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value | url stringlengths 53 65 | alphabet stringclasses 20
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values | correct int64 0 467 | total int64 0 485 | editorials listlengths 1 6 | task_content stringlengths 28 1.49k |
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OMC147 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc147/tasks/296 | C | OMC147(C) | 200 | 347 | 354 | [
{
"content": "ãæ¡ä»¶ã«ãã $P$ ã® $x$ 座æšã¯ $Q$ ã® $x$ 座æšãã倧ããããšã«çæããïŒçŽç· $PR$ ãš $x$ 軞ãšã®äº€ç¹ã $A$ïŒçŽç· $QR$ ãš $y$ 軞ã®äº€ç¹ã $B$ ãšããã°ïŒäžè§åœ¢ $OAP$ ããã³äžè§åœ¢ $OBQ$ ã®é¢ç©ã¯ãšãã« $24$ ã§ããïŒäžæ¹ã§åè§åœ¢ $OARB$ ã¯é·æ¹åœ¢ã§ããããïŒçŽç· $OR$ ã¯åè§åœ¢ $OPRQ$ ã®é¢ç©ãäºçåããïŒããªãã¡çŽç· $OR$ ãåŒ $y=\\dfrac{x}{2}$ ã§è¡šãããããšããïŒ$R$ ã®åº§æšã¯ $(12,6)$ ã§äžãããïŒæ±ããã¹ã $P$ ã® $x$ 座æšã¯ $\\textbf{12}$ ã§ã... | ã$O$ ãåç¹ãšããçŽäº€åº§æšå¹³é¢ã«ãããŠïŒæ²ç· $xy=48\ (x\gt 0)$ äžã« $2$ ç¹ $P,~Q$ ããããŸãïŒ$P$ ãéã $y$ 軞ã«å¹³è¡ãªçŽç·ãšïŒ$Q$ ãéã $x$ 軞ã«å¹³è¡ãªçŽç·ã®äº€ç¹ $R$ ãæ²ç· $xy=72$ äžã«ããïŒçŽç· $y=\dfrac{x}{2}$ ãåè§åœ¢ $OPRQ$ ã®é¢ç©ãäºçåãããšãïŒ$P$ ã® $x$ 座æšãæ±ããŠãã ããïŒ |
OMC147 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc147/tasks/283 | D | OMC147(D) | 300 | 127 | 307 | [
{
"content": "ãç·åã®é·ãã $1$ ãšããŠããïŒ$r$ ã $1$ æªæºã®æ£ã®æçæ°ãšãããšãïŒå·Šç«¯ãã $r$ ã®äœçœ®ã«ããç¹ã«åããŠæå°ãä»ãã®ã¯ïŒ$r$ ãæ¢çŽåæ°ã§è¡šãããšãã®åæ¯ã $a$ ãšãããšã $a$ åç®ã®æäœã§ããïŒåŸã£ãŠïŒ$f(n)$ 㯠$n$ 以äžã®æ£æŽæ°ã§ $n$ ãšäºãã«çŽ ãªãã®ã®åæ°ã«çãã(Eulerã®ããŒã·ã§ã³ãé¢æ°)ïŒ$n$ ã®çŽ å æ°å解ã $n=p_1^{a_1}\\cdots p_k^{a_k}$ ãšãããšã\r\n$$f(n)=(p_1^{a_1}-p_1^{a_1-1})\\cdots(p_k^{a_k}-p_k^{a_k-1})$$\r\nãæãç«ã€ïŒ... | ãç·å $S$ ãããïŒä»¥äžã®æäœã $n=2,3,4,\ldots$ ã®é ã«è¡ããŸãïŒ
- $S$ ã $n$ çåãã $n-1$ åã®ç¹ããããã«ã€ããŠïŒããã«æå°ãä»ããŠããªããšãæ°ãã«æå°ãä»ããïŒ
- ãã®ãšãïŒæ°ãã«ä»ããæå°ã®æ°ã $f(n)$ ãšããïŒ
äŸãã°ïŒ$f(2)=1$ïŒ$f(3)=2$ïŒ$f(4)=2$ïŒ$f(5)=4$ïŒ$f(6)=2$ ã§ãïŒ\
ã$f(x)=32$ ãã¿ãã $2$ 以äžã®æŽæ° $x$ ã®ç·åãæ±ããŠãã ããïŒ |
OMC147 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc147/tasks/3355 | E | OMC147(E) | 400 | 74 | 137 | [
{
"content": "**è£é¡.**ãä»»æã®æ£ã®å¶æ° $m$ ã«ã€ããŠ\r\n\r\n$$\r\n\\sum_{k = 0}^{m - 1} \\cos \\frac{2k\\pi}{m} = 0.\r\n$$\r\n\r\n**蚌æ.**ã$m$ ãå¶æ°ã§ããããšãã次ã®ããã«èšç®ã§ããïŒ\r\n\r\n$$\r\n\\sum_{k = 0}^{m - 1} \\cos \\frac{2k\\pi}{m} = \\sum_{k = 0}^{(m\\/2) - 1} \\bigg(\\cos \\frac{2k\\pi}{m} + \\cos \\bigg(\\frac{2k\\pi}{m} + \\pi\\b... | ã$0 \le a \lt b \lt 123456$ ãªãæŽæ°ã®çµ $(a, b)$ ãã¹ãŠã«ã€ããŠïŒ
$$\biggl(\cos\dfrac{2a\pi}{123456}\biggr)\biggl(\cos\dfrac{2b\pi}{123456}\biggr)$$
ã足ãåãããå€ã® $2$ ä¹ãæ±ããŠäžããïŒ |
OMC147 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc147/tasks/3017 | F | OMC147(F) | 400 | 9 | 66 | [
{
"content": "ãåãã¹ãé ç¹ãšãïŒäžå¿ãè·é¢ $\\sqrt{5}$ ã®é¢ä¿ã«ãã $2$ ãã¹ã®éã«ç¡å蟺ã匵ã£ãã°ã©ã $G_n$ ã«ãããŠïŒEulerè·¯ããã€éšåã°ã©ãã®èŸºæ°ãæ倧åããã°ããïŒãŸãïŒ$G_n$ ã®èŸºæ° $T(n)$ ã«ã€ããŠïŒ$T(n)=4(n-1)(n-2)$ ãæç«ããããšããããïŒããã¯ïŒ$4$ çš®é¡ã®ãåŸããããããããã€èŸºã«ã€ããŠèããããšã§ç¢ºèªã§ããïŒ\\\r\nã$G_3$ ã¯ãã¹ãŠã®é ç¹ã®æ¬¡æ°ãå¶æ°ã§ããããEuleréè·¯ããã€ïŒ$n\\geq 4$ ã«å¯ŸãïŒ$G_n$ ã¯æ¬¡æ° $3$ ã®é ç¹ã $8$ åæã€ããïŒå°ãªããšã $6$ 蟺ãåãé€ããªããã°äžçæžã... | ã$n\times n$ ã®ãã¹ç®ããããŸãïŒ\
ãOMCåã¯ãã€ãã®é§ã $1$ ã€æã£ãŠããïŒãããé©åœãªãã¹ $A_0$ ã«çœ®ããŸãïŒããããïŒä»¥äžã®æ¡ä»¶ãã¿ããããã«**ãã€ãã®ç§»å**ãã¡ããã© $m$ åç¹°ãè¿ããŸãïŒ
- $k=1,\ldots,m$ ã«ã€ããŠïŒãã€ãã $k$ å移åããæç¹ã§äœçœ®ãããã¹ã $A_k$ ãšãããšãïŒ$m$ åã®éå $\\{A_0,A_1\\}, \\{A_1,A_2\\}, \ldots, \\{A_{m-1},A_m\\}$ ã¯çžç°ãªãïŒ
ããã®ãããªç§»åãå¯èœãª $m$ ãšããŠããããæ倧ã®å€ã $M(n)$ ãšãããŸãïŒ\
ã$n=3,4,\ldots,999... |
SOMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/somc001/tasks/1928 | A | SOMC001(A) | 100 | 193 | 218 | [
{
"content": "ãåçŽç· $PO$ äžã« $OR=10$ ãªãç¹ $R$ ããšããšïŒ$PXO$ ãš $PQR$ ã¯çžäŒŒãªäžè§åœ¢ã§ããïŒçžäŒŒæ¯ã¯ $1:6$ ã§ããããïŒç¹ $Q$ ã¯ã€ãã« $RQ=18$ ãã¿ããïŒéã«ãã®ãã㪠$Q$ ã¯ãã¹ãŠæ¡ä»¶ãã¿ããããïŒæ±ããè»è·¡ã¯ååŸ $18$ ã®ååšã§ããïŒè§£çãã¹ãå€ã¯ $\\textbf{18}$ ã§ããïŒäžè¬ã«ïŒãã®èšŒæãšåæ§ã«ããŠåã $1$ ç¹ãäžå¿ãšããŠçžäŒŒæ¡å€§ããŠåŸãããå³åœ¢ã¯ã€ãã«åã§ããããšãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contes... | ãå¹³é¢äžã«ïŒå®ç¹ $O$ ãäžå¿ãšããååŸ $3$ ã®å $C$ ãš$OP=2$ ãªãå®ç¹ $P$ ããããŸãïŒ\
ãç¹ $Q$ ã以äžã®æ¡ä»¶ããšãã«ã¿ãããªããåããšãïŒãã®è»è·¡ã¯ååšããªããŸãïŒãã®ååŸãæ±ããŠãã ããïŒ
- $Q$ ã¯å $C$ ã®å€éšïŒåšäžãå«ãŸãªãïŒã«ããïŒ
- ç·å $PQ$ ãšå $C$ ã®äº€ç¹ã $X$ ãšãããšïŒ$PX:QX=1:5$ ã§ããïŒ |
SOMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/somc001/tasks/3331 | B | SOMC001(B) | 200 | 191 | 219 | [
{
"content": "ãäžå€®ã®ãã¹ç®ã« $4$ ã®åæ°ïŒ$4$ ãŸã㯠$8$ïŒãæžã蟌ãåŠãã§å Žååããè¡ãïŒ\r\n- æžã蟌ããšãïŒä»ã®ãã¹ç®ãžã®æžã蟌ã¿æ¹ã«ãããæ¡ä»¶ãã¿ããããïŒ$2 \\times 8!$ éã.\r\n- æžã蟌ãŸãªããšãïŒæ¡ä»¶ãã¿ãããã㪠$4$ ãš $8$ ã®æžã蟌ã¿æ¹ã¯ $4$ éãã§ããããïŒ$4 \\times 7!$ éãïŒ\r\n\r\n以äžã«ããïŒæ¡ä»¶ãã¿ããæžã蟌ã¿æ¹ã¯ $\\bf{ 100800 }$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/... | ã$3 \times 3$ ã®ãã¹ç®ã®åãã¹ã«ïŒæ¬¡ã®æ¡ä»¶ãã¿ããããã« $1$ ä»¥äž $9$ 以äžã®æŽæ°ãäžåºŠãã€æžã蟌ãæ¹æ³ã¯äœéããããŸããïŒãããã§ïŒå転ãè£è¿ãã«ãã£ãŠäžèŽããæžã蟌ã¿æ¹ãåºå¥ããŸãïŒ
- ã©ã® $2 \times 2$ ã®éšåãã¹ç®ã«ã€ããŠãïŒãã®äžã« $4$ ã®åæ°ãæžããããã¹ãå«ãŸããŠããïŒ
ããªãïŒã$2\times 2$ ã®éšåãã¹ç®ããšã¯ïŒé£æ¥ãã $2$ è¡ãšé£æ¥ãã $2$ åã®å
±ééšåãšãªã $4$ ãã¹ã®ããšããããŸãïŒ |
SOMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/somc001/tasks/1711 | C | SOMC001(C) | 200 | 152 | 212 | [
{
"content": "ã$t=x^2+x+1$ ãšãããšïŒäžåŒã¯ $t$ ã®é¢æ°ãšããŠä»¥äžã®ããã«æžããããããïŒ\r\n$$\\biggl(f(t):=\\biggr)t+1+\\dfrac{1}{9t}$$\r\nãã㧠$t$ 㯠$3\\/4$ 以äžã®å®æ°å€ããšãããšã«çæããïŒããã§ïŒ$a\\gt b\\geq 3\\/4$ ã«ã€ããŠ\r\n$$f(a)-f(b)=\\dfrac{(a-b)(9ab-1)}{9ab}\\gt 0$$\r\nã«ããïŒãã®ç¯å²ã§ $f$ ã¯å調å¢å ã§ããããïŒæ±ããæå°å€ã¯ $f\\left(\\dfrac{3}{4}\\right)=\\dfrac{205}{108}$... | ã$x$ ãå®æ°å
šäœãåããšãïŒä»¥äžã®ãšãããæå°ã®å€ãæ±ããŠãã ããïŒ
$$x^2+x+2+\dfrac{1}{9x^2+9x+9}$$
ãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
SOMC001 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/somc001/tasks/1345 | D | SOMC001(D) | 300 | 113 | 165 | [
{
"content": "ãå æ°å®çã«ããïŒ$P(x)$ 㯠$l+m+n=1345$ ãªãæ£æŽæ° $l, m, n$ ã«ãã£ãŠ\r\n$$P(x)=(x-4)^l (x-6)^m (x-9)^n $$\r\nãšè¡šããïŒç¹ã« $p=2l+m$ ãšããã°ïŒ$3\\leq p\\leq 2687$ ã§ããïŒ$P(x)$ ã®å®æ°é
㯠$-2^p\\times 3^{2690-p}$ ãšè¡šããïŒ\\\r\nãéã« $p$ ã¯ãã®ç¯å²ãã¹ãŠããšãããããïŒç¹ã«æ±ããã¹ãå Žåã®æ°ã¯ $\\textbf{2685}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemat... | ãæé«æ¬¡ä¿æ°ã $1$ ã§ãããããªæŽæ°ä¿æ° $1345$ 次å€é
åŒ $P(x)$ ã«ã€ããŠïŒ$x$ ã®æ¹çšåŒ $P(x)=0$ ãè€çŽ æ°ã®ç¯å²ã§è§£ããšïŒãã®è§£ã¯ $x=4$ïŒ$x=6$ïŒ$x=9$ ã®ã¡ããã© $3$ ã€ãšãªããŸããïŒ$P(x)$ ã®å®æ°é
ãšããŠããããå€ã¯ããã€ãããŸããïŒ |
OMC146 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/1871 | A | OMC146(A) | 300 | 226 | 273 | [
{
"content": "ãéè² æŽæ° $n$ ã«å¯Ÿã $n^{n}$ ã $5$ ã§å²ã£ãããŸãã $f(n)$ ã§è¡šãïŒæŽæ° $q,r\\~(0\\leq r\\leq 4)$ ãçšã $n=5q+r$ ãšè¡šããããšãïŒ$f(n)$ 㯠$r^{5q+r}$ ã $5$ ã§å²ã£ãããŸãã«çããïŒ\\\r\nã$r=0,1,\\ldots,4$ ã«å¯ŸããŠïŒ$q$ ã $0,1,2,\\dots$ ãšåããããšã $r^{5q+r}$ ã $5$ ã§å²ã£ãããŸãã¯\r\n- $r = 0$ ã®ãšãïŒ$0, 0, 0, 0, 0, 0, 0, 0, 0, \\cdots$\r\n- $r = 1$ ã®ãšãïŒ$1, 1,... | ã$2021$ åã®æŽæ° $1, 2, \ldots ,2021$ ããçžç°ãªã $2$ 〠$a\lt b$ ãéžã¶æ¹æ³ã§ãã£ãŠïŒ$a^{a} + b^{b}$ ã $5$ ã§å²ãåãããã®ã¯ããã€ãããŸããïŒ |
OMC146 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/2640 | B | OMC146(B) | 500 | 28 | 104 | [
{
"content": "ã$2$ åæŠãçµãã£ãæç¹ã§ïŒ$2$ é£åäžã®éžæã $1$ 人ïŒ$1$ é£åäžã®åå è
ã $2$ 人ããïŒãã® $3$ 人ã $3$ åæŠã®ç¬¬ $1$ è©Šåã§å¯ŸæŠããïŒ$2$ åæŠãçµãã£ãæç¹ã®æ瞟ã«å¿ããŠïŒä»¥äžã®ããã«æåãããïŒ\r\n\r\n- $x$ïŒ$2$ é£åäžã®éžæãåªåãã確ç\r\n- $y$ïŒ$1$ é£åäžã®éžæãããããåªåãã確ç\r\n- $z$ïŒ$3$ åæŠã®ç¬¬ $2,3$ äŒå Žã«åºå Žããéžæãããããåªåãã確ç\r\n- $w$ïŒ$3$ åæŠã§åŸ
æ©ããéžæãåªåãã確ç\r\n\r\nããã®ãšãïŒæ¬¡ãæãç«ã€.\r\n$$ x=\\frac{1}{3}... | ã$A_1$ ãã $A_{10}$ ãŸã§ã® $10$ 人ã®éžæãïŒæ¬¡ã®èŠé ã§è©Šåãè¡ããŸãïŒ
- äžåºŠã®è©Šåã«ã¯ $3$ 人ãåå ãïŒ$1,2,3$ äœãšéè€ãªãé äœãä»ãïŒ
- $10$ 人ã®éžæã®å®åã¯äºãã«å¯Ÿçã§ïŒåè©Šåã«ã€ã㊠$3$ 人ã®é äœã®çµã¿åãã $6$ éãã¯åçã«çŸããïŒ
- $1$ åæŠã®ç¬¬ $j$ äŒå ŽïŒ$j=1,2,3$ïŒã§ã¯ $A_{3j-2}$, $A_{3j-1}$, $A_{3j}$ ã® $3$ éžæã察æŠãïŒ$A_{10}$ ã¯åŸ
æ©ããïŒ
- $n$ åæŠã®çµæã«å¿ããŠïŒ$n+1$ åæŠã®å²ãæ¯ããäžå³ã®ããã«è¡ãïŒ$n\geq 1$ïŒïŒ
 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/1804 | C | OMC146(C) | 500 | 29 | 64 | [
{
