãããããã, æ±ããæ倧å€ã¯ $\\textbf{5}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url... | ã次ã®æ¡ä»¶ãã¿ããå¹³é¢äžã®çžç°ãªã $n$ ç¹ãååšãããããª, æ£æŽæ° $n$ ã®æ倧å€ãæ±ããŠãã ããïŒ
- ä»»æã® $3$ ç¹ã®ãªãäžè§åœ¢ã«ã€ããŠ, é·ãã $1$ ãã倧ãã蟺㚠$1$ ããå°ãã蟺ããšãã«ååšãã. |
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