"content": "ã$\\angle A=3\\theta$ïŒ$\\angle PAT=\\alpha$ ãšããã°ïŒ\r\n$$90^\\circ=\\angle ABT=\\angle B+\\theta-\\alpha,\\quad 45^\\circ=\\angle ACT=\\angle C+\\alpha$$\r\nãæç«ããããïŒ$\\angle B+\\angle C+3\\theta=180^\\circ$ ãšããã㊠$\\theta=22.5^\\circ$ïŒããªãã¡ $\\angle A=67.5^\\circ$ ãåŸãïŒ\\\r\nãããã«ããïŒååšè§ãèããããšã§ $\\angle... | ã$BC=6$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠïŒè§ $A$ ã®ïŒå
è§ã®ïŒäžçåç·ãšèŸº $BC$ ã®äº€ç¹ã®ãã¡ïŒ$B,C$ ã«è¿ãæ¹ããããã $P,Q$ ãšããŸãïŒäžè§åœ¢ $ABQ,ACP$ ã®å€æ¥åããããã $\Omega,\Gamma$ ãšãïŒããããã®äžå¿ã $V,W$ ãšãããšãïŒ$\Omega$ ãš $\Gamma$ ã®äº€ç¹ã®ãã¡ $A$ ã§ãªããã® $T$ ã«ã€ããŠïŒ$V$ ã¯ç·å $AT$ äžã«ããïŒ$W$ 㯠$\Omega$ äžã«ãããŸããïŒ$\Gamma$ ãšèŸº $AB$ ã®äº€ç¹ã®ãã¡ $A$ ã§ãªããã®ã $K$ ïŒ$\Omega$ ãšèŸº $AC$ ã®äº€ç¹ã®ãã¡ $A$ ã§ãªããã®ã $L$ ãšãïŒçŽç· $... |
OMC146 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/5048 | D | OMC146(D) | 500 | 16 | 47 | [
{
"content": "ã$2n = 5758, a_0 = 0$ ãšããïŒãŸãïŒæ¬¡ã®è£é¡ã瀺ãïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ**$a_k\\in S$ ãªãã°ïŒ$m$ ã $\\max S_i=a_k$ ãªã $i$ ã®åæ°ïŒ$a_{l}$ ã $S$ ã®èŠçŽ ã§ãã£ãŠ $a_k$ æªæºã®ãã®ã®ãã¡æ倧ã®ãã®ïŒååšããªãå Žå㯠$l=0$ïŒãšãããšãïŒ\r\n$$f(S\\setminus \\\\{a_k\\\\})-f(S)=2m(a_k-a_l)-(2n-k+1)$$\r\nãæãç«ã€ïŒç¹ã«\r\n$$f(S\\setminus \\\\{a_k\\\\})-f(S)\\geq 2(a_... | ãæŽæ° $a_1, a_2, \ldots, a_{5758}$ 㯠$1\leq a_1\lt a_2 \lt\cdots\lt a_{5758}\leq 3001^2$ ãã¿ãããŠããŸãïŒãã®ãšãïŒ$U=\\{a_1,a_2,\ldots, a_{5758}\\}$ ãšãïŒ$U$ ã®éšåéå $S$ ã«å¯ŸãïŒæŽæ° $f(S)$ ã以äžã®ããã«å®ããŸãïŒ
- å $i=1,2,\dots,5758$ ã«å¯ŸãïŒ$S_i$ 㧠$S$ ã®èŠçŽ ã®ãã¡ $a_i$ 以äžã®ãã®å
šäœã®ãªãéåãè¡šããšãïŒ$$f(S)=\sum_{i=1}^{5758}\bigl(\vert S_i \vert-2\max S_i\bigr)$$
ãšã... |
OMC146 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/2539 | E | OMC146(E) | 700 | 11 | 25 | [
{
"content": "**è£é¡1.**ã$BC$ ã®äžç¹ã $M$ ãšãããšãïŒ$4$ ç¹ $X,Y,M,D$ ã¯å
±åã§ããïŒ\\\r\n**蚌æ.**ã$AB \\neq AC$ ãªã®ã§çŽç· $EF$ ãšçŽç· $BC$ ã®äº€ç¹ããšããïŒããã $Z$ ãšãããšïŒæ¹ã¹ãã®å®çãã\r\n$$ZD\\times ZM=ZE\\times ZF=ZB\\times ZC=ZX\\times ZY$$\r\nãã瀺ãããïŒãã ã, $B,C,E,F$ ããã³ $D,E,F,M$ ããããã®å
±åïŒä¹ç¹åïŒãçšãã.\r\n----\r\n**è£é¡2.**ã$\\triangle{ABC}$ ã®å€å¿ã $O$ ãšãããšã... | ã$AB\neq AC$ ãªãéè§äžè§åœ¢ $ABC$ ã«ãããŠïŒ$A$, $B$, $C$ ãã察蟺ã«ããããåç·ã®è¶³ããããã $D,E,F$ ãšãïŒäžè§åœ¢ $ABC$ ã®å€æ¥åãšçŽç· $EF$ ã® $2$ ã€ã®äº€ç¹ã $X$, $Y$ ãšããŸãïŒãã ãïŒ$X$, $E$, $F$, $Y$ ã®é ã§äžŠã¶ãã®ãšããŸãïŒïŒçŽç· $DE$, $DF$, $DA$ ãšäžè§åœ¢ $DXY$ ã®å€æ¥åã®äº€ç¹ã®ãã¡ïŒ $D$ ã§ãªãæ¹ããããã $P$, $Q$, $R$ ãšãããšïŒ
$$AP=26,\quad AQ=34,\quad AR=19$$
ãæãç«ã¡ãŸããïŒãã®ãšãïŒç·å $EF$ ã®é·ãã¯æ£æŽæ° $a$, $b$, $c$, $... |
OMC146 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc146/tasks/2405 | F | OMC146(F) | 800 | 28 | 101 | [
{
"content": "ã$n$ ãçŽ ã¹ã $p^k$ ã®ãšã,\r\n$$d_1 = p , \\quad d_2 = p^k , \\quad d_3 = p^2 , \\quad d_4 = p^{k - 1}, \\ldots$$\r\nãšããããšã§ïŒLTEã®è£é¡ã«ããé¡æãã¿ããããšããããïŒãã£ãŠïŒãã®å Žåã®æ¡ä»¶ãæºãã $n$ ã®ç·åã¯\r\n$$\r\n3^2 + \\cdots + 3^5 + 5^ 2 + 5^3 + 7^2 + 11^ 2 + 13^2 + 17^2 = 1138.\r\n$$\r\nã以äžïŒ$n$ ãçŽ ã¹ãã§ãªãå ŽåãèããïŒ$n$ ã®æå°ã®çŽ å æ°ã $p$ ãšããïŒ\r\... | ã以äžã®æ¡ä»¶ãã¿ããïŒ$1$ ä»¥äž $300$ 以äžã®**å¥æ°ã®åææ°** $n$ ã®ç·åãæ±ããŠãã ããïŒ
- $n$ ã® $1$ ã§ãªãæ£ã®çŽæ°ãã¹ãŠãé©åã«äžŠã³æ¿ã㊠$d_1,d_2,\ldots,d_m$ ãšããããšã§ïŒä»¥äžãæç«ããïŒ
$$
(d_1 + 1)^{d_2} \equiv (d_2 + 1)^{d_3} \equiv \cdots \equiv (d_{m-1} + 1)^{d_{m}} \equiv 1 \pmod n
$$ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/4490 | A | OMC145(A) | 100 | 377 | 381 | [
{
"content": "ãé·ãã $4$ ã®èŸºãš $50$ ã®èŸºã®ãªãè§ã $\\theta$ ãšããã°ïŒ$100=4\\times 50\\times (\\sin\\theta)\\/2$ ã§ããããïŒ$\\theta=90^\\circ$ ã§ããïŒãã£ãŠïŒæ±ããå€ã¯äžå¹³æ¹ã®å®çãã $4^2+50^2=\\mathbf{2516}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc145/editorial/4490"
}
] | ãé·ã $4$ ã®èŸºãšé·ã $50$ ã®èŸºããã¡ïŒé¢ç©ã $100$ ã§ããäžè§åœ¢ã«ãããŠïŒãã $1$ 蟺ã®é·ãã® $2$ ä¹ãæ±ããŠãã ããïŒ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/4376 | B | OMC145(B) | 200 | 342 | 378 | [
{
"content": "ãäžäžã®äœãšäžã®äœã®ã»ãã« $1$ ãšãªãåŸãã®ã¯çŸã®äœã®ã¿ã§ããïŒãã®ãããªãã®ã¯ $9^2=81$ éãååšããïŒããã§ãªããã®ã«ã€ããŠã¯ïŒãŸãçŸã®äœã $9$ éãååšãïŒããããã«ã€ããŠæ®ãã®äœãç¬ç«ã« $8$ éããã€ååšããããïŒå
šäœã§ã¯ $81+9\\times8^2=\\textbf{657}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc145/editorial/4376"
}
] | ãåé²æ³è¡šèšã§ä»¥äžã®æ¡ä»¶ãã¿ãã $5$ æ¡ã®æ£æŽæ°ã¯ããã€ãããŸããïŒ
- äžäžã®äœãšäžã®äœã¯ãšãã« $1$ ã§ããïŒ
- ä»»æã®é£ãåã $2$ æ¡ã«ã¯çžç°ãªãæ°ã䞊ã¶ïŒ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/3059 | C | OMC145(C) | 200 | 317 | 352 | [
{
"content": "ã$n$ ãå¶æ°ïŒããªãã¡ $n=2k$ãšè¡šãããšã\r\n$$n^{n}\\equiv (-1)^{2k}\\equiv 1\\pmod{2k+1} $$\r\nãæãç«ã¡ïŒäžæ¹ã§ $n$ ãå¥æ°ïŒããªãã¡ $n=2k-1$ ãšè¡šãããšã\r\n$$n^{n}\\equiv (-1)^{2k-1}\\equiv -1\\equiv 2k-1\\pmod{2k}$$\r\nãæãç«ã€ïŒä»¥äžã«ãã, æ±ããå€ã¯\r\n$$(1+1)+(3+1)+(5+1)+\\cdots +(99+1)=\\sum_{k=1}^{50} 2k=\\textbf{2550}$$\r\nã§ããïŒãªãïŒæåŸã®ç·å... | ãæ£ã®æŽæ° $n$ ã«å¯ŸããŠïŒ$n^{n}$ ã $n+1$ ã§å²ã£ãäœãã $f(n)$ ã§è¡šããšãïŒ
$$f(1)+f(2)+\cdots +f(99)+f(100)$$
ãæ±ããŠãã ããïŒ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/4773 | D | OMC145(D) | 300 | 192 | 248 | [
{
"content": "$$n^4-5n^2+4=(n+2)(n+1)(n-1)(n-2)$$\r\nã«æ³šæããã°ïŒ\r\n$$\\dfrac{6n}{n^4-5n^2+4} = \\dfrac{1}{n-2}-\\dfrac{1}{n-1}-\\dfrac{1}{n+1}+ \\dfrac{1}{n+2} $$\r\nãšå€åœ¢ã§ããããïŒ\r\n$$\\begin{aligned}\\sum_{n=3}^{1002}\\dfrac{n}{n^4-5n^2+4}&= \\dfrac{1}6\\bigg(\\dfrac{1}1-\\dfrac{1}2-\\dfrac{1} 4 +\\dfrac{1} 5 +\\cdo... | ã以äžã®ç·åãèšç®ããŠãã ããïŒ
$$\displaystyle \sum_{n=3}^{1002}\dfrac{n}{n^4-5n^2+4}$$
ãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}b$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/2957 | E | OMC145(E) | 300 | 117 | 190 | [
{
"content": "ã$AD\\neq 9$ ã§ããïŒ$AD\\gt9$ ã®ãšãïŒçŽç· $AB$ ãšçŽç· $CD$ ã®äº€ç¹ $O$ ã«ã€ã㊠$BO=7x, CO=8x$ ãšããïŒ\r\n$$\\triangle OBC:\\triangle OPQ:\\triangle OAD=56x^2:(7x+3)(8x+5):(7x+7)(8x+8)$$\r\nãæãç«ã€ïŒããã§ïŒäžããããæ¡ä»¶ã¯\r\n$$\\triangle OAD-\\triangle OPQ=\\triangle OPQ-\\triangle OBC$$\r\nãšåå€ã§ããããšããïŒ$x=\\\\dfrac{13}{3}\\$ãšãããïŒãã®ãš... | ãåè§åœ¢ $ABCD$ 㯠$AD\parallel BC$ ãªãå°åœ¢ã§ããïŒ
$$AB=7,\quad BC=9,\quad CD=8$$
ãã¿ãããŠããŸãïŒããã§ïŒç¹ $P$ ã蟺 $AB$ äžã«ïŒç¹ $Q$ ã蟺 $CD$ äžã«ïŒ$BP=DQ=3$ ãšãªãããã«ãšã£ããšããïŒç·å $PQ$ ã¯åè§åœ¢ $ABCD$ ãé¢ç©ãåã $2$ ã€ã®åè§åœ¢ã«åããŸããïŒãã®ãšãïŒç·å $AD$ ã®é·ãã¯äºãã«çŽ ãªæ£ã®æŽæ° $p,q$ ãçšã㊠$\\dfrac{p}{q}\$ ãšè¡šããã®ã§ïŒ$p+q$ ã解çããŠãã ããïŒ |
OMC145 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc145/tasks/5260 | F | OMC145(F) | 400 | 41 | 118 | [
{
"content": "ã$a_{2n-1}=a_{n}+2$ ããã³ $a_{2n}=a_n+1$ ã«æ³šæããã°ïŒ$S_n=a_{2^{n-1}+1}+\\cdots+a_{2^n}$ ã«ã€ããŠ\r\n\r\n$$ S_{n+1}=2S_{n}+3\\cdot 2^{n-1}$$\r\n\r\nãæç«ããïŒ$S_{1}=1$ ãèžãŸããã°ïŒäžè¬é
㯠$S_{n}=(3n-1)2^{n-2}=2^{n-1} + 3(n-1)2^{n-2}$ ãšãªãïŒãã£ãŠïŒ\r\n\r\n$$ \\sum_{n=1}^{M} S_n = (2^M-1)+3\\bigl(2^{M-1}M-2^{M}+1\\bigr)=( 3M-4... | ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒä»¥äžã®æäœã $n$ ã $1$ ã«ãªããŸã§ç¹°ãè¿ããŸãïŒ
- $n$ ãå¥æ°ãªãã°ïŒ$n$ ã« $1$ ã足ãïŒ
- $n$ ãå¶æ°ãªãã°ïŒ$n$ ã $2$ ã§å²ãïŒ
ãã®ãšãïŒ$a_n$ ã $n$ ã $1$ ã«ãªããŸã§ã«å¿
èŠãªæäœã®åæ°ã§å®ããŸãïŒ\
ãäŸãã° $5$ ã¯ä»¥äžã®ããã«æäœãããã®ã§ïŒ$a_5=5$ ã§ãïŒ
$$5\rightarrow6\rightarrow3\rightarrow4\rightarrow2\rightarrow1.$$
ããã®ãšãïŒä»¥äžã®ç·åãçŽ æ° $1021$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ
$$a_2+a_3+a_4+\cdots... |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/6825 | A | OMC144(A) | 100 | 354 | 362 | [
{
"content": "ãæ¡ä»¶ãã¿ãã $n$ æ°ã¯ã©ã® $2$ ã€ãçŽ å æ°ãå
±æããªãããïŒ$29$ 以äžã®çŽ æ°ã $10$ åã§ããããšãèæ
®ãããšïŒ$n$ ã¯ïŒ$1$ ã®ååšã«æ³šæããŠïŒ$11$ 以äžã§ããïŒéã«ïŒ$29$ 以äžã®çŽ æ°ãš $1$ ãéžã¹ã° $n=11$ ãšã§ããããïŒè§£çãã¹ãå€ã¯ $\\bf 11$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc144/editorial/6825"
}
] | ã$1$ ä»¥äž $29$ 以äžã®æŽæ°ã®ãã¡çžç°ãªã $n$ åãéžãã ãšããïŒãã®ãã¡ã©ã®çžç°ãªã $2$ ã€ã«ã€ããŠãäºãã«çŽ ã§ããïŒãã®ãšãïŒ$n$ ãšããŠããããæ倧å€ãæ±ããŠãã ããïŒ |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/6765 | B | OMC144(B) | 200 | 259 | 303 | [
{
"content": "ãæ¡ä»¶ããåè§åœ¢ $ABDE$ ã¯çèå°åœ¢ã§ããããïŒPtolemyã®å®çãã $BE^2 - AB^2 = AE \\cdot BD$ ã§ããïŒ$\\angle{BCE}=90°$ ã§ããããäžå¹³æ¹ã®å®çãã $BC^2=BE^2-EC^2=BE^2-AB^2=AE\\cdot BD$ ã§ããïŒåŸã£ãŠïŒæ±ããå€ã¯ $BC^2=7\\cdot 3=\\mathbf{21}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc144/editorial/6765"
}
] | ãåžäºè§åœ¢ $ABCDE$ ã¯ä»¥äžã®æ¡ä»¶ãã¿ãããŸãïŒ
$$\begin{aligned}
&AE\parallel BD, \quad AB=EC=ED, \\\\
&AE=3, \quad BD=7, \quad \angle{BCE}=90^\circ.
\end{aligned}$$
ãã®ãšãïŒ$BC$ ã®é·ãã®äºä¹ã解çããŠäžãã. |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/6683 | C | OMC144(C) | 300 | 295 | 321 | [
{
"content": "ãç®±ã®äžã«å
¥ã£ãŠããçã®æ°ãïŒå·Šããé ã« $a_1,a_2,a_3,a_4$ åãšããïŒãã®ãšãïŒäžããããæ¡ä»¶ã¯ä»¥äžã®ããã«æŽçã§ããïŒ\r\n$$a_1+a_2=2019,\\quad a_3+a_4=7981,\\quad 0\\leq a_1\\lt a_2\\lt a_3\\lt a_4$$\r\n\r\nãããŸïŒ$a_2$ ã $1010\\leq a_2\\leq 2019$ ã®ç¯å²ã§åºå®ãããšãïŒã¿ããã¹ãæ¡ä»¶ã¯\r\n$$a_2\\lt a_3\\lt 7981-a_3 \\iff a_2\\lt a_3 \\leq 3990$$\r\nã®ããã«æžããããããïŒãã£ãŠïŒæ¡... | ãäºãã«åºå¥ã§ãã $4$ ã€ã®ç®±ãå·Šå³äžåã«äžŠãã§ããïŒãããã®äžŠã³é ã¯åºå®ãããŠããŸãïŒãããã®ç®±ã«åèš $10^4$ åã®åºå¥ã§ããªãçãå
¥ã£ãŠããŸãïŒãããã®ç®±ãçãïŒä»¥äžã®æ¡ä»¶ãã¿ãããŠãããšãïŒããããã®ç®±ã«å
¥ã£ãŠããçã®æ°ã®çµã¿åãããšããŠãããããã®ã¯äœéããããŸããïŒ
- ã©ã®çžç°ãªã $2$ ã€ã®ç®±ã«å¯ŸããŠãïŒå·ŠåŽã®ç®±ã«å
¥ã£ãŠããçã®æ°ãããå³åŽã®ç®±ã«å
¥ã£ãŠããçã®æ°ã®æ¹ãå€ãïŒ
- å·Šååã® $2$ åã®ç®±ã«å
¥ã£ãŠããçã®æ°ã®åèšã¯ $2019$ åã§ããïŒ
ããã ãïŒçã $1$ åãå
¥ã£ãŠããªãç®±ããã£ãŠãæ§ããŸããïŒ |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/4969 | D | OMC144(D) | 400 | 75 | 180 | [
{
"content": "ãäžè¬æ§ã倱ãã $AB=1$ ãšã§ããããšã«æ³šæããïŒãã®ãšãïŒ$CA=x, CB=y$ ãšããã°ïŒ\r\n$$x^2+y^2\\lt1,\\quad x+y\\gt1,\\quad x\\gt0,\\quad y\\gt0$$\r\nã§äžããããé åïŒäžå³ã®éè²éšïŒãçŽç· $y=-k(x-2)-2$ ãééãããã㪠$k$ ã®ç¯å²ãèããããšã«åž°çãããïŒãã®çŽç·ã¯ã€ãã«å®ç¹ $(2,-2)$ ãéãããšã«æ³šæããŠäžå³ããå€æããã°ïŒä»¥äžã®ããšããããïŒ\r\n\r\n- çŽç·ã $(0,1)$ ãéãå ŽåãïŒåŸãã®äžé $-3\\/2$ ãäžããïŒ\r\n- çŽç·ãåäœåã®äžååãš... | ã以äžãã¿ãããããªå®æ° $s$ ãšããŠããããæ倧å€ã $m$ ãšãïŒå®æ° $t$ ãšããŠããããæå°å€ã $M$ ãšããŸãïŒãã ãïŒååšã¯ä¿èšŒãããŸãïŒïŒ
- $\angle C\gt 90^\circ$ ãã¿ãããããªãäžè§åœ¢ $ABC$ ã«å¯ŸããŠãïŒä»¥äžãæãç«ã€ïŒ$$s\lt \dfrac{2AB+CB}{2AB-CA} \lt t$$
ããã®ãšãïŒ$|M-m|$ ãæ ¹ã«ãã€æŽæ°ä¿æ° $2$ 次å€é
åŒ $P(x)$ ãååšããŸãïŒãã®ãã¡ïŒä¿æ°ã®æ倧å
¬çŽæ°ã $1$ ã§ãããã®ã«ã€ããŠïŒ$\lvert P(100)\rvert$ ã解çããŠãã ããïŒ |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/6615 | E | OMC144(E) | 500 | 24 | 60 | [
{
"content": "ã$f(1)=a, f(2)=b$ ãšããïŒãã®ãšã\r\n$$f(3)=f(1)^2+f(2)^2=a^2+b^2, \\quad f(4)=f(1)f(2)+f(2)f(3)=a^2b+ab+b^3$$\r\nã§ããïŒãŸãïŒ$f(5)=f(1)f(3)+f(2)f(4)=f(2)f(2)+f(3)f(3)$ ã§ããããïŒä»£å
¥ããŠè§£ãããšã§ $a=1$ ãåããïŒåŸã£ãŠïŒäžåŒã« $m=1$ ã代å
¥ãïŒ\r\n$$f(1)=1,\\quad f(2)=b,\\quad f(n+2)=f(n)+bf(n+1)$$\r\nãä»»æã®æ£ã®æŽæ° $n$ ã«ã€ããŠæç«ããããšããããïŒéã«ïŒããããæ... | ãæ£æŽæ°ã«å¯ŸããŠå®çŸ©ããïŒæ£æŽæ°å€ããšãé¢æ° $f$ ãïŒä»»æã®æ£æŽæ° $m,n$ ã«å¯ŸããŠ
$$f(m+n+1)=f(m)f(n)+f(m+1)f(n+1)$$
ãã¿ãããŸãïŒãã®ãã㪠$f$ ã®ãã¡ïŒ$\left| f(4)-10^6\right|$ ãæãå°ãããªããã®ãã¹ãŠã«ã€ããŠïŒ$\dfrac{f(1012)}{f(1010)}$ ã®**æŽæ°éšåã®ç·å**ã解çããŠãã ããïŒ |
OMC144 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc144/tasks/6725 | F | OMC144(F) | 600 | 15 | 71 | [
{
"content": "ãã«ã¿ãã ãªåã $n$ åç®ã«èŸºãäŒã£ãŠç§»åããæã« $X$ ããã®è·é¢ã $1,2,3,4$ ã®é ç¹ã«ããå Žåã®æ°ãããããïŒ$a_n, b_n, c_n, d_n$ ãšãããšïŒæ±ããå Žåã®æ°ã¯ $d_{9999}$ ã§ããïŒãŸãïŒä»¥äžã®æŒžååŒãæãç«ã€ïŒ\r\n$$\r\na_{n+1}=b_{n}, \\quad\r\nb_{n+1}=2a_{n}+b_{n}+c_{n}, \\quad\r\nc_{n+1}=b_{n}+c_{n}+2d_{n}, \\quad\r\nd_{n+1}=c_{n}\r\n$$\r\nããã§ïŒ$s_n = a_n + d_n, t_n = a_n - ... | ãæ£åäºé¢äœã®é ç¹ã®äžã€ã $X$ ãšãïŒäžå¿ã«é¢ã㊠$X$ ãšå¯Ÿç§°ãªé ç¹ã $Y$ ãšããŸãïŒ\
ãã«ã¿ãã ãªåã¯ïŒ$X$ ããã¹ã¿ãŒãããŠïŒèŸºã§çµã°ããé ç¹ã«ç§»åããããšã $10^4$ åç¹°ãè¿ããŸãïŒããã§ïŒçŽåã«ããé ç¹ã«åŒãè¿ãããšãå¯èœã§ãïŒãã®ãšãïŒä»¥äžã®æ¡ä»¶ãã¿ãã移åæ¹æ³ã®ç·æ°ãïŒçŽ æ° $5003$ ã§å²ã£ãããŸããæ±ããŠãã ããïŒ
ããã
- äžåºŠã $X$ ã«æ»ããïŒã〠$10^4$ åç®ã®ç§»åã§åã㊠$Y$ ã«å°éããïŒ
ããã ãïŒå転ããŠäžèŽãããããªç§»åæ¹æ³ãç°ãªããã®ãšããŠæ°ããŸãïŒ |
OMCT004 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct004/tasks/3112 | A | OMCT004(A) | 100 | 102 | 171 | [
{
"content": "ãåè§åœ¢ $ABCD$ ã®é¢ç©ã¯äžè§åœ¢ $ABD$ ã®é¢ç©ã® $2$ åã§ããïŒäžè§åœ¢ $ABD$ ã®é¢ç©ã¯ $\\angle BAD=90^{\\circ}$ ã®ãšãã«æå€§å€ $55$ ããšãïŒ$\\angle BAD$ ãé©åœãªéè§ã«ããããšã«ãã£ãŠäžè§åœ¢ $ABD$ ã®é¢ç©ã $55$ æªæºã®ä»»æã®æ£ã®å®æ°ã«ããããšãã§ããïŒãã®ãšãïŒåè§åœ¢ $ABCD$ ã¯åžã§ããïŒïŒãã£ãŠïŒåžåè§åœ¢ $ABCD$ ã®é¢ç©ã¯ $1,2,\\ldots, 110$ ã® $\\bf{110}$ éãã®æ£æŽæ°å€ãåããã.",
"text": "å
¬åŒè§£èª¬",
"url": "https:... | ãåžåè§åœ¢ $ABCD$ ã以äžãã¿ãããŸãïŒ
$$AB=BC=10,\quad CD=DA=11$$
ãã®ãšãïŒåè§åœ¢ $ABCD$ ã®é¢ç©ãšããŠããããæ£æŽæ°å€ã¯äœéããããŸããïŒ |
OMCT004 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct004/tasks/2928 | B | OMCT004(B) | 200 | 138 | 161 | [
{
"content": "ã$2$ ã€ã®çµ $(p_1, p_2, p_3, p_4), (p_2, p_4, p_1, p_3)$ ã«ã€ããŠïŒããããã®ã¹ã³ã¢ã®å㯠\r\n$$\\begin{aligned}\r\n(p_1p_2+p_2p_3+p_3p_4)+(p_2p_4+p_4p_1+p_1p_3)\r\n&=p_1p_2+p_1p_3+p_1p_4+p_2p_3+p_2p_4+p_3p_4 \\\\\\\\\r\n&=0\\cdot1+0\\cdot2+0\\cdot3+1\\cdot2+1\\cdot3+2\\cdot3 \\\\\\\\\r\n&=11\r\n\\end{aligned}$$\r\n... | ãå®æ° $4$ ã€ã®çµ $(a,b,c,d)$ ã«å¯ŸããŠïŒãã®**ã¹ã³ã¢**ã以äžã§å®ããŸãïŒ
$$ab+bc+cd$$
$0,1,2,3$ ããããäžã€ãã€ãããªãçµã¯ $4!$ éããããŸããïŒããããã«ã€ããŠã¹ã³ã¢ãæ±ãïŒãã®ç·åã解çããŠãã ããïŒ |
OMCT004 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct004/tasks/4126 | C | OMCT004(C) | 200 | 153 | 176 | [
{
"content": "ã$k$ 㯠$(103^2-1)-(101^2-1)=204\\cdot2$ ãš $(109^2-1)-(107^2-1)=216\\cdot2$ ã®å
¬çŽæ°ïŒããªãã¡ $24$ ã®çŽæ°ã§ããïŒãŸãïŒ$100$ 以äžã®çŽ æ° $p$ 㯠$p=6a\\pm1$ ã®åœ¢ã§è¡šããïŒ$(6a\\pm1)^2-1=12a(3a\\pm1)$ ãããã㯠$24$ ã®åæ°ã§ããïŒãããã£ãŠïŒæ±ããæ倧㮠$k$ 㯠$\\textbf{24}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/om... | ã次ã®åœé¡ãçãšãªããããªæ倧ã®æ£æŽæ° $k$ ãæ±ããŠãã ããïŒ
- ä»»æã® $100$ 以äžã®çŽ æ° $p$ ã«å¯ŸãïŒ$p^2-1$ 㯠$k$ ã®åæ°ã§ããïŒ |
OMCT004 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct004/tasks/3495 | D | OMCT004(D) | 200 | 81 | 100 | [
{
"content": "ã$P$ ãéã $PD$ ã«åçŽãªçŽç·ãšèŸº $AB, AC$ ã®äº€ç¹ã $X, Y$ ãšãïŒçŽç· $PE, PF$ ãšèŸº $BC$ ã®äº€ç¹ã $Z, W$ ãšããïŒ$XY \\parallel BC, EZ \\parallel AB, FW \\parallel AC$ ã«çæããã°ïŒä»¥äžãæç«ããïŒ\r\n\r\n$$\r\n\\begin{aligned}\r\nAF &= PE = 6,&& AE = PF = 8\\\\\\\\\r\nFX &= \\frac{PF}{\\sqrt{3}} = \\frac{8}{\\sqrt{3}},&& EY = \\sqrt{3}PE =... | ãäžè§åœ¢ $ABC$ ã¯
$$\angle A = 90^\circ, \quad \angle B= 60^\circ, \quad \angle C = 30^\circ$$
ãã¿ãããŸãïŒäžè§åœ¢ $ABC$ ã®å
éšã®ç¹ $P$ ãã蟺 $BC, CA, AB$ ã«ããããåç·ã®è¶³ããããã $D, E, F$ ãšãããšïŒä»¥äžãæãç«ã¡ãŸããïŒ
$$PD = 5, \quad PE = 6, \quad PF = 8$$
ãã®ãšãïŒäžè§åœ¢ $ABC$ ã®é¢ç©ã¯æ£ã®æŽæ° $a, b, c$ïŒ$c$ ã¯å¹³æ¹å åããããªãïŒãçšã㊠$a + b\sqrt{c}$ ãšè¡šãããã®ã§ïŒ$a + b + c$ ã解ç... |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/3262 | A | OMCT003(A) | 100 | 212 | 218 | [
{
"content": "ã$x$ ã $3$ ã§å²ããåæ°ã $f(x)$ ã§è¡šãã°ïŒ$3261$ ããã³ $3264$ ã $3$ ã®åæ°ã§ããããšããïŒ\r\n$$\\cdots=f(3260!)\\lt f(3261!)=f(3262!)=f(3263!)\\lt f(3264!)=\\cdots$$\r\nãæç«ããïŒãããã£ãŠïŒæ±ããç·å㯠$3261+3263=\\textbf{6524}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct003/editorial/3262"
}
... | ã$n!$ ãš $3262!$ ã $3$ ã§å²ãåããæ倧ã®åæ°ãçãããããªïŒ$3262$ 以å€ã®æ£ã®æŽæ° $n$ ã®ç·åãæ±ããŠãã ããïŒ |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/3412 | B | OMCT003(B) | 300 | 148 | 173 | [
{
"content": "ãçŽç· $AB$ äžã«ããïŒ$x$ 座æšã $t$ ã§ããç¹ã $D$ ãšããïŒ$CD$ ãåºèŸºãšããŠèããã° \r\n$$\r\n\\triangle{ABC}=\\frac{1}{2} \\times CD \\times \\bigl( 25-(-13) \\bigl)\r\n$$\r\nã§ããããïŒç·å $CD$ ã®é·ãã«ã€ããŠèããã°ããïŒ$D$ ã® $y$ 座æšã¯ $12t+325$ ã§ããããïŒ\r\n$$CD=|t^2-12t-325|=|(t-6)^2-361|$$\r\nã§ããïŒãããæ倧ãšãªã $t$ ã®å€ã $3$ ã€ååšããã®ã¯ïŒæ倧å€ã $361$ ãšãªããšãã§... | ã$a, b$ ã¯æ£ã®å®æ°ã§ãïŒåº§æšå¹³é¢äžã® $3$ ç¹
$$A:(-13, 13^2),\quad B:(25, 25^2),\quad C:(t, t^2)$$
ã«ã€ããŠïŒ$-a \leq t \leq b$ ã®ç¯å²ã§ç¹ $C$ ãåããšãïŒäžè§åœ¢ $ABC$ ã®é¢ç©ãæ倧ãšãªã $t$ ã®å€ã¯ã¡ããã© $3$ ã€ååšããŸããïŒãã®ãšãïŒ$ab$ ã®å€ãæ±ããŠãã ããïŒ |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/3178 | C | OMCT003(C) | 300 | 131 | 160 | [
{
"content": "ãäžåŒã $2$ åŒãã€è¶³ãåãããããšã§ïŒæ¡ä»¶ã¯ä»¥äžãšåå€ã§ããïŒ\r\n$$(a+b)(c+d)=143,\\quad (a+c)(b+d)=140,\\quad (a+d)(b+c)=135$$\r\n$143=11\\times 13$ ãèžãŸããŠç¬¬ $1$ åŒã«çç®ããã°ïŒ$a+b+c+d$ ã¯ã€ãã« $11+13=24$ ã§ããïŒããã«ãã\r\n$$ \\\\{a+b,~ c+d\\\\} = \\\\{11,13\\\\}, \\quad \\\\{a+c,~ b+d\\\\} = \\\\{10,14\\\\}, \\quad \\\\{a+d,~ b+c\\\\}=\\... | ã以äžããã¹ãŠã¿ããæ£ã®æŽæ°ã®çµ $(a,b,c,d)$ ãã¹ãŠã«ã€ããŠïŒ$a+b+c+d$ ã®ç·åãæ±ããŠãã ããïŒ
$$ab+cd=66,\quad ac+bd=69,\quad ad+bc=74$$ |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/4230 | D | OMCT003(D) | 300 | 44 | 80 | [
{
"content": "$$\\angle DBA=\\angle DBI-\\angle ABI=\\angle BID-\\angle IBC=\\angle BCI=\\angle ICA=\\angle DCA$$\r\nã«ããïŒ$D$ ã¯äžè§åœ¢ $ABC$ ã®å€æ¥åäžã«ããïŒãªãïŒåŒ§ $AB$ ã®äžç¹ãäžè§åœ¢ $AIB$ ã®å€å¿ã§ãããšããæåäºå®ãèžãŸããã°ïŒçŽæ¥ã®è§åºŠè¿œè·¡ãè¡ãããšãããã $D$ ã§ããããšããããïŒïŒç¹ $E$ ã«ã€ããŠãåæ§ã§ããïŒãããã£ãŠïŒäžè§åœ¢ $IBC$ ãšäžè§åœ¢ $IED$ ã¯çžäŒŒãªã®ã§ïŒ $IB:ID=BC:DE=4:5$ ãš $BD=ID$ ã«ãã\r\n$$ \... | ã$AB=3,BC=4$ ãªãäžè§åœ¢ $ABC$ ãããïŒãã®å
å¿ã $I$ ãšããŸãïŒç·å $BI$ ã®åçŽäºçåç·ãšçŽç· $CI$ ã®äº€ç¹ã $D$ ãšãïŒç·å $CI$ ã®åçŽäºçåç·ãšçŽç· $BI$ ã®äº€ç¹ã $E$ ãšãããšïŒ$DE=5$ ãæãç«ã¡ãŸããïŒãã®ãšãïŒäžè§åœ¢ $ABC$ ã®å€æ¥åã®é¢ç©ã¯ïŒäºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\displaystyle \frac{a}{b}\pi$ ãšè¡šããŸãïŒ$a + b$ ã®å€ãæ±ããŠãã ããïŒ |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/4250 | E | OMCT003(E) | 300 | 114 | 136 | [
{
"content": "ãå®éã«æäœãå®è¡ããããšã§ïŒæ¬¡ã®ããšããããïŒ\r\n\r\n- ããããã®æäœã®æç¹ã«ãããŠïŒå人ãéžã³ããè¡ãŸãã¯åã¯ãã¹ãŠäžèŽããïŒç¹ã«ïŒåèšã§ $17$ åç®ã®æäœä»¥éã¯ïŒå€ªéããã¯ãã¹ãŠ $1$ ã®è¡ãïŒæ¬¡éããã¯ãã¹ãŠ $0$ ã®åãéžã³ç¶ããããšã«ãªãïŒïŒ\r\n\r\nãããèžãŸããã°ïŒå€ªéããã¯éžã³ããè¡ã®ãã¡æãå·Šã«ãããã®ãïŒæ¬¡éããã¯éžã³ããåã®ãã¡æãäžã«ãããã®ãéžã³ç¶ãããšããŠãããïŒãã®å¶çŽã®ããšã§å®éã«æäœãå®è¡ããã°ïŒããã«æ¬¡ã®ããšããããïŒ\r\n\r\n- åèšã§ $16$ ã®åæ°åã®æäœãçµããçŽåŸã®ãã¹ç®ã¯ïŒå·ŠäžååïŒå¯Ÿè§ç·ãå«ãïŒããã¹ãŠ $... | ã$8\times8$ ã®ãã¹ç®ãããïŒã¯ãããã¹ãŠã®ãã¹ã«ã¯ $1$ ãæžãããŠããŸãïŒãã®ãã¹ç®ã«å¯ŸããŠïŒå€ªéãããšæ¬¡éããã次ã®ãããªæäœãïŒå€ªéãããå
æãšããŠäº€äºã« $5000$ åãã€è¡ããŸããïŒ
- 倪éããã¯ïŒçžŠã®è¡ã®ãã¡æžãããŠããæ°ã®ç·åãæã**倧ãã**ãã®ã®äžããé©åœã«äžã€ãéžã³ïŒãã®è¡ã®ãã¹ããã¹ãŠ $0$ ã«æžããããïŒ
- 次éããã¯ïŒæšªã®åã®ãã¡æžãããŠããæ°ã®ç·åãæã**å°ãã**ãã®ã®äžããé©åœã«äžã€ãéžã³ïŒãã®åã®ãã¹ããã¹ãŠ $1$ ã«æžããããïŒ
ãã¹ãŠã®æäœãçµãã£ãããšïŒãã¹ç®å
šäœã«æžãããæ°ã®ç·åãæ±ããŠãã ããïŒ\
ããã ãïŒæ±ããå€ã¯æäœã«ãããäžæã«å®ãŸãããš... |
OMCT003 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct003/tasks/3271 | F | OMCT003(F) | 500 | 7 | 47 | [
{
"content": "ã$AD=BC=a, AC=BD=b$ ãšããïŒçèå°åœ¢ $ABCD$ ã¯åã«å
æ¥ããã®ã§ïŒPtolemyã®å®çãã\r\n$$a^2+2\\times10^{200}=b^2 \\tag{1}$$\r\nãæç«ããïŒãŸãïŒ$DA+AB+BC\\gt DC$ ãã\r\n$$a\\gt\\dfrac12 \\times 10^{100}\\tag{2}$$\r\nãæç«ããïŒéã«ïŒãã® $2$ æ¡ä»¶ãæç«ããŠãããšãïŒæ¡ä»¶ãã¿ããçèå°åœ¢ãååšããïŒãããã£ãŠïŒäžã® $2$ æ¡ä»¶ãã¿ããæ£ã®æŽæ°ã®çµ $(a, b)$ ãæ°ããã°ããïŒ\\\r\nã$(1)$ ãæžãæãããš $(b+a)(... | ã$AB\parallel DC$ ãªãçèå°åœ¢ $ABCD$ ã¯ïŒ $AB=10^{100}, CD=2\times10^{100}$ ãã¿ãããŠããïŒç·å $AC, AD$ ã®é·ãã¯ãšãã«æ£æŽæ°å€ã§ãïŒãã®ãšãïŒãã®çèå°åœ¢ã®åœ¢ç¶ãšããŠãããããã®ã¯äœéããããŸããïŒããã ãïŒ$\log_{10}2$ ãå°æ°ç¬¬ $5$ äœã§åæšäºå
¥ããå€ãšã㊠$\log_{10}2\approx0.3010$ ãä¿èšŒãããŸãïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/6454 | A | OMCT002(A) | 100 | 313 | 320 | [
{
"content": "ãæåŸã®ã²ãŒã 㯠$A$ ããã®åã¡ã§ããããšã«æ³šæããã°ïŒæ®ãã® $9$ ã²ãŒã ãã $A$ ããã®åã¡ã $5$ ã€éžã¹ã°ããããïŒæ±ããå Žåã®æ°ã¯ ${}\\_{9}\\mathrm{C}\\_{5}=\\mathbf{126}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omct002/editorial/6454"
}
] | ã$A$ ãããš $B$ ããã®äºäººãïŒå
ã«ã©ã¡ããã $6$ åãããçµäºãšããã«ãŒã«ã§ïŒãªã»ãã§å¯Ÿæ±ºããŸããïŒæçµçã« $A$ ããã $6$ åïŒ$B$ ããã $4$ åããŠçµäºãïŒåŒãåãã¯ç¡ãã£ããšãïŒå
š $10$ åã®ã²ãŒã ã®åæã®çµã¿åãããšããŠãããããã®ã¯äœéããããŸããïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/4036 | B | OMCT002(B) | 200 | 226 | 302 | [
{
"content": "ãæ¡ä»¶ã¯ïŒ$4$ ã€ã®ç®ãçžç°ãªãïŒããã« $1,4$ ãããããŠé«ã
$1$ åããåºãªãããšãšèšããããããïŒ$4$ ã€ã®ç®ãçžç°ãªããã®ã¯ $6Ã5Ã4Ã3=360$ éãããïŒãã®ãã¡ $1,4$ ãåæã«åºããã®ã¯ ${}\\_{4}\\mathrm{C}\\_{2}Ã4!=144$ éãã§ããããïŒæ±ããå€ã¯ $\\dfrac{360-144}{6^4}=\\dfrac{1}{6}$ ã§ããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{7}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/... | ãã©ã®ç®ãç確çã§åºããããªäžè¬çãªå
é¢äœã®ãµã€ã³ãã $4$ åæ¯ãïŒåºãç®ãé ã« $a_1, a_2, a_3, a_4$ ãšãããšãïŒä»»æã® $1\leq{i}\lt{j}\leq4$ ã«å¯Ÿã $a_{i}a_{j}$ ãå¹³æ¹æ°ãš**ãªããªã**確çãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $s,t$ ãçšã㊠$\dfrac{s}{t}$ ãšè¡šããã®ã§ïŒ$s+t$ ã解çããŠãã ããïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/1986 | C | OMCT002(C) | 200 | 181 | 215 | [
{
"content": "ãç¹ $Q$ ãåºå®ãããšãïŒç·å $PQ$ ã®ååšãåŸãé åã¯äžè§åœ¢ $BGQ$ ã§ããããšã«çæããã°ïŒæ±ããé å㯠$B$ ãé ç¹ãšããŠïŒåååããçŽè§äºç蟺äžè§åœ¢ãé€ããå³åœ¢ãåºé¢ãšããéäœã§ããïŒãã®äœç©ã¯\r\n$$\\dfrac{1}{3}\\times 6\\times\\left( 9\\pi-18 \\right)=18\\pi-36$$\r\nã§äžããããããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\textbf{648}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/o... | ãäžèŸºã®é·ãã $6$ ã®ç«æ¹äœ $ABCD-EFGH$ ãåºå®ãããŠããïŒç·å $BG$ äžãåãç¹ $P$ ãšïŒ$C$ ãäžå¿ãšã㊠$D,G$ ãéãåã®å£åŒ§ $DG$ äžãåãç¹ $Q$ ããããŸãïŒãã®ãšãïŒç·å $PQ$ ã®ééãããé åã®äœç©ãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯æŽæ° $a,b$ ã«ãã£ãŠ $a\pi+b$ ãšè¡šããã®ã§ïŒ$|ab|$ ã解çããŠãã ããïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/6762 | D | OMCT002(D) | 300 | 183 | 231 | [
{
"content": "ãæ¡ä»¶ã¯ïŒ$25p\\le n^2\\lt25p+5$ ãšåå€ã§ããïŒ$n^2$ ã $5$ ã§å²ã£ãäœã㯠$0,1,4$ ã®ããããã§ããïŒ$25p$ ã¯å¹³æ¹æ°ã«ãªãããªãããšã«æ³šæããã°ïŒ$n^2=25p+1$ ãŸã㯠$n^2=25p+4$ ã®ãããããæç«ããïŒ\r\n\r\n- $n^2=25p+1$ ã§ããå Žå\\\r\nã$25p=(n-1)(n+1)$ ã§ããïŒ$n-1$ ãš $n+1$ ã®ãã¡ $5$ ã®åæ°ã§ããã®ã¯é«ã
äžã€ã§ããïŒ$n-1$ ãš $n+1$ ã®å·®ã¯ $2$ ã§ããããïŒ$\\\\{n-1,n+1\\\\} = \\\\{25,p\\\\}$ ã§ãã... | ã$\left\lfloor\dfrac{n^2}{5}\right\rfloor=5p$ ãã¿ããæŽæ° $n$ ãååšãããããªçŽ æ° $p$ ã®ç·åãæ±ããŠãã ããïŒ\
ããã ãïŒå®æ° $x$ ã«å¯Ÿã㊠$\lfloor x\rfloor$ 㧠$x$ ãè¶
ããªãæ倧ã®æŽæ°ãè¡šããŸãïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/231 | E | OMCT002(E) | 300 | 64 | 158 | [
{
"content": "ã$T_n$ ã® $3$ é ç¹ã $A_n,B_n,C_n$ ãšãããšã, å
è§ã«ã€ããŠä»¥äžã容æã«ç¢ºããããã.\r\n\r\n- $T_n$ ãéè§äžè§åœ¢ã§ãããšã,$$\\angle A_{n+1}=180^\\circ-2\\angle A_n,\\ \\ \\angle B_{n+1}=180^\\circ-2\\angle B_n,\\ \\ \\angle C_{n+1}=180^\\circ-2\\angle C_n.$$\r\n- $T_n$ ãéè§äžè§åœ¢ã§ãããšã, äŸãã° $\\angle A_n$ ãéè§ã§ãããªãã°,$$\\angle A_{n+1}=2\\ang... | ãäžè§åœ¢ $T_0$ ã¯ïŒ$3$ ã€ã®å
è§ã®å€§ããã床æ°æ³ã«ãããŠãã¹ãŠæ£æŽæ°å€ã§ãïŒ\
ãããŸïŒä»¥äžã®ããã«äžè§åœ¢ã®å $T_0,T_1,T_2,\dots$ ãå®ããŸãïŒ
- éè² æŽæ° $n$ ã«å¯ŸãïŒ$T_n$ ã®åé ç¹ãã察蟺ïŒãŸãã¯ãã®å»¶é·ïŒã«ããããåç·ã®è¶³ã $3$ é ç¹ãšããäžè§åœ¢ã $T_{n+1}$ ãšããïŒãã ãïŒ$T_n$ ãçŽè§äžè§åœ¢ã§ããå Žåã¯ïŒ$T_{n+1}=T_{n}$ ãšããïŒ
ãããéè² æŽæ° $N$ ãååšããŠïŒ$T_N$ ãæ£äžè§åœ¢ã§ãããã㪠$T_0$ ã«ã€ããŠïŒãã®æ倧ã®å
è§ã®å€§ãããšããŠããããå€ã床æ°æ³ã§ãã¹ãŠæ±ãïŒãããã®ç·åã解çããŠãã ããïŒ |
OMCT002 (åäœãã¹ã) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omct002/tasks/1457 | F | OMCT002(F) | 400 | 81 | 132 | [
{
"content": "ãçã«æžãããæ°ã $x_i\\ (i=1,2,\\cdots,100)$ ãšãïŒããããåºå®ãããšãåŸç¹ã®æåŸ
å€ã $E$ ãšããïŒãã®ãšã\r\n$$E=\\frac{1}{{}\\_{100}\\mathrm{P}\\_2}\\sum\\_{i\\neq j}\\frac{x_i}{x_j}=\\frac{1}{9900}\\left\\\\{\\left(\\sum\\_{i=1}^{100}x_i\\right)\\left(\\sum\\_{j=1}^{100}\\frac{1}{x_j}\\right)-100\\right\\\\}.$$\r\nãã㧠$\\display... | ã$100$ åã®çã $1$ ã€ã®ç®±ã«å
¥ã£ãŠããŸãïŒããããã®çã«ã¯ïŒ$1$ ä»¥äž $100$ 以äžã®æŽæ°ã®ãã¡ $1$ ã€ãæžã蟌ãŸããŠããŸãïŒçžç°ãªããšã¯éããŸããïŒïŒOMCåã¯ããããçšãã次ã®ã²ãŒã ãèããŸããïŒ
- ç®±ã®äžãã $2$ åç¶ããŠçãåãåºãïŒ$1,2$ åç®ã«åãåºããçã«æžãããæ°ããããã $a,b$ ãšããïŒ
- ãã®ãšãïŒ$\displaystyle\frac{a}{b}$ ãOMCåã®åŸç¹ãšãªãïŒ
- ãã ãïŒ$1$ åç®ã«åãåºããçã¯ç®±ã«æ»ããªããšããïŒ
ãçã«æžã蟌ãŸããæ°ãå€åãããšãïŒãã®ã²ãŒã ã«ãããOMCåã®åŸç¹ã®æåŸ
å€ãšããŠããããæ倧å€ã¯ïŒäºãã«çŽ ãªæ£æŽæ° $... |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/5549 | A | OMC143(A) | 100 | 325 | 341 | [
{
"content": "ãçžå çžä¹å¹³åã®äžçåŒãã\r\n$$\\displaystyle \\Big( x+\\frac{20}{x} \\Big)\\Big( x+\\frac{500}{x} \\Big) = x^2+ \\frac{100^2}{x^2}+520 \\geq 2\\sqrt{x^2\\times \\frac{100^2}{x^2}}+520=720$$\r\nãæãç«ã€ïŒ$x = 10$ ã®ãšãçå·ãæç«ããã®ã§ïŒæ±ããçã㯠$\\bf{720}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/cont... | ãæ£ã®å®æ° $x$ ã«å¯ŸãïŒæ¬¡ã®åŒããšãããæå°å€ãæ±ããŠãã ããïŒ\
$$\displaystyle \bigg( x+\frac{20}{x} \bigg)\bigg( x+\frac{500}{x} \bigg)$$ |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/4762 | B | OMC143(B) | 200 | 304 | 341 | [
{
"content": "ãçŸã®äœãšäžã®äœã®å¶å¥ãäžèŽããã°ïŒé©ããåã®äœãäžæã«å®ãŸãïŒãšãã«å¥æ°ã§ãããã®ã¯ $5^2$ éãïŒãšãã«å¶æ°ã§ãããã®ã¯ïŒäžã®äœã¯ $0$ ã§ãè¯ãããšã«çæããŠïŒ$4\\times 5$ éãã§ããããïŒå
šäœã§ã¯ $\\mathbf{45}$ éãã§ããïŒãªãïŒå
¬å·®ãšããŠããåŸãã®ã¯ $-4$ ä»¥äž $4$ 以äžã§ããããšããïŒããããæ°ãäžããŠãããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc143/editorial/4762"
}
] | ã$3$ æ¡ïŒ$100$ ä»¥äž $999$ 以äžïŒã®æ£æŽæ°ã®ãã¡ïŒçŸã®äœã»åã®äœã»äžã®äœããã®é ã«çå·®æ°åããªããã®ã¯ããã€ãããŸããïŒ |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/3396 | C | OMC143(C) | 200 | 294 | 310 | [
{
"content": "ã$\\triangle ADE\\equiv\\triangle ADF$ ãæãç«ã€ããïŒç°¡åãªè°è«ã«ãã $AD=5,AD\\perp EF$ ããããïŒ\\\r\nãããããåè§åœ¢ $AEDF$ ã®é¢ç©ã«ã€ããŠïŒæ¬¡ãæãç«ã€ïŒ\r\n$$2\\times(\\triangle ADE ã®é¢ç©)=12=\\dfrac{AD\\times EF}{2}=\\dfrac{5}{2}EF$$\r\nãããã $EF=\\dfrac{24}{5}$ ãåŸãããããïŒè§£çãã¹ãå€ã¯ $\\textbf{29}$ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https:... | ãäžè§åœ¢ $ABC$ ã«ãããŠïŒ$\angle{A}$ ã®å
è§ã®äºçåç·ãšèŸº $BC$ ã®äº€ç¹ã $D$ ãšãïŒ$D$ ãã蟺 $AB,AC$ ã«äžãããåç·ã®è¶³ããããã $E,F$ ãšããŸãïŒ
$$AF=4, \quad DE=3$$
ã®ãšãïŒç·å $EF$ ã®é·ãã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/3193 | D | OMC143(D) | 300 | 209 | 279 | [
{
"content": "ã$k$ 人座ããšãïŒåº§ã£ãŠããåžã®çªå·ãå°ããé ã« $a_1,a_2,\\dots,a_k$ ãšããïŒ\r\n$b_i=a_i-4(i-1)$ ãšããã°ïŒ\r\n$$1\\leq b_1\\lt b_2\\lt\\cdots\\lt b_k\\leq 20-4(k-1)$$\r\nãªãæŽæ°ã®çµ $(b_1,b_2,\\dots,b_k)$ ãèããããšãšç䟡ã§ããïŒãã®ãããªçµ $(b_1,b_2,\\dots,b_k)$ 㯠$\\binom{24-4k}{k}$ çµååšããããïŒæ±ããåæ°ã¯æ¬¡ã§æ±ããããïŒ\r\n$$\\binom{20}{1}+\\binom{16}{2}+\\b... | ãã¹ããŒãžã«åãã£ãŠ $20$ åã®æ€
å $1,2,\ldots, 20$ ããã®é ã«æšªäžåã«äžŠãã§ããŸãïŒããŸïŒãããã«ã客ããã座ããããã®ã§ããïŒãœãŒã·ã£ã«ãã£ã¹ã¿ã³ã¹ãä¿ã€ããïŒé£ã®äººãšã¯æäœã§ãæ€
å $4$ ååã¯ç©ºããŠããããã§ãïŒããšãã°ïŒæ€
å $2,7,15,20$ ãéžãã§åº§ãããããšã¯å¯èœã§ããïŒæ€
å $10,14$ ãéžãã§åº§ãããããšã¯ã§ããŸããïŒ$1$ 人以äžã®ã客ããã座ããããšãïŒäººã座ã£ãŠããæ€
åã®éåãšããŠãããããã®ã¯ããã€ãããŸããïŒ |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/5428 | E | OMC143(E) | 300 | 229 | 274 | [
{
"content": "$$a+b+c=s,\\quad ab+bc+ca=t,\\quad abc=u$$ãšãããšïŒæ¡ä»¶ã¯\r\n$$\r\n\\begin{cases}\r\n\\{s + t + u = 0}\\\\\\\\\r\n\\{4s + 2 t + u = -4}\\\\\\\\\r\n\\{9s + 3 t + u = -18}\\\\\\\\\r\n\\end{cases}\r\n$$\r\nãšæžããã®ã§ïŒããã解ã㊠$s = -5, ~ t = 11, ~ u = -6$ ã§ããïŒåŸã£ãŠïŒæ±ããå€ã¯\r\n$$(a + 100)(b + 100)(c + 100) = 100^3 + 1... | ã$3$ ã€ã®è€çŽ æ° $a,b,c$ ã以äžãã¿ãããŠããŸãïŒ\
$$
\begin{cases}
\(a+1)(b+1)(c+1)=1^2\\\\
\(a+2)(b+2)(c+2)=2^2\\\\
\(a+3)(b+3)(c+3)=3^2\\\\
\end{cases}
$$
ãã®ãšãïŒ$(a+100)(b+100)(c+100)$ ãæ±ããŠãã ããïŒ |
OMC143 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc143/tasks/1914 | F | OMC143(F) | 400 | 66 | 142 | [
{
"content": "ãçµè«ããè¿°ã¹ããš, 以äžã®ããã«åå²ãããããšãåããïŒ\r\n\r\n- äžã»å段ç®ã«ã€ããŠ, äœç© $1\\/6$ ã $6$ ã€, äœç© $1\\/2$ ã $2$ ã€, äœç© $2\\/3$ ã $2$ ã€ã§ãã.\r\n- äºã»äžæ®µç®ã«ã€ããŠ, äœç© $1\\/6$ ã $6$ ã€, äœç© $2\\/3$ ã $4$ ã€, äœç© $5\\/6$ ã $2$ ã€, äœç© $1$ ã $2$ ã€ã§ãã.\r\n\r\nããããã®äœç©ã®å°ç«äœã¯, ãã®äžé¢ããã³äžé¢ã®åœ¢ç¶ã«ãã£ãŠä»¥äžã®ããã«å€æå¯èœã§ããïŒ\r\n\r\n- äœç© $1\\/6$ ã®ãã®ïŒ$1$ ç¹ããã³çŽè§äºç蟺äž... | ãäžèŸº $4$ ã®ç«æ¹äœ $ABCD-EFGH$ ãïŒäžèŸº $1$ ã®ç«æ¹äœ $64$ åã«åå²ãããšïŒæ£åé¢äœ $BDEG$ ãè€æ°ã®ç«äœã«åå²ãããŸãïŒãã®ãšãïŒæ£åé¢äœ $BDEG$ ãåå²ãããçµæãšããŠåŸãããããããã®ç«äœã®äœç©ã®**ç·ç©**ã¯ïŒçžç°ãªãçŽ æ° $p,q,r$ ããã³æŽæ° $s,t,u$ ã«ãã£ãŠ $p^sq^tr^u$ ãšè¡šããŸãïŒæ¬¡ã®å€ã解çããŠäžããïŒ
$$p\times |s|+q\times |t|+r\times|u|.$$
ãããã§ïŒç«äœã¯æ£ã®äœç©ãæã€ãã®ã®ã¿ãæããã®ãšããŸãïŒ |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2725 | A | OMC142(A) | 200 | 319 | 332 | [
{
"content": "ã$a=142857$ ãšããã°ïŒ$a,2a,3a,4a,5a,6a$ ã®äžããéè€ãèš±ããŠéžã¶æ¹æ³ã®ãã¡ïŒç·åã $7a$ ã§ãããã®ã®ç·æ°ãæ±ããããšã«ãªãïŒ$7$ ã $2$ ã€ä»¥äžã®æ£ã®æŽæ°ã®åã«åå²ããæ¹æ³ã®ç·æ°ïŒ $7$ ã®åå²æ°ãã $1$ åŒãããã®ïŒãèããŠïŒæ±ããå Žåã®æ°ã¯ $\\textbf{14}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc142/editorial/2725"
}
] | ã以äžã® $6$ ã€ã®æ°ã®äžããïŒéè€ãèš±ããŠããã€ããéžã¶æ¹æ³ã®ãã¡ïŒãããã®åã $999999$ ãšãªããããªãã®ã¯äœéããããŸããïŒ
$$142857, \quad 285714, \quad 428571, \quad 571428, \quad 714285, \quad 857142$$
ããã ãïŒæ°ãéžã¶é çªã¯èæ
®ããŸããïŒ |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2726 | B | OMC142(B) | 200 | 254 | 296 | [
{
"content": "ãäžè§åœ¢ $ABT$ ãïŒ$A$ ãäžå¿ã«ïŒåæèšåãã« $120^\\circ$ å転ãããšäžè§åœ¢ $AFP$ ãšäžèŽããïŒäžè§åœ¢ $AFP$ 㯠$AF$ ãåºèŸºãšèŠããšé«ãã $\\dfrac{5}{6}AC=\\dfrac{5\\sqrt{3}}{2}$ ã§ããããïŒ $X = \\dfrac{1}{2}\\cdot3\\cdot\\dfrac{5\\sqrt{3}}{2}=\\dfrac{15\\sqrt{3}}{4}$ ã§ããïŒãã£ãŠïŒ$X^2=\\dfrac{675}{16}$ ããïŒç¹ã«è§£çãã¹ãæ°å€ã¯ $\\textbf{691}$ ãšãªãïŒ",
"text": ... | ãäžèŸºã®é·ãã $3$ ã®æ£å
è§åœ¢ $ABCDEF$ ã«ãããŠïŒèŸº $BC$ äžã« $BP=2$ ãªãç¹ $P$ ããšãïŒæ£å
è§åœ¢ $APQRST$ ãäœããŸããïŒããã§ïŒ$A,B,C,D,E,F$ ã¯ãã®é ã«**åæèšåã**ã«äžŠã³ïŒ$A,P,Q,R,S,T$ ã¯ãã®é ã«**æèšåã**ã«äžŠãã§ãããã®ãšããŸãïŒ\
ãäžè§åœ¢ $ABT$ ã®é¢ç©ã $X$ ãšãããšãïŒ$X^2$ ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠãã ããïŒ |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2727 | C | OMC142(C) | 300 | 222 | 254 | [
{
"content": "ã$a_1\\lt a_2\\lt \\cdots\\lt a_{54}$ ãšããŠäžè¬æ§ã倱ããïŒãã®ãšã䞊ã¹æ¿ãäžçåŒããäžåŒã¯ $b_1\\gt b_2\\gt\\cdots\\gt b_{54}$ ã®ãšãã«æ倧å€ããšãïŒããªãã¡ïŒæ±ããæ倧å€ã¯\r\n$$\\dfrac{1}{6300}+\\dfrac{2}{3150}+\\cdots+\\dfrac{6300}{1}$$\r\nã§ããïŒããã¯ä»¥äžã®ããã«èšç®ã§ããããïŒè§£çãã¹ãæ°å€ã¯ $\\textbf{19749}$ ã§ããïŒ\r\n$$\\sum_{d\\mid6300}\\frac{d^2}{6300}=\\dfrac{(... | ã$6300$ ã¯æ£ã®çŽæ°ãå
šéšã§ $54$ åãã¡ãŸãïŒ\
ã $a_1,a_2,\ldots,a_{54}$ ããã³ $b_1,b_2,\ldots,b_{54}$ ãïŒããããçžç°ãªã $54$ åã® $6300$ ã®æ£ã®çŽæ°ã§ãããšãïŒä»¥äžã®ãšãããæ倧å€ãæ±ããŠãã ããïŒ
$$\dfrac{a_1}{b_1}+\dfrac{a_2}{b_2}+\cdots+\dfrac{a_{54}}{b_{54}}$$
ãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $m,n$ ãçšã㊠$\dfrac{m}{n}$ ãšè¡šãããã®ã§ïŒ$m+n$ ã®å€ã解çããŠãã ããïŒ |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2728 | D | OMC142(D) | 400 | 87 | 142 | [
{
"content": "ã $a=\\sqrt[3]{2x-y-z}, ~ b=\\sqrt[3]{2y-z-x}, ~ c=\\sqrt[3]{2z-x-y}$ ãšãããšïŒä»¥äžãæãç«ã€ïŒ\r\n $$a+b+c=3, \\quad \\dfrac{1}{a}+\\dfrac{1}{b}+\\dfrac{1}{c}=\\dfrac{3}{10},\\quad a^3+b^3+c^3=0$$\r\nãããã $ab+bc+ca=3k, ~ abc=10k$ ãšããããïŒãã®ãšã\r\n$$\\begin{aligned}\r\na^3+b^3+c^3&=(a+b+c)\\bigl((a+b+c)^2-3(ab+b... | ãå®æ° $x,y,z$ ã以äžã® $2$ åŒãã¿ãããšããŸãïŒ\
$$\begin{cases}
\sqrt[3]{2x-y-z}+\sqrt[3]{2y-z-x}+\sqrt[3]{2z-x-y}=3\\\\
\\\\
\dfrac{1}{\sqrt[3]{2x-y-z}}+\dfrac{1}{\sqrt[3]{2y-z-x}}+\dfrac{1}{\sqrt[3]{2z-x-y}}=\dfrac{3}{10}
\end{cases}$$
ãã®ãšãïŒ${(x-y)}^2+{(y-z)}^2+{(z-x)}^2$ ã®å€ãæ±ããŠãã ããïŒ
<details><summary>$3$ ä¹æ ¹ã«ã€ããŠ<\/summary>
... |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2730 | E | OMC142(E) | 500 | 39 | 144 | [
{
"content": "ãç¹ $P$ ãçŽç· $x=2000$ ããå
ã«çŽç· $y=2000$ ã«ãã©ãçã確çã¯å¯Ÿç§°æ§ãã $\\dfrac{1}{2}$ ã§ããïŒãã®ããšã¯ïŒç¹ $P$ ãç¹ $(1999,2000)$ ãéã確çã $\\dfrac{1}{2}$ ã§ãããšèšãæããããïŒäžæ¹ã§ïŒç¹ $P$ ãç¹ $(2000,1999)$ ã«çã確ç㯠$\\dfrac{\\_{3999}\\mathrm{C}\\_{1999}}{2^{3999}}$ ã§ããããïŒåããç¯å²ã $x\\geq 0,y\\geq 0$ å
šäœãšããŠèããã®ãšåãïŒïŒç¹ $P$ ãç¹ $(2000,2000)$ ãéã確çã¯ä»¥äž... | ã座æšå¹³é¢äžã«ç¹ $P$ ãããïŒã¯ããã¯åç¹ $(0,0)$ ã«ãããŸãïŒ\
ã以äžã®èŠåã«åŸã£ãŠ $P$ ã $4001$ ååãããšïŒæçµçã« $P$ ã¯ç¹ $(2001,2000)$ ã«éããŸãïŒãã®ãšãïŒ$P$ ãéäžã§ç¹ $(2000,2000)$ ãéã確çãæ±ããŠãã ããïŒ
- $P$ ã $x=2001$ äžã«ãããšãïŒ$P$ ã $y$ 軞ã®æ£ã®æ¹åã« $1$ ã ãåããïŒ
- $P$ ã $y=2000$ äžã«ãããšãïŒ$P$ ã $x$ 軞ã®æ£ã®æ¹åã« $1$ ã ãåããïŒ
- $P$ ããã以å€ã®é åã«ãããšãïŒ$P$ ã $x$ 軞ã®æ£ã®æ¹åã« $1$ ã ãåããæäœãš $y$ 軞ã®æ£ã®æ¹åã«... |
OMC142 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc142/tasks/2729 | F | OMC142(F) | 500 | 13 | 34 | [
{
"content": "ãå $\\omega$ ãšå $\\Omega$ ã®çžäŒŒã®äžå¿ã¯ $T$ ã§ããããïŒ$P$ ã«ããã $\\Omega$ ã®æ¥ç·ã¯ $D$ ã«ããã $\\omega$ ã®æ¥ç·ïŒããªãã¡çŽç· $AB$ ãšå¹³è¡ã§ããïŒåŸã£ãŠïŒ$P$ ã¯åŒ§ $AB$ ã®äžç¹ã§ããïŒ åæ§ã«ããŠïŒ$Q$ ã¯åŒ§ $AC$ ã®äžç¹ã§ããã®ã§ïŒ$BQ$ ãš $CP$ ã®äº€ç¹ãå
å¿ $I$ ã§ããïŒãã¹ã«ã«ã®å®çãã $3$ ç¹ $D,I,E$ ã¯åäžçŽç·äžã«ããïŒ\\\r\nããŸãïŒ$PA=PI,QA=QI$ ãªã®ã§ïŒ$PQ$ 㯠$AI$ ã®åçŽäºçåç·ãšãªãïŒãã£ãŠïŒ$AI$ ãš $PQ$ ã®äº€ç¹ã $M$ ... | ãäžè§åœ¢ $ABC$ ãããïŒãã®å€æ¥åã $\Omega$ïŒå
å¿ã $I$ ãšããŸãïŒå $\Omega$ ã«å
æ¥ãïŒãã€èŸº $AB$ ãšç¹ $D$ ã§æ¥ãïŒèŸº $AC$ ãšç¹ $E$ ã§æ¥ããåã $\omega$ ãšãïŒãã® $\Omega$ ãšã®æ¥ç¹ã $T$ ãšããŸãïŒçŽç· $TD,TE$ ãšå $\Omega$ ã®äº€ç¹ã®ãã¡ $T$ ã§ãªãæ¹ããããã $P,Q$ ãšããŸãïŒãŸãïŒç·å $PQ$ ãšç·å $AB,AC$ ã®äº€ç¹ããããã $R,S$ ãšããŸãïŒãããšïŒ
$$RS=11, \quad AI=23$$
ãšãªãïŒããã« $3$ ç¹ $T,I,S$ ã¯åäžçŽç·äžã«ãããŸããïŒãã®ãšãïŒ$PQ$ ã®é·ãã¯æ£æŽæ°... |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/3409 | A | OMC141(A) | 100 | 343 | 347 | [
{
"content": "ãã³ãã $C$ ã®ã³ãŒããŒã®éãšãã«ã¯ã®éã«ã€ããŠ, 以äžã®åŒãæãç«ã€.\r\n$$\r\n\\left(100 \\times \\dfrac{11}{14} + 200 \\times \\dfrac{9}{14} \\right):\\left( 100 \\times \\dfrac{3}{14} + 200 \\times \\dfrac{5}{14} \\right)= 29 : 13\r\n$$\r\nããããã, çããã¹ãå€ã¯ $\\mathbf{42}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemat... | ãããã§ã¯**ã³ãŒããŒ**ãš**ãã«ã¯**ã®æ··åç©ã**ã«ãã§ãªã¬**ãšãã¶ããšãšããŸãïŒ\
ã$2$ ã€ã®ã³ãã $A, B$ ãããïŒããããã«ãã§ãªã¬ã $100 \mathrm{mL}, ~ 200 \mathrm{mL}$ å
¥ã£ãŠããŸãïŒã³ãŒããŒãšãã«ã¯ã®äœç©ã®æ¯ã¯ïŒã³ãã $A$ 㯠$11:3$ïŒã³ãã $B$ 㯠$9:5$ ã§ãïŒ\
ãããããã®äžèº«ãå®å
šã«ã³ãã $C$ ã«ç§»ããŠïŒ$300 \mathrm{mL}$ ã®ã«ãã§ãªã¬ãäœããŸããïŒãã®ãšãïŒãã®ã³ãŒããŒãšãã«ã¯ã®äœç©ã®æ¯ã¯ïŒäºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$a:b$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/3411 | B | OMC141(B) | 200 | 303 | 335 | [
{
"content": "ã$1, A, A, A, A, B, B, B, B$ ã® $9$ æåã䞊ã¹æ¿ããŠæååãäœã, $A$ ã«å·Šãã $2, 3, 4, 5$ ã®é ã«æ°åãåœãŠã¯ã, $B$ ã«å·Šãã $6, 7, 8, 9$ ã®é ã«æ°åãåœãŠã¯ããã°, é¡æãæºããæŽæ°ãéè€ãªããã¹ãŠäœãããšãã§ãã. ãããã£ãŠ, ãããã®æåã®äžŠã³æ¿ãã®ç·æ°ãèããã°ãã, 解çãã¹ãå€ã¯\r\n$$\r\n\\frac{9!}{(4!)^2}=\\mathbf{630}\r\n$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/c... | ã次ã®æ¡ä»¶ããã¹ãŠã¿ãã $9$ æ¡ã®æ£æŽæ°ã¯ããã€ãããŸããïŒ
- $1$ ä»¥äž $9$ 以äžã®æŽæ°ãïŒåæ¡ã«äžåºŠãã€çšããããŠããïŒ
- $2, 3, 4, 5$ ã®ã¿ãæãåºãããšãïŒå·Šãããã®é ã§äžŠãã§ããïŒ
- $6, 7, 8, 9$ ã®ã¿ãæãåºãããšãïŒå·Šãããã®é ã§äžŠãã§ããïŒ |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/2867 | C | OMC141(C) | 200 | 239 | 323 | [
{
"content": "ã$1+2+\\cdots+n=n(n+1)\\/2$ ã $3$ ã®åæ°ã«ãªãå¿
èŠãããããïŒ$n$ 㯠$3$ ã§å²ã£ãŠ $0$ ãŸã㯠$2$ äœãïŒ\\\r\nãéã«ïŒããã« $n\\geq 5$ ãªãã°æ¡ä»¶ãã¿ããããšããããããïŒæ±ããåæ°ã¯ $666-2=\\textbf{664}$ ã§ããïŒ\\\r\nãå
·äœçã«ã¯ïŒ$1+6=2+5=3+4$ ã«æ³šç®ã㊠$n-6$ ã®å Žåã«åž°çããããšã§ç€ºãããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc141/editoria... | ãOMCå㯠$n$ åã®ã¢ã¡ãæã£ãŠããŸãïŒãããã®ã¢ã¡ã¯ãã¹ãŠéããç°ãªãïŒãããã $1,2,\ldots, n$ ã§ãïŒåäœã¯ã°ã©ã ïŒïŒOMCåã¯ãããã®ã¢ã¡ãïŒOåã»Måã»Cåã® $3$ 人ã«éäžè¶³ãªãé
ãããã§ãïŒ$3$ 人ããããããããåã«ã€ããŠïŒéãã®ç·åãçãããªãããã«ã§ãããšãïŒ$n$ ãšããŠé©ãã $1$ ä»¥äž $1000$ 以äžã®æŽæ°ã¯ããã€ãããŸããïŒ |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/2961 | D | OMC141(D) | 200 | 249 | 282 | [
{
"content": "**解æ³1.**ã蟺 $AB$ äžã« $AQ:QB=1:3$ ãšãªãç¹ $Q$ ããšããšïŒ$BQ=3, PQ=7\\/4$ ã§ããïŒäºèŸºæ¯å€Ÿè§çžçããäžè§åœ¢ $PBQ$ ãš $ABP$ ã¯çžäŒŒã§ããããïŒ$AP=7\\sqrt{3}\\/6$ïŒãã£ãŠ $AC^2=\\dfrac{196}{3}$ ã§ããïŒæ±ããå€ã¯ $196+3=\\mathbf{199}$ïŒ\r\n\r\n**解æ³2.**ãStewart ã®å®çããã®ãŸãŸé©çšããããšãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests... | ã$AB=4,~BC=7$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠïŒèŸº $AC$ ã $1:3$ ã«å
åããç¹ $P$ ã $BP=2\sqrt{3}$ ãã¿ãããŸããïŒãã®ãšãïŒ$AC$ ã®é·ãã® $2$ ä¹ã¯ïŒäºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/3415 | E | OMC141(E) | 300 | 132 | 195 | [
{
"content": "ãæ¡ä»¶ãã, $i \\leq j$ ãªãä»»æã® $i, j$ ã«å¯ŸããŠ\r\n$$\r\n\\frac{10^{i^2-i}}{a_i+a_{i+1}}=\\frac{10^{j^2-j}}{a_j+a_{j+1}}\r\n$$\r\nãæãç«ã€. ç¹ã« $i=1$ ãšããã°, ä»»æã®æ£æŽæ° $n$ ã«å¯Ÿã\r\n$$\r\n\\frac{10^{n^2-n}}{a_n+a_{n+1}}=\\frac{1}{a_1+a_2}=\\frac{1}{10}\r\n$$\r\nãšãªããã, $a_n+a_{n+1}=10^{n^2-n+1}$ ã§ãã. ãããã, æ£æŽæ° $k$ ã«å¯ŸããŠ\r... | ãå®æ°å $\\{a_n\\}_{n=1,2,\ldots}$ ã¯ïŒ$i \leq j$ ãªãä»»æã®æ£æŽæ° $i, j$ ã«å¯ŸããŠä»¥äžãã¿ãããŸãïŒ
$$
\frac{a_j+a\_{j+1}}{a_i+a\_{i+1}}=10 ^ {(j-i)(j+i-1)}
$$
$a_1=a_2=5$ ã§ãããšãïŒ$a\_{100}$ ã¯æ£æŽæ°ã«ãªãã®ã§ïŒ$a\_{100}$ ã®ïŒåé²æ³è¡šèšã§ã®ïŒåæ¡ã®åãæ±ããŠãã ããïŒ |
OMC141 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc141/tasks/3261 | F | OMC141(F) | 400 | 88 | 185 | [
{
"content": "ã$2$ ããã³ $3$ ã§å²ã£ãäœããèããããšã§ïŒ$p,q,r,s$ ã«ã¯ $2$ ããã³ $3$ ãå«ãŸããïŒ\\\r\nã$p=2$ ã〠$q=3$ ã®ãšãïŒ$(s+r)(s-r)=24$ ã§ããããïŒä»¥äžãåŸãïŒ\r\n$$(p, q, r, s)=(2, 3, 5, 7)$$ \r\nã$p=2$ ã〠$r=3$ ã®ãšãïŒ$8q=(s+3)(s-3)$ ã§ããããïŒä»¥äžãåŸãïŒ\r\n$$(p, q, r, s)=(2, 2, 3, 5),(2, 5, 3, 7)$$\r\nã$q=2$ ã®ãšãïŒ$s^2-r^2=2p^3$ ã§ãããïŒ$r^2,s^2$ ã¯ãšãã« $4$ ... | ãçŽ æ°ã®çµ $(p, q, r, s)$ ã§ãã£ãŠïŒä»¥äžã®çåŒ
$$p^3q+r^2=s^2$$
ãã¿ãããã®ãã¹ãŠã«ã€ããŠïŒ $p+q+r+s$ ã®**ç·ç©**ãæ±ããŠãã ããïŒ |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/4442 | A | OMC140(A) | 100 | 354 | 359 | [
{
"content": "ãæ¡ä»¶ãã¿ããé·æ¹åœ¢ã®çžŠã®é·ãã $x$, 暪ã®é·ãã $y$ ãšãããš, $$x+y = 4\\sqrt{6},\\quad xy= 10$$ ãæç«ãã. ãããã£ãŠ, æ±ããçãã¯ä»¥äžã®ããã«èšç®ã§ãã. \r\n$$ x^2 + y^2 = (x+y)^2- 2xy=96-20=\\textbf{76} $$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc140/editorial/4442"
}
] | ãããé·æ¹åœ¢ã«ã€ããŠïŒãã®åšé·ã $8\sqrt{6}$ïŒé¢ç©ã $10$ ã®ãšãïŒå¯Ÿè§ç·ã®é·ãã® $2$ ä¹ãæ±ããŠãã ããïŒ |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/1547 | B | OMC140(B) | 200 | 301 | 336 | [
{
"content": "ã$2048=2^{11}$ ã§ããããšã«çæããã°, CMOåã解çãã¹ãæ£æŽæ°å€ãšãªãåŸããã®ãå°ããé ã«äžŠã¹ããš\r\n$$1+2=3,\\quad 1+4=5,\\quad 3+4=7,\\quad 1+8=9,\\quad 3+8=11,\\quad \\cdots$$\r\nåæ§ã«ããŠ, å
šäœã§ããåŸããã®ã¯ $4096$ æªæºã®å¥æ°ãã¹ãŠãã $1$ ãé€ãããã®ã§ããããšã容æã«ããã, ç¹ã«CMOåãæåºããåæ°ãšããŠããåŸãæ倧å€ã¯ $\\textbf{2047}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlin... | ãCMOåã¯OMCã®ããã³ã³ãã¹ãã«åå ããŠããŸããïŒã©ã®åé¡ããããèããŠãããããªãã®ã§ïŒAåé¡ãããã¯åœãŠãã£ãœãã§æ£è§£ããããšæããŸããïŒ\
ãCMOåãAåé¡ã«ã€ããŠããã£ãŠããã®ã¯ïŒä»¥äžã® $2$ ç¹ã®ã¿ã§ãïŒ
- çã㯠**$1$ æªæº** ã®æ£ã®æçæ°å€ã§ããïŒãã® $2048$ åã¯æŽæ°å€ã§ããïŒ
- çããäºãã«çŽ ãªæ£æŽæ° $p,q$ ãçšã㊠$p\/q$ ãšè¡šãããšãïŒ$p+q$ ã解çããïŒ
ããã®ãšãïŒCMOåã¯è§£çãã¹ãæ£æŽæ°å€ãšãªãåŸããã®ãå°ããæ¹ããé ã«ã¡ããã©äžåãã€ïŒCAãåºããŸã§æåºããããšã«ããŸããïŒCMOåãAåé¡ã§**æåºããåæ°**ãšããŠããåŸãæ倧ã®å€ãæ±ããŠãã ... |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/3094 | C | OMC140(C) | 200 | 282 | 338 | [
{
"content": "ãæ¡ä»¶ãã¿ããæ°ã**è¯ãæ°**ãšåŒã¶ïŒãŸãïŒæ°å $a,b,c$ ããã®é ã«äžŠã¹ãŠã§ãã $3$ æ¡ã®æ£æŽæ°ã $\\overline{abc}$ ãšè¡šãããšã«ããïŒ$4$ æ¡ã®å Žåãåæ§ïŒïŒ$3$ æ¡ã®è¯ãæ°ã¯ïŒ$a\\neq b$ ãã¿ãã $1\\leq a\\leq 9$ ããã³ $0\\leq b\\leq 9$ ãçšããŠ\r\n$$\\overline{abb},\\quad \\overline{aab},\\quad \\overline{aba}$$\r\nãšè¡šããããïŒ$3\\times 9\\times 9=243$ åååšããïŒãŸãïŒ$2022$ 以äžã® $4$ ... | ã$100$ ä»¥äž $2022$ 以äžã§ãã£ãŠïŒåé²æ³ã«ãããŠã¡ããã©äžã€ã®æ¡ã ããä»ãšç°ãªãæ£æŽæ°ã¯ããã€ãããŸããïŒãã ãïŒå
é ã® $0$ ã¯èããªããã®ãšããŸãïŒ |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/3579 | D | OMC140(D) | 300 | 128 | 298 | [
{
"content": "ããäºãã«æãæãåããã¢ããŸã決ã, ãã®ããšã«ãã以å€ã®æ§æãèãããšãã.\\\r\nãæãæãåããã¢ã®çµã¿åãã㯠${}_7\\mathrm{C}_2\\cdot{}_5\\mathrm{C}_2 \\div2$ éãååšãã. ãã®ãã¡äžã€ãåºå®ãããšã, ãã¢ã§ã¯ãªã人ããã¢ã§ãã人ãæå·®ããŠãã人æ°ã§å Žååãããããš,\r\n- $0$ 人ã®ãšã, $3$ 人ã§äžåšããããã«æãå¿
èŠããã, $2$ éã\r\n- $1$ 人ã®ãšã, æ®ã $2$ 人ãæãåã£ãŠã¯ãªããªãããšã«æ³šæããŠ, $3\\cdot4\\cdot3=36$ éã\r\n- $2$ 人ã®ãšã, $3... | ã$7$ 人ã®çåŸãããïŒèªèº«ä»¥å€ã®ãã $1$ 人ãäžæã«ç¡äœçºã«æãããšãïŒãäºãã«æãããåã£ãŠãã
$2$ 人çµãã¡ããã© $2$ çµååšãã確çãæ±ããŠãã ããïŒãã ãïŒæ±ãã確çã¯äºãã«çŽ ãªæ£æŽæ° $a$ , $b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/2448 | E | OMC140(E) | 300 | 143 | 256 | [
{
"content": "ã次ãã¿ããæ£æŽæ°ã®åæ°ãæ±ããã°ããïŒ\r\n- (a) åé²è¡šèšã§ã¡ããã© $4$ æ¡ã§ããïŒ\r\n- (b1) $3$ ã®åæ°ã§ããïŒ\r\n- (b2) äžã®äœã®æ°åãå¶æ°ã§ããïŒ\r\n- (c) åé²è¡šèšãããšãã«åæ¡ã«çŸããæ°ã¯ã¡ããã© $2$ çš®é¡ã§ããïŒ\r\n\r\nããæ£æŽæ°ã (a), (b1), (c) ãã¿ãããšãïŒåæ¡ã«çŸããæ°ã¯æ¬¡ã®ããããã«åé¡ã§ããïŒ\r\n- (i) ãã¹ãŠ $3$ ã®åæ°\r\n- (ii) $3$ ã®åæ°(1å)ãš $3$ ã§å²ãåããªãæ°(3å)\r\n- (iii) $3$ ã§å²ããš $1$ äœãæ°(2å)ãš $2$ äœãæ°... | ã$2022$ ã¯æ¬¡ã®æ¡ä»¶ããã¹ãŠã¿ããæ£æŽæ°ã§ãïŒ
- åé²æ°è¡šèšã§ïŒã¡ããã© $4$ æ¡ã§ããïŒããªãã¡ $1000$ ä»¥äž $9999$ 以äžïŒ
- åé²æ°è¡šèšã§ïŒåæ¡ã«çŸããæ°ã¯ã¡ããã© $2$ çš®é¡ã§ããïŒ
- $6$ ã®åæ°ã§ããïŒ
ãã®æ¡ä»¶ããã¹ãŠã¿ããæ£æŽæ°ã¯ïŒ$2022$ ãå«ããŠããã€ãããŸããïŒ |
OMC140 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc140/tasks/4032 | F | OMC140(F) | 400 | 91 | 131 | [
{
"content": "$BC$ ã®äžç¹ã $M$, äžè§åœ¢ $ABC$ ã®åå¿, å€å¿ããããã $H,O$ ãšããã°, $HD:OM=DE:ME=3:1$ ã§ãã, well-known-factãšã㊠$AH:OM=2:1$ ã§ããã®ã§\r\n$$AD:HD=5:3$$\r\nã§ãã. ãŸã, \r\n$$\\angle{HBD}=90^{\\circ}-\\angle{BHD}= 90^{\\circ}-\\angle{ACB} =\\angle{DAC}$$\r\n ã§ããããäžè§åœ¢ $DBH$ ãšäžè§åœ¢ $DAC$ ã¯çžäŒŒã§ããããšãããã, $AD=x$ ãšããã°ä»¥äžãåŸã. \r\n$$... | ãäžè§åœ¢ $ABC$ ã«ãããŠïŒ$A$ ãã $BC$ ã«äžãããåç·ã®è¶³ã $D$ ãšãïŒäžè§åœ¢ $ABC$ ã®[ãªã€ã©ãŒç·](https:\/\/onlinemathcontest.com\/terms)ãšçŽç· $BC$ ã®äº€ç¹ãååšããã®ã§ããã $E$ ãšããã°ïŒ$4$ ç¹ $B,D,E,C$ ã¯ãã®é ã«äžŠã³ãŸããïŒ
$$BD=DE=3, \quad EC=4$$
ã§ãããšãïŒäžè§åœ¢ $ABC$ ã®é¢ç©ã® $2$ ä¹ãæ±ããŠãã ããïŒ |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/2467 | A | OMC139(A) | 300 | 179 | 207 | [
{
"content": "**è£é¡.**ãéè² æŽæ° $n$ ã«ã€ããŠïŒ$10^n \\equiv 9n+1 \\pmod{81}$ïŒ\\\r\n**蚌æ.**ã$n=0$ ã§ã¯æãç«ã€ããïŒéå·®ããšã£ãŠ $9\\times 10^n \\equiv 9 \\pmod{81}$ ã瀺ãã°ããïŒãã㯠$10^n \\equiv 1 \\pmod{9}$ ããæç«ããïŒ\r\n\r\n---\r\nãè£é¡ããïŒåé¡ã¯æ¬¡ã®ããã«èšããããããïŒ\r\n\r\n---\r\n\r\n**åé¡â .**ã$1,10,19,28,\\ldots$ ããããã€ããéžãã§ïŒãããã®åã $81$ ã®åæ°ã«ããïŒ\r\n\r\n---... | ã$81$ ã§å²ãåãïŒãã€åé²æ³ã§è¡šèšãããšãåæ¡ã $0$ ãŸã㯠$1$ïŒäžæ¹ã®ã¿ã§ãããïŒã§ãããããªæ£æŽæ°ã®ãã¡ïŒ$3$ çªç®ã«å°ãããã®ãæ±ããŠãã ããïŒ |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/5138 | B | OMC139(B) | 300 | 168 | 192 | [
{
"content": "**è£é¡.**ãä»»æã®å®æ° $x$ ã«ã€ããŠïŒä»¥äžãæç«ããïŒ\r\n$$\\displaystyle\\lfloor x\\rfloor\r\n+\\bigg\\lfloor x+\\frac{1}{99}\\bigg\\rfloor\r\n+\\bigg\\lfloor x+\\frac{2}{99}\\bigg\\rfloor\r\n+\\cdots\r\n+\\bigg\\lfloor x+\\frac{98}{99}\\bigg\\rfloor\r\n=\\lfloor99x\\rfloor$$\r\n**蚌æ.**ã$x$ ã®å°æ°éšåã $\\dfrac{k}{99}$ ä»¥äž ... | ã以äžã®çåŒãã¿ããæ£ã®å®æ° $x$ ã®ç·åãæ±ããŠãã ããïŒ
$$\lfloor x\rfloor
+\bigg\lfloor x+\frac{1}{99}\bigg\rfloor
+\bigg\lfloor x+\frac{2}{99}\bigg\rfloor
+\cdots
+\bigg\lfloor x+\frac{98}{99}\bigg\rfloor
=9999x-999$$
ãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/2756 | C | OMC139(C) | 400 | 137 | 169 | [
{
"content": "ããŸãïŒæ¡ä»¶ $\\gcd(a,b)\\times\\gcd(b,c) =\\gcd(a,c)$ ã«ã€ããŠèãããïŒ$a,b,c$ ã®å°ãªããšãäžã€ãå²ãåãé©åœãªçŽ æ° $p$ ã«ã€ããŠïŒãããããå²ãåãåæ°ã $x,y,z$ ãšããïŒäžè¬æ§ã倱ãã $x\\leq z$ ãšããã°ïŒæ¡ä»¶ã¯\r\n$$\\min(x,y)+\\min(y,z)=x.$$\r\nããã㧠$y\\geq z$ ã®ãšãïŒ$x\\leq z$ ãã $\\min(x,y)=0$ ãå¿
èŠïŒãã㯠$x=z=0$ ãæå³ããïŒãŸã $y\\lt z$ ã®ãšãïŒ$x\\lt y$ ãªãã° $y=0$ ãšãªãççŸïŒããª... | ãæ£æŽæ° $a,b,c$ ã¯ä»¥äžã®æ¡ä»¶ãã¿ãããŸã.
$$a+b+c=2022,\quad \gcd(a,b) Ã \gcd(b,c) =\gcd(a,c)$$
ãã®ãšãïŒä»¥äžã®åŒã®ãšãããæå°å€ãæ±ããŠãã ããïŒ
$$\dfrac{a^2+b^2+c^2}{\gcd(a,b,c)}$$ |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/5270 | D | OMC139(D) | 500 | 97 | 121 | [
{
"content": "ã äžè§åœ¢ $ABC$ ã®å€æ¥åã®ååŸã $R$ ãšãïŒ$BC$ ã®äžç¹ã $M$ ãšããïŒäžè§åœ¢ $HBC$ ã«ãããŠäžç·å®çãã\r\n$$HM = \\sqrt{\\frac{1}{2}(HB^2 + HC^2) - BM^2} = 6$$\r\nãåŸãïŒ$M$ ã«é¢ã㊠$H$ ãšå¯Ÿç§°ãªç¹ã $D$ ãšãããšïŒããã¯äžè§åœ¢ $ABC$ ã®å€æ¥åäžã«ããïŒãã£ãŠïŒäžè§åœ¢ $ODH$ ã«ãããŠäžç·å®çãã\r\n$$25 + R^2 = 2\\times(36 + OM^2)$$\r\nãåŸãïŒãŸãïŒäžå¹³æ¹ã®å®çãã\r\n$$OM^2 = R^2 - \\frac{361}{4}$$... | ãäžè§åœ¢ $ABC$ ã«ã€ããŠïŒãã®åå¿ã $H$ïŒå€å¿ã $O$ ãšãããšïŒ
$$OH=5,\quad HB^2+HC^2=\frac{505}{2},\quad BC=19$$
ãæãç«ã¡ãŸããïŒãã®ãšãïŒäžè§åœ¢ $ABC$ ã®å€æ¥åã®ååŸã® $2$ ä¹ã¯ïŒäºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/2442 | E | OMC139(E) | 700 | 9 | 37 | [
{
"content": "**è£é¡.**ã$p,k,d$ ãããããçŽ æ°ïŒæ£æŽæ°ïŒæŽæ°ãšããïŒãã ãïŒ$p=2$ ã®ãšã㯠$k\\geq3$ ãšããïŒ$d\\equiv a^2-b^2\\pmod{p^k}$ ãªã $p$ ãšäºãã«çŽ ãªæŽæ° $a,b$ ãååšããããã®ïŒ$p,k,d$ ã«é¢ããå¿
èŠååæ¡ä»¶ã¯ïŒ\r\n- ã$p=2$ ã〠$d\\equiv0 \\pmod8$ããŸã㯠ã$p=3,5$ ã〠$d\\not\\equiv \\pm1\\pmod{p}$ããŸãã¯ã$p\\geq7$ã\r\n\r\n**蚌æ.**ããŸã $p=2$ ã®ãšãïŒ$a,b$ ã¯å¥æ°ã ãã $a^2\\equiv b^2... | ãçŽäº€åº§æšå¹³é¢ã«ãããŠïŒæ¬¡ãã¿ããæ Œåç¹ $(x,y)$ ã**è¶
æ Œåç¹**ãšãã¶ããšãšããŸãïŒ
- $x\equiv a^2, ~ y\equiv b^2 \pmod{2022!}$ ãã¿ãããããªïŒ$2022!$ ãšäºãã«çŽ ãªæŽæ° $a,b$ ãååšããïŒ
ããã¹ãŠã®é ç¹ãè¶
æ Œåç¹ã§ããïŒãã€å蟺ã®é·ãããã¹ãŠæ£æŽæ°å€ã§ãããããªïŒééåãªïŒäžè§åœ¢ã«ã€ããŠïŒãã®åšé·ãšããŠããããæå°å€ãæ±ããŠãã ãã. |
OMC139 (for experts) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc139/tasks/2349 | F | OMC139(F) | 800 | 19 | 48 | [
{
"content": "ãã¹ã³ã¢ã®ç·å $S$ ã¯æ¬¡ã®åŒã§äžããããïŒãã ã $a_1,\\ldots,a_6$ ã¯éè² æŽæ°ãåãïŒ$S\\bmod{p}$ ã«ã€ããŠèãããïŒ\r\n$$S=\\sum_{\\sum a_i=N}\\left(\\dfrac{N!}{\\prod a_i!}\\times \\prod a_i^i\\right)$$\r\nãããã§**äžééä¹åª** $x^{\\underline{n}}$ ãå®æ° $x$ ããã³æ£æŽæ° $n$ ã«å¯Ÿã $x^{\\underline{n}}=x(x-1)\\cdots(x-n+1)$ ãšå®ããïŒ\r\nãã®ãšã $x$ ãå€æ°ãšã¿ãã° $x^{... | ãçŽ æ° $p=10^9+7$ ã«ã€ããŠïŒ$N=4p+7$ ãšããŸãïŒ\
ã$1$ ä»¥äž $6$ 以äžã®æŽæ°ã®çµ $D=(d_1, d_2, \dots, d_N)$ ã«å¯ŸãïŒãã®äžã«å«ãŸãã $i$ ã®åæ°ã $a_i$ ãšãããšãïŒ$D$ ã®**ã¹ã³ã¢**ã
$$a_1\times a_2^2\times a_3^3\times a_4^4\times a_5^5\times a_6^6$$
ãšå®ããŸãïŒãã®ãšãïŒ$D$ ãšããŠãããã $6^N$ éããã¹ãŠã«ã€ããŠïŒã¹ã³ã¢ã®ç·åã $p$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/3510 | A | OMC138(A) | 100 | 276 | 284 | [
{
"content": "ã$2022^{n}$ ã®äžäžæ¡ã¯ $2$, $4$, $8$, $6$, $2$, $ \\ldots$ ãšããåšæãç¹°ãè¿ã. ããŸèããã®ã¯ $n=\\underbrace{2022^{2022^{\\cdot^{\\cdot^{\\cdot^{2022}}}}}}_{2021åã®2022}$ ã§ãã, ãã® $n$ 㯠$4$ ã®åæ°ã§ãããã, æ±ããå€ã¯ $\\mathbf{6}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc138/editorial/3510... | ã以äžã®æ°ãåé²æ³è¡šèšãããšãïŒäžã®äœãæ±ããŠãã ããïŒ
$$\underbrace{2022^{2022^{\cdot^{\cdot^{\cdot^{2022}}}}}}_{2022åã®2022}$$
ããã ãïŒææ°ã¯å³äžã«ãã $2$ æ°ããé ã«èšç®ããŸãïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/4106 | B | OMC138(B) | 200 | 236 | 277 | [
{
"content": "ã$10$ ã§äœåå²ãåãããã¯çŽ å æ° $2,5$ ã®åæ°ã§æ±ºãŸã. $2$ ã¯ååå€ãã®ã§, æ±ããå€ã¯çŽ å æ° $5$ ã®åæ°ãšçãã. \r\n$$\\begin{aligned}\r\n\\prod_{k=1}^{100} k! &= 1! \\times 2! \\times \\ldots \\times 100! \\\\\\\\\r\n&= 1^{100} \\times 2^{99} \\times \\ldots \\times 100^1 \\\\\\\\\r\n&= \\prod_{n=1}^{100} n^{101 - n}\r\n\\end{aligned}$$ ... | ã$1! \times 2!\times 3!\times \cdots \times99!\times 100!$ 㯠$10$ ã§æ倧äœåå²ãåããŸããïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/1754 | C | OMC138(C) | 200 | 216 | 252 | [
{
"content": "ã$B$ ãã $AC$ ã«äžãããåç·ã®è¶³ã $H$ ãšãããšã, äžå¹³æ¹ã®å®çãã\r\n$$BA^2-AH^2=BH^2=BC^2-(5-AH)^2$$\r\nããã解ã㊠$AH=\\dfrac{7}{5},BH=\\dfrac{24}{5}$ ãåŸã. ããã«æ¹ã¹ãã®å®çãã\r\n$$\\frac{7}{5} \\times \\frac{18}{5} =AH\\times CH=BH\\times DH=\\dfrac{24}{5}DH$$\r\nãã£ãŠ $BD=BH+DH=\\dfrac{117}{20}$ ã§ãã, 解çãã¹ãå€ã¯ $\\textbf{137}$ ã§ãã.... | ã$AB=AC=5,BC=6$ ãªãäžè§åœ¢ $ABC$ ã«ãããŠïŒãã®å€æ¥åãš $B$ ãã $AC$ ã«ããããåç·ã®äº€ç¹ã $D ~ (\neq B)$ ãšãããšãïŒ$BD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $m,n$ ã«ãã£ãŠ $\dfrac{m}{n}$ ãšè¡šãããŸãïŒ$m+n$ ã解çããŠãã ããïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/3171 | D | OMC138(D) | 200 | 224 | 252 | [
{
"content": "ãçžå ã»çžä¹å¹³åã®é¢ä¿ããïŒä»¥äžã®ããã«è©äŸ¡ã§ããïŒ\r\n$$\r\n\\frac{n^2}{5}+\\frac{200}{n}=\\frac{n^2}{5}+\\frac{100}{n}+\\frac{100}{n}\\geq 3 \\sqrt[3]{\\frac{n^2}{5}\\cdot \\frac{100}{n}\\cdot \\frac{100}{n}}=30\\sqrt[3]{2}\r\n$$\r\nçå·æç«æ¡ä»¶ã¯ $n=\\sqrt[3]{500}\\approx 7.9$ ã§ããïŒãã®å€ãå¢ã«äžåŒã¯å調ã«å€åããããšã確èªã§ããïŒåŸ®åãå®è¡ããŠãããïŒïŒãããã£ãŠïŒ$n... | ã**æ£æŽæ°** $n$ ã«å¯ŸããŠïŒ$\dfrac{n^2}{5}+\dfrac{200}{n}$ ã®ãšãããæå°å€ã¯ïŒäºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/4295 | E | OMC138(E) | 300 | 125 | 145 | [
{
"content": "ãåè§åœ¢ $BCED$ ãå
æ¥åãæã€ããšã¯ïŒ$BC + DE = BD + CE$ ãšåå€ã§ããïŒåŸã£ãŠïŒäžè§åœ¢ $ADE$ ã®åšé·ã¯\r\n$$(AB - BD) + (AC - CE) + DE = AB + AC - (BD + CE - DE) = AB + AC - BC = 30$$\r\nã§ããïŒãŸãïŒäžè§åœ¢ $ABC$ ã®åšé·ã¯ $AB + BC + CA = 44$ ã§ããïŒäžè§åœ¢ $ABC$ ãšäžè§åœ¢ $AED$ ã¯çžäŒŒã§ããããïŒãã®çžäŒŒã®çžäŒŒæ¯ã¯ $22 : 15$ ã§ããïŒåŸã£ãŠ\r\n$$BD = AB - AD = 19 - 18\\times\\frac{... | ã$AB=19,BC=7,CA=18$ ãªãäžè§åœ¢ $ABC$ ã®èŸº $AB$ äžã«ç¹ $D$ïŒèŸº $AC$ äžã«ç¹ $E$ ããããŸãïŒåè§åœ¢ $BCED$ ãå€æ¥åãšå
æ¥åããšãã«ãã€ãšãïŒ $BD$ ã®é·ãã¯äºãã«çŽ 㪠$a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC138 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc138/tasks/2721 | F | OMC138(F) | 400 | 98 | 159 | [
{
"content": "ãæ¡ä»¶ã«åãè¯ãæ°ã $10$ é²æ³è¡šèšã§ $n=\\overline{ a_1a_2a_3\\ldots a_{18} }$ ãšããïŒ$n$ 㯠$20$ ã®åæ°ã§ã¯ãªãã®ã§ $\\overline{ a_{17}a_{18} }$ ã$20$ã®åæ°ã§ã¯ãªãïŒãã㯠$a_{18}=2$ ãšåå€ã§ããïŒãŸãïŒ$n$ 㯠$22$ ã®åæ°ã§ãªãã®ã§ïŒ$11$ ã®åæ°å€å®æ³ãã $a_1+a_3+a_5+\\cdots +a_{17}\\neq a_2+a_4+a_6+\\cdots+a_{16}+2$ ã§ããïŒãã®åŒã®çå·ãæºãããã㪠$( a_1,a_2,\\ldots ,a_{17}... | ã$2022$ ã®ããã«ïŒåé²æ³è¡šèšã§åæ¡ã®æ°ã $0$ ãŸã㯠$2$ ã ãïŒäžæ¹ã®ã¿ã§ãããïŒãããªãæ£æŽæ°ã**è¯ãæ°**ãšãã¶ããšãšããŸãïŒ\
ã$20$ ã§ã $22$ ã§ãå²ãåããªã $10^{18}$ æªæºã®**è¯ãæ°**ã¯ããã€ãããŸããïŒ |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/4734 | A | OMC137(A) | 100 | 281 | 286 | [
{
"content": "ã$2$ ãš $4$ ãé£ãåããã®ãé€ãã°ããããïŒ$5!-2Ã4!=\\mathbf{72}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc137/editorial/4734"
}
] | ã$1, 2, 3, 4, 5$ ã® $5$ æ°ãå·Šå³äžåã«äžŠã¹ãæ¹æ³ã§ãã£ãŠïŒã©ã®é£ããã $2$ æ°ãäºãã«çŽ ã§ãããã®ã¯ããã€ãããŸããïŒãã ãïŒå·Šå³å転ã§äžèŽãããã®ãåºå¥ããŸãïŒ |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/1992 | B | OMC137(B) | 200 | 255 | 279 | [
{
"content": "ãçŽ å æ°å解 $10!=2^8\\times 3^4\\times 5^2\\times7$ ã«çæããã°, ããã¯æ£ã®çŽæ°ã $270$ åãã€ãã, $10!$ ãå¹³æ¹æ°ã§ãªãããšãšäœµããŠãããã®ç©ã¯ $(10!)^{135}$ ã§ãã. ãã£ãŠ, ããããã€æ£ã®çŽæ°ã®åæ°ã¯\r\n$$(8\\times 135+1)(4\\times 135+1)(2\\times 135+1)(1\\times 135+1)=\\textbf{21554162776}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.co... | ã$10!$ ã®æ£ã®çŽæ°ã®ç·ç©ã«ã€ããŠïŒãã®æ£ã®çŽæ°ã®åæ°ãæ±ããŠãã ãã. |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/5565 | C | OMC137(C) | 300 | 239 | 252 | [
{
"content": "ãäžè¬ã«æ£æŽæ° $n$ ã«å¯ŸããŠä»¥äžã®ç·å\r\n$$ \\sum_{k=0}^{n-1} \\dfrac{{}\\_{n-1}\\mathrm{C}\\_{k}}{k+1} $$\r\nãæ±ãããïŒ\r\n\r\n----\r\n\r\n**解æ³1ïŒ**\r\näºé
ä¿æ°ã«é¢ãã次ã®äºã€ã®çåŒã«æ³šæããïŒ\r\n$$\\dfrac{{}\\_{n-1}\\mathrm{C}\\_{k}}{k+1}=\\dfrac{{}\\_{n}\\mathrm{C}\\_{k+1}}{n},\\quad\\sum_{k=0}^{n}{}\\_{n}\\mathrm{C}\\_{k}=2^n$$\r\næ±... | ã以äžã®ç·åã¯ïŒäºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ
$$ \dfrac{{}\_{15}\mathrm{C}\_{0}}{1} + \dfrac{{}\_{15}\mathrm{C}\_{1}}{2} + \cdots + \dfrac{{}\_{15}\mathrm{C}\_{14}}{15} + \dfrac{{}\_{15}\mathrm{C}\_{15}}{16} $$ |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/1814 | D | OMC137(D) | 300 | 149 | 201 | [
{
"content": "ãäžè§åœ¢ $HEB$ ã«ã€ããŠ, $â EHB$ ã®å€è§ã®äºçåç·ã $DH$ ã§ãããã,\r\n$$EH:BH=ED:BD=1:2$$\r\nãåãã. $EH=x$ ãšããã° $EB=ED= \\sqrt{3}x$ ã§, ãŸãäžè§åœ¢ $EHD$ ãš $EAC$ ã®çžäŒŒã«ãã\r\n$$EC= \\sqrt{3}EA= \\sqrt{3}Ã2 \\sqrt{3}x=6x$$\r\nãåŸãã®ã§ $HC=5x$ ã§ãã, $A, E, F, C$ ã®å
±åããæ¹ã¹ãã®å®çãã\r\n$$AHÃHF=EHÃHC=5x^2$$\r\nãæãç«ã€. äžæ¹, äžè§åœ¢ $AEH$ ã«äžå¹³æ¹ã®å®çãçšã... | ã$H$ ãåå¿ãšããéè§äžè§åœ¢ $ABC$ ã«ãããŠïŒè§ $BHC$ ã®äºçåç·ãšèŸº $AB$ ã亀ãã£ãã®ã§ãã®äº€ç¹ã $D$ ãšãïŒ$CH$ ãšèŸº $AB$ ã®äº€ç¹ã $E$ ãšãããšïŒ
$$AD=DE=EB$$
ãæãç«ã¡ãŸããïŒ$AH$ ãš $BC$ ã®äº€ç¹ $F$ ã«ã€ããŠïŒ$AF=18$ ã§ãããšãïŒ$AB^2$ ãæ±ããŠãã ãã. |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/2716 | E | OMC137(E) | 500 | 49 | 151 | [
{
"content": "ãæ¡ä»¶ãã¿ããæ£æŽæ° $x$ ã¯æ£æŽæ° $a,b$ ãçšããŠ\r\n$$x=a(a+p)=b(b+2p)$$\r\nããªãã¡\r\n$$x=\\left( a+\\frac p2 \\right)^2 -\\frac{p^2}{4} = (b+p)^2-p^2$$\r\nãšè¡šã, ãããå€åœ¢ããããšã§\r\n$$\\left( a+b+\\frac{3p}{2} \\right) \\left( b-a+\\frac p2 \\right) = \\frac{3p^2}{4}$$\r\n$p=2$ ã¯æããã«äžé©ã§ãããã, 巊蟺ã®å æ°ã¯ããããåæŽæ°ã§ãã, ããåŸãå解ã¯ä»¥äžã®ãããã... | ãéè² æŽæ° $n$ ã«å¯ŸãïŒå·®ã $n$ ã§ãã $2$ ã€ã®æ£æŽæ°ã®ç©ãšããŠè¡šããæ°ã $n$ **çæ°**ãšãã¶ããšã«ããŸãïŒäŸãã° $15=3\times 5=1\times 15$ ããïŒ$15$ 㯠$2$ çæ°ã〠$14$ çæ°ã§ãïŒ\
ããã®ãšãïŒä»¥äžã®æ¡ä»¶ãã¿ããæå°ã®çŽ æ° $p$ ãæ±ããŠãã ããïŒ
- $p$ çæ°ã〠$2p$ çæ°ã§ããïŒããã« $2022$ ã®åæ°ã§ããæ£æŽæ°ãååšããïŒ |
OMC137 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc137/tasks/2254 | F | OMC137(F) | 600 | 8 | 76 | [
{
"content": "ãåž $(n+2)$ è§åœ¢ã«äºãã«äº€ãããªããã察è§ç·ã $n-1$ æ¬åŒã $n$ åã®äžè§åœ¢ã«åå²ããå Žåã®æ°ã¯ïŒã«ã¿ã©ã³æ°\r\n$$C_n=\\dfrac{(2n)!}{n!(n+1)!}$$\r\nã§ããããšãç¥ãããŠããïŒ\\\r\nãããããïŒæ£åè§åœ¢ã«äºãã«äº€ãããªããã察è§ç·ã $7$ æ¬åŒã $8$ åã®äžè§åœ¢ã«åå²ããæ¹æ³ã¯ $C_8=1430$ éãããïŒ\r\nãã®åå² ((a)ãšãã) ãã察è§ç·ã $2$ æ¬åé€ããŠã§ããåå² (ååå²ã«ã€ã $21$ éã) ã¯æ¬¡ã® $2$ çš®ã«åé¡ã§ããïŒ\r\n\r\n- åå²(b)ïŒ$4$ åã®äžè§åœ¢ãš $2$ åã®å... | ãæ£åè§åœ¢ã«ãããŠïŒäºãã«äº€ãããªãããã«ïŒç«¯ç¹ã¯å
±æããŠãããïŒ$5$ æ¬ã®å¯Ÿè§ç·ãåŒãïŒ$4$ åã®äžè§åœ¢ãš $2$ åã®åè§åœ¢ã«åå²ããæ¹æ³ã¯äœéããããŸããïŒ\
ããã ãïŒãã¹ãŠã®é ç¹ã¯åºå¥ããŠèããŸãïŒããªãã¡ïŒå転ãå転ã«ãã£ãŠäžèŽãããã®ãåºå¥ããŠæ°ããŸãïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/3295 | A | OMC136(A) | 100 | 192 | 253 | [
{
"content": "ãé·æ¹åœ¢ã®çžŠãšæšªã®é·ãã $a,b$ ãšããïŒ$ab=1000$ ããïŒçžå ã»çžä¹å¹³åã®é¢ä¿ãã\r\n$$2(a+b)\\geq4\\sqrt{ab}=40\\sqrt{10}\\approx 126.5$$\r\nãªã®ã§ïŒããããåšé·ã®æå°ã®æŽæ°å€ã¯ $\\textbf{127}$ ã§ããïŒãªãïŒãã®ãããªé·æ¹åœ¢ã¯ç¢ºãã«ååšããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc136/editorial/3295"
}
] | ãé¢ç©ã $1000$ ã§ããé·æ¹åœ¢ã«ã€ããŠïŒåšé·ã®ãšãåŸãæå°ã®æŽæ°å€ãæ±ããŠãã ããïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/2334 | B | OMC136(B) | 200 | 212 | 230 | [
{
"content": "ãæ£æŽæ° $a,b$ ã«ãã£ãŠïŒäžèŸºã®é·ãã $20-a\\ (a\\lt 20),20,20+b$ ãšè¡šãã°ïŒäžè§äžçåŒããå¿
èŠååæ¡ä»¶ã¯\r\n$$ 20+b \\lt 20+(20-a) \\iff (2\\leq)a+b\\lt 20$$\r\nãããã£ãŠïŒæ±ããå Žåã®æ°ã¯ $1+2+\\cdots+18=\\textbf{171}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc136/editorial/2334"
}
] | ãäžèŸºã®é·ããçžç°ãªãæ£æŽæ°å€ã§è¡šãããäžè§åœ¢ã®ãã¡ïŒ$2$ çªç®ã«é·ã蟺ã®é·ãã $20$ ã§ãããã®ã¯äœéããããŸããïŒãã ãïŒåå㪠$2$ ã€ã®äžè§åœ¢ã¯ãã¹ãŠåããã®ã§ãããšã¿ãªããŸãïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/3199 | C | OMC136(C) | 200 | 185 | 231 | [
{
"content": "**ç¥è§£**\\\r\nã$100\\times100\\times1$ ã®çŽæ¹äœã®è¡šé¢ã®ãã¡ïŒ$2$ ã€ã® $100\\times100$ ã®é¢ã«ã¯ $3$ ãŸã㯠$4$ ãé
眮ãïŒæ®ã $4$ ã€ã®é¢ã®ãã¡ $2$ ã€ã®ïŒå¯Ÿé¢ããŠããïŒé¢ã«ã¯ $6$ ãïŒæ®ãã® $2$ ã€ã®é¢ã«ã¯ $5$ ãé
眮ããããšãã§ããïŒãã®é
眮ã®æ¹æ³ã¯èªæãªäžçã§ããããïŒçã㯠$(3+4)\\times 100+6\\times2\\times 100+5\\times2\\times 100=\\mathbf{72200}$ ã§ããïŒ\r\n\r\n**解説**\\\r\nã以äžïŒ$i,j$ 㯠$... | ãäžèŸºã®é·ãã $1$ ã§ããäžè¬çãªå
é¢äœã®ããããã $10000$ åããïŒãããã $100\times100\times1$ ã®çŽæ¹äœç¶ã«ãã£ã€ããŠäžŠã¹ãŸãïŒããã§ïŒäºãã«æ¥è§ŠããŠãã $2$ ã€ã®é¢ã«ã€ããŠïŒããããã®ç®ãçãããªãããã«ããŸãïŒãã®ãšãïŒè¡šé¢ïŒä»ã®é¢ãšæ¥è§ŠããŠããªãé¢ïŒå
šäœã«ã€ããŠïŒãã®ç®ã®åã®æ倧å€ãæ±ããŠãã ããïŒ\
ããã ãïŒäžè¬çãªå
é¢äœã®ããããã«ãããŠïŒ$6$ ã€ã®é¢ã®ç®ã¯ãããã $1,2,3,4,5,6$ ã§ããïŒå察åŽã«äœçœ®ãã $2$ ã€ã®é¢ã®ç®ã®å㯠$7$ ã§ãããšããŸãïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/3277 | D | OMC136(D) | 300 | 76 | 146 | [
{
"content": "ã$N$ ã $k$ é²æ°è¡šèšããéã«æ«å°Ÿã« $0$ ãã¡ããã© $(n-k+1)$ å䞊ã¶ããšã¯ïŒ$N$ ã $k$ ã§æ倧 $(n-k+1)$ åå²ãåãããšèšãæããããïŒåäžã® $N$ ã«ã€ããŠã¯æ¡ä»¶ãã¿ãã $n$ ã¯é«ã
$1$ åã§ããããšã«æ³šæããïŒ\\\r\nã$N$ ã $2$ ã§å²ãåããæ倧ã®åæ° $v_2(N)$ ã¯ïŒ$k=2$ ãèããã° $v_2(N)=n-1$ïŒäžæ¹ã§ïŒ$n\\geq4$ ã®ãšã $k=4$ ãèããã° $v_2(N)=2(n-3)$ ãŸã㯠$v_2(N)=2(n-3)+1$ ã§ããïŒããããäžèŽããããšããïŒ$n\\leq 5$ ã«ã€ããŠ... | ã次ã®æ¡ä»¶ãæºãã $2$ 以äžã®æŽæ° $n$ ãååšãããããªïŒ$10000$ 以äžã®æ£æŽæ° $N$ ã¯äœåãããŸããïŒ
- $2\leq k \leq n$ ãªãä»»æã®æŽæ° $k$ ã«ã€ããŠïŒ$N$ ã $k$ é²æ³ã§è¡šèšãããšæ«å°Ÿã« $0$ ãã¡ããã© $(n-k+1)$ å䞊ã¶ïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/3281 | E | OMC136(E) | 300 | 55 | 86 | [
{
"content": "ã$AB$ ã®äžç¹ã $M$ïŒ$AC$ ã®äžç¹ã $N$ ãšãããšïŒäžç·å®çãã\r\n$$\\begin{aligned}4AP^2+3BP^2+CP^2&=3(AP^2+BP^2)+(AP^2+CP^2)\\\\\\\\\r\n&=6(MP^2+AM^2)+2(NP^2+AN^2)\\\\\\\\\r\n&=2(3MP^2+NP^2)+\\frac{3}{2}AB^2+\\frac{1}{2}AC^2\r\n\\end{aligned}$$\r\nãããã«ïŒ$MN$ ã®äžç¹ã $K$ïŒ$MK$ ã®äžç¹ã $L$ ãšãããšïŒäžç·å®çãç¹°ãè¿ãçšããŠ\r\n$$\\begin{aligne... | ã$BC=4,CA=5,AB=6$ ãªãäžè§åœ¢ $ABC$ ããããŸãïŒæ¬¡ãã¿ããç¹ $P$ ãååšãããé åã®é¢ç© $S$ ãæ±ããŠãã ããïŒ
$$4AP^2+3BP^2+CP^2\leq100$$
ããã ã $S$ ã¯äºãã«çŽ ãªæ£æŽæ° $p,q$ ãçšã㊠$S=\dfrac{p}{q}\pi$ ãšè¡šããã®ã§ïŒ**ç©** $pq$ ã解çããŠãã ããïŒ |
OMC136 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc136/tasks/2348 | F | OMC136(F) | 400 | 24 | 71 | [
{
"content": "ãæ£ $100$ è§åœ¢ã®é ç¹ããäžã€ãéžãã§ã§ããäžè§åœ¢ã®å
è§ã¯ïŒãã¹ãŠ $180^\\circ\\/100$ ã®æŽæ°åã§ããããšã«çæããïŒ\\\r\nãããŸïŒå¯Ÿç§°æ§ãã $A_1$ãæ倧ã®å
è§ $k\\times 180^\\circ\\/100$ (ã®äžã€) ããšããããªäžè§åœ¢ãèãããšïŒæ®ãã®å
è§ã«ã€ããŠ\r\n\r\n- $k=50,\\cdots,98$ ã®ãšãïŒæ®ãã®äºã€ã®å
è§ã®çµã¯ $99-k$ éãããïŒ\r\n- $k=34,\\cdots,49$ ã®ãšãïŒæ®ãã®äºã€ã®å
è§ã®çµã¯ $3k-99$ éãããïŒ\r\n- $k\\leq 33$ ã®ãšãïŒ$A_1$ ã¯æ倧ã®... | ãæ£ $100$ è§åœ¢ $A_1A_2\cdots A_{100}$ ã® $3$ é ç¹ãçµãã§ã§ãããã¹ãŠã®äžè§åœ¢ã«å¯ŸããŠïŒ$3$ ã€ã®å
è§ã®ãã¡æ倧ã®ãã®ã®ç·åã床æ°æ³ã§æ±ããŠãã ããïŒãã ãïŒæ·»ãåã®é åºãéãã ãã®äžè§åœ¢ã¯åºå¥ããïŒæ倧ã®å
è§ãè€æ°ããå Žåã¯ãã®ãã¡é©åœãªäžã€ãæ倧ã®ãã®ãšããŸãïŒ |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/2301 | A | OMC135(A) | 100 | 246 | 248 | [
{
"content": "ã$2$ æ°ã $m,m+1000$ ãšããã°ïŒäºé€æ³ã«ãã $\\gcd (m,m+1000)=\\gcd (1000,m)$ ã§ããïŒéã«é©åœã« $m$ ãéžã¹ã°æ倧å
¬çŽæ°ãšã㊠$1000$ ã®ä»»æã®æ£ã®çŽæ°ãå®çŸå¯èœã§ããïŒ$1000=2^3\\times 5^3$ ã®æ£ã®çŽæ°ã¯ $\\textbf{16}$ åã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc135/editorial/2301"
}
] | ãå·®ã $1000$ ã§ãã $2$ ã€ã®æ£æŽæ°ã®æ倧å
¬çŽæ°ãšããŠããããå€ã¯ããã€ãããŸããïŒ |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/3161 | B | OMC135(B) | 200 | 224 | 237 | [
{
"content": "ãç·åãæ£åäºé¢äœã®é¢äžã«ãªããã°è¯ãïŒ$2$ ã€ã®é ç¹ãçµã¶æ¹æ³ã¯ ${}\\_{20}\\mathrm{C}\\_{2}=190$ éãã§ããïŒæ£åäºé¢äœã®èŸºã®æ°ã¯ $30$ïŒåé¢ã«é·ã $\\phi$ ã®å¯Ÿè§ç·ã $5$ æ¬ãã€åŒããããïŒè§£ç㯠$190-30-5\\times 12=\\bf{100}$ ãšãªã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc135/editorial/3161"
}
] | ã äžèŸºã $1$ ã§ããæ£äºè§åœ¢ã®å¯Ÿè§ç·ã®é·ãã $\phi$ ãšããŸãïŒäžèŸºã $1$ ã®æ£åäºé¢äœã® $2$ é ç¹ãçµã¶ç·åã§ãã£ãŠïŒãã®é·ãã $\phi$ ãã倧ãããã®ã¯äœæ¬ãããŸããïŒ\
ããã ãïŒæ£åäºé¢äœã¯æ£äºè§åœ¢ãåé¢ã«ãã¡ãŸãïŒ |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/2504 | C | OMC135(C) | 400 | 144 | 199 | [
{
"content": "ãããæ¢çŽåæ° $\\dfrac{p}{q}$ ãæéå°æ°ã§ãããšãïŒ$q$ 㯠$2,5$ 以å€ã®çŽ å æ°ãæããªãããšã«æ³šæãã.\\\r\nã$m,n$ ã®çŽ å æ°å解ã«çŸãã $5$ 以å€ã®å¥çŽ æ°ãïŒé©åœã« $p_1,p_2,\\cdots$ ãšããïŒãã®ãšã\r\n$$m=5^s p_1^{a_1}p_2^{a_2}\\cdots,\\quad n=5^t p_1^{b_1} p_2^{b_2}\\cdots$$\r\nãšè¡šãã°ïŒ$s,t$ ã®å°ãªããšãäžæ¹ã¯ $0$ ã§ããïŒããã§ïŒ$\\dfrac{1}{m}+\\dfrac{1}{n}=\\dfrac{m+n}{mn}$ ãæé... | ãæ£ã®**å¥æ°**ã®çµ $(m,n)$ ã以äžã®æ¡ä»¶ããã¹ãŠã¿ãããŸãïŒ
- $m\lt n\lt 1000m$ïŒ
- $m$ ãš $n$ ã®å°ãªããšãäžæ¹ã¯ $5$ ã§å²ãåããªãïŒ
- $\dfrac{1}{m}+\dfrac{1}{n}$ ã¯ïŒåé²æ³è¡šèšã§ïŒæéå°æ°ãšããŠè¡šçŸã§ããïŒ
ãã®ãšãïŒ$m$ ãšããŠããããå€ã®ç·åãæ±ããŠãã ããïŒ |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/4115 | D | OMC135(D) | 400 | 74 | 129 | [
{
"content": "ãäžåŒãã¿ãã $f(x),g(x)$ ãäžã€èŠã€ããããšã§ïŒãã¹ãŠã®è§£ãè¡šçŸã§ããïŒããã¯Euclidã®äºé€æ³ã®çµæãå©çšããŠäžæ¬¡äžå®æ¹çšåŒã®è§£ãåŸãã®ãšãŸã£ããåãã§ããïŒããã§ã¯ïŒäŸãã°\r\n$$f_0(x)=2x^2-x+4,\\quad g_0(x)=-2x^3+x^2-2x-1$$\r\nãšããã°\r\n$$(x^4+x+1)f_0(x)+(x^3+x+1)g_0(x)=3$$\r\nã§ããããïŒãããçšããã°\r\n$$(x^4+x+1)(f(x)-f_0(x))+(x^3+x+1)(g(x)-g_0(x))=0.$$\r\n$x^4+x+1$ ãš $x^3+x+1$ ã¯å
±... | ãå®æ°ä¿æ°å€é
åŒ $f(x),g(x)$ ãïŒ$f(-1)+g(-1)=57$ ããã³
$$(x^4+x+1)f(x)+(x^3+x+1)g(x)=3$$
ãã¿ãããŸãïŒããã« $f(x)$ ã $3$ 次åŒã®ãšãïŒ$g(2)-f(2)$ ãæ±ããŠãã ããïŒ |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/2098 | E | OMC135(E) | 500 | 46 | 74 | [
{
"content": "ãäžè§åœ¢ $APQ$ ã®å
æ¥åãš $AP,AQ$ ãšã®æ¥ç¹ããããã $M,N$ ãšãã,\r\n$$AM=x,\\quad PL=y,\\quad QL=z$$\r\nãšããã°,\r\n$$PQ=y+z=41,\\quad KL=y-z=1$$\r\nã§ããããïŒäžè§åœ¢ $APQ$ ã®é¢ç©ã $S$ ãšããã° $S=41\\triangle AKL=41\\times 6\\sqrt{7}$ ãåŸãïŒäžæ¹ã§Heronã®å
¬åŒãã\r\n$$S=\\sqrt{xyz(x+y+z)}=\\sqrt{420x(x+41)}.$$\r\nããã解ã㊠$x=\\dfrac{-205+41\\sqr... | ã$AB=AC$ ãªãäºç蟺äžè§åœ¢ $ABC$ ã«ãããŠïŒçŽç· $AB,AC$ ãšãããã $B,C$ ã§æ¥ããåã $\Gamma$ ãšããŸãïŒ$\Gamma$ ã®å£åŒ§ $BC$ äžã« $BK\lt CK$ ãªãç¹ $K$ ããšãïŒ$K$ ã«ããã $\Gamma$ ã®æ¥ç·ã $\ell$ ãšãïŒ$\ell$ ãšçŽç· $AB,AC$ ã®äº€ç¹ããããã $P,Q$ ãšããŸãïŒããã«ïŒäžè§åœ¢ $APQ$ ã®å
æ¥åãš $\ell$ ã®æ¥ç¹ã $L$ ãšããã°ïŒä»¥äžãæç«ããŸããïŒ
$$BP=20,\quad CQ=21,\quad \triangle AKL=6\sqrt{7}$$
ãã®ãšãïŒ$\Gamma$ ã®ååŸ $r$ ã¯æ£æŽ... |
OMC135 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc135/tasks/4856 | F | OMC135(F) | 500 | 13 | 20 | [
{
"content": "ã$N = 7$ ãšããïŒ$i \\in \\\\{1, 2, 3\\\\}, j \\in \\\\{0, 1, 2\\\\}$ ã«å¯ŸãïŒ$n_i$ ã $3$ é²æ°è¡šèšãããšãã«çŸãã $j$ ã®åæ°ã $G(i, j)$ ãšè¡šããšïŒä»¥äžãæãç«ã€ïŒ\r\n- ä»»æã® $i \\in \\\\{1, 2\\\\}, j \\in \\\\{0, 1, 2\\\\}$ ã«å¯ŸãïŒ$F(i, j) = G(i + 1, j) - G(i, j) $\r\n- ä»»æã® $i \\in \\\\{1, 2, 3\\\\}$ ã«å¯ŸãïŒ$G(i, 0) + G(i, 1) + G(i, 2) = N... | ã$0$ ä»¥äž $3^7-1$ 以äžã®æŽæ° $3$ ã€ãããªãå $(n_1, n_2, n_3)$ ãäžããããŠããŸãïŒ\
ããã®åããã³ $i \in \\{1, 2\\}, ~ j \in \\{0, 1, 2\\}$ ã«å¯ŸãïŒ$F(i, j)$ ã以äžã§å®ããŸãïŒ
- $n_{i+1}$ ã® $3$ é²æ³è¡šèšïŒ $7$ æ¡ïŒã§ $j$ ãçŸããåæ°ããïŒ$n_{i}$ ã® $3$ é²æ³è¡šèšïŒ $7$ æ¡ïŒã§ $j$ ãçŸããåæ°ãåŒããŠåŸãããå€ïŒãã ãïŒã $3$ é²æ³è¡šèšïŒ $7$ æ¡ïŒããšã¯ïŒå¿
èŠãªãã°å
é ã« $0$ ãå ããŠã¡ããã© $7$ æ¡ãšããŠè¡šèšããããšãæå³ããïŒ
<details><sum... |
OMC134 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc134/tasks/2977 | A | OMC134(A) | 100 | 255 | 262 | [
{
"content": "ãæ£äºåé¢äœã®åé¢ã¯æ£äžè§åœ¢ã§ãã.\\\r\nããããã£ãŠ, äžèŸºã®é·ãã $a$ ãšãããš, è¡šé¢ç©ã«ã€ããŠæ¬¡ã®åŒãæãç«ã€ã®ã§, $a^4=\\textbf{12}$ ã§ããïŒ\r\n$$\\displaystyle 30=\\frac{\\sqrt{3}}{4}a^2\\times20$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc134/editorial/2977"
}
] | ãè¡šé¢ç©ã $30$ ã®æ£äºåé¢äœã«ã€ããŠïŒãã®äžèŸºã®é·ãã® $4$ ä¹ãæ±ããŠãã ããïŒ |
OMC134 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc134/tasks/2979 | B | OMC134(B) | 200 | 226 | 261 | [
{
"content": "**解æ³1.**ããŸããã¹ãéžã°ããè¡ããã³åã $3$ ã€ãã€éžã¶. ãã®ãšã, åè£ãšãªã $3\\times3=9$ ã€ã®ãã¹ããå®éã« $3$ ã€ã®ãã¹ãéžã¶æ¹æ³ã¯ $3!$ éãååšãã. ãã£ãŠ, æ±ããå Žåã®æ°ã¯ ${}_5\\mathrm{C}_3\\times{}_5\\mathrm{C}_3\\times 3!=\\textbf{600}$ ã§ãã.\r\n\r\n**解æ³2.**ããŸãäžã€ç®ã®ãã¹ã $5^2$ åããèªç±ã«éžã¶. ãã®ãšã, äºã€ç®ã®ãã¹ãšããŠéžã¹ããã®ã¯ $4^2$ åã§ãã, ããã«äžã€ç®ã®ãã¹ãšããŠéžã¹ããã®ã¯ $3^2$ éãã§ãã. åãã... | ã$5\times5$ ã®ãã¹ç®ãæ§æãã $25$ åã®ãã¹ããïŒæ¬¡ã®æ¡ä»¶ãã¿ããããã«çžç°ãªã $3$ ã€ã®ãã¹ãéžã¶æ¹æ³ã¯äœéããããŸããïŒ
- éžãã ãã¹ã®ãã¡ïŒã©ã® $2$ ã€ã®ãã¹ã«ã€ããŠãåãè¡ã«ãåãåã«ããªã
ãã ãïŒå転ãå転ã«ãã£ãŠäžèŽãããã®ãåºå¥ãããã®ãšãïŒãã¹ãéžã¶é åºã¯èããŸããïŒ |
OMC134 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc134/tasks/2981 | C | OMC134(C) | 200 | 186 | 253 | [
{
"content": "ã$x=6$ ã®ãšãäžæ¹çšåŒãã¿ãã $y$ ã¯ååšããªãïŒä»¥äž, $x\\ne 6$ ãšããïŒ\\\r\nãäžæ¹çšåŒã次ã®ããã« $y$ ã«ã€ããŠè§£ãããšã§, $x-6$ 㯠$25$ ã®çŽæ°ã§ããããšããããïŒ\r\n$$\\displaystyle y=x+6+\\frac{25}{x-6}$$\r\nãããã£ãŠ $x$ ã®åè£ã¯ $x=1,5,7,11,31$ ã§ãã,ãã®ãã¡ $y\\gt 0$ ãšãªããã®ã¯ $x=1,7,11,31$ ã§ããïŒ\\\r\nããã£ãŠæ±ãã $x$ ã®ç·å㯠$\\textbf{50}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
... | ãæ£ã®æŽæ°ã®çµ $(x,y)$ ã次ã®åŒãã¿ãããšãïŒ$x$ ã®å€ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ
$$x^2-xy+6y-11=0$$ |
OMC134 (for beginners) | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc134/tasks/2980 | D | OMC134(D) | 200 | 178 | 199 | [
{
"content": "ãæ¹çšåŒ $f(x)=0$ ã®è€çŽ æ°è§£ã $x=1,5$ ã®ã¿ã§ããããšãã,æ£ã®æŽæ° $m,n$ ãçšããŠ\r\n$$f(x)=(x-1)^m(x-5)^n$$\r\nãšè¡šããïŒãã®åŒã« $x=3,9$ ã代å
¥ããŠæ¬¡ã® $2$ åŒãåŸãïŒ\r\n$$(-1)^n 2^{m+n}=2^{1000},\\quad 2^{3m+2n}=2^{2022}$$\r\nãããã£ãŠ $n\\equiv0\\pmod 2,m+n=1000,3m+2n=2022$ ãã $(m,n)=(22,978)$ ãåŸãïŒããªãã¡\r\n$$f(x)=(x-1)^{22}(x-5)^{978}$$\r\nãã,å®æ°... | ãæŽæ°ä¿æ°å€é
åŒ $f(x)$ ã¯æé«æ¬¡ã®ä¿æ°ã $1$ ã§ããïŒ
$$f(3)=2^{1000},\quad f(9)=2^{2022}$$
ãã¿ãããŠããŸãïŒ$x$ ã®æ¹çšåŒ $f(x)=0$ ã®è€çŽ æ°è§£ãïŒéè€åºŠã蟌ããŠïŒ $x=1,5$ ã®ã¿ã§ãããšãïŒ$f(x)$ ã®å®æ°é
ã $5$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
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