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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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OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/10674 | A | OMC225(A) | 100 | 293 | 324 | [
{
"content": "ã$r = \\lfloor r \\rfloor + \\lbrace r \\rbrace$ ãçšããŠäžåŒãå€åœ¢ãããšïŒ\r\n$$\r\n\\begin{aligned}\r\n\\dfrac{1}{\\lbrace r \\rbrace} + \\dfrac{1}{\\lfloor r \\rfloor} = \\dfrac{25}{4r}\r\n&\\iff\\dfrac{r}{\\lbrace r \\rbrace} + \\dfrac{r}{\\lfloor r \\rfloor} = \\dfrac{25}{4} \\\\\\\\\r\n&\\iff\\dfrac{\\lf... | ãæŽæ°ã§ãªãïŒ$1$ 以äžã®å®æ° $r$ ã§ãã£ãŠïŒä»¥äžã®çåŒãã¿ãããã®ã®ç·åãæ±ããŠãã ããïŒ
$$\dfrac{1}{\lbrace r \rbrace} + \dfrac{1}{\lfloor r \rfloor} = \dfrac{25}{4r}$$
ããã ãïŒçãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a + b$ ã®å€ã解çããŠäžããïŒãŸãïŒæ£ã®å®æ° $x$ ã«ã€ã㊠$\lfloor x \rfloor$ 㧠$x$ ã®æŽæ°éšåïŒ$\lbrace x \rbrace$ 㧠$x$ ã®å°æ°éšåãè¡šããã®ãšããŸãïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/10611 | B | OMC225(B) | 400 | 82 | 129 | [
{
"content": "ãäžè§åœ¢ $ABD, ACE, BEH, CDH$ ã¯ããããçŽè§äºç蟺äžè§åœ¢ã§ããïŒ$BE = a, ~ CD = b$ ãšãããšïŒ\r\n$$ AB = 2a + \\sqrt2b, \\quad AC = \\sqrt2a + 2b $$\r\nã§ããïŒåè§åœ¢ $AEHD$ ã®é¢ç©ã¯\r\n$$ 5 = \\frac12 (a + \\sqrt2b)^2 - \\frac12 b^2 = \\frac12(a^2 + b^2) + \\sqrt2 ab$$\r\nã§ããïŒãŸãïŒäžè§åœ¢ $ABC$ ã®å€å¿ã $P$ïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãããšïŒåè§åœ¢ $BPCO$ ã¯æ£æ¹... | ã$\angle A = 45^\circ$ ã§ãããããªéè§äžè§åœ¢ $ABC$ ã®åå¿ã $H$ ãšããŸãïŒ$B, C$ ãã察蟺ã«äžãããåç·ã®è¶³ããããã $D, E$ ãšãïŒäžè§åœ¢ $BHC$ ã®å€å¿ã $O$ ãšãããšïŒ$AO = 7$ ã§ããïŒãã€åè§åœ¢ $AEHD$ ã®é¢ç©ã $5$ ãšãªããŸããïŒãã®ãšãïŒ$BC^{2}$ ã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠäžããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9342 | C | OMC225(C) | 400 | 75 | 124 | [
{
"content": "ãæ±ããã¹ãã¯ïŒ$100 - 4x \\lt z + 2y \\lt 100 + 4x$ ã〠$4x - 100 \\lt 2y - z \\lt 100 - 4x$ ãæºããéè² æŽæ° $(x,y,z)$ ã®çµã®æ°ã§ããïŒãã㧠$x$ ã®å€ãåºå®ããŠïŒæ¡ä»¶ãæºããç¹ $(y, z)$ ã®é åã $yz$ 座æšå¹³é¢äžã«å³ç€ºããããšãèããïŒ$4x - 100 \\lt 100 - 4x$ ãã $0 \\lt x \\lt 25$ ã®å Žåã®ã¿èããã°ããïŒãã®ãšãç¹ $(y, z)$ ã®é å㯠$4$ çŽç·\r\n$$z = -2y + 100 - 4x, \\quad z = -2y ... | ãéè² æŽæ°ã®çµ $(x, y, z)$ ã§ãã£ãŠïŒ
$$\lvert 100 - 4x - 2y \rvert \lt z \lt 100 - \lvert 4x - 2y \rvert$$
ãæºãããã®ã¯ããã€ãããŸããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/8064 | D | OMC225(D) | 400 | 97 | 172 | [
{
"content": "ãç·å $AC$ ã $4$ çåããç¹ãïŒ$A$ ã«è¿ãæ¹ãã $A_1,A_2,A_3$ ãšãïŒ$A_4=C$ ãšããïŒ$A$ ãã $A_i$ ãŸã§ç·ã®äžã®ã¿ãéã£ãŠæçã§ç§»åããæ¹æ³ã®ç·æ°ã $f(i)$ ã§è¡šãïŒ$f(4)$ ãæ±ããã°ããïŒ\\\r\nããŸãïŒ$A_1$ ãéãæ¹æ³ã¯ïŒ$A$ ãã $A_1$ ãŸã§ã $f(1)$ éãïŒ$A_1$ ãã $C$ ãŸã§ã $f(3)$ éãã§ïŒãããã¯ç¬ç«ãªã®ã§å
šäœã§ã¯ $f(1)f(3)$ éãã§ããïŒæ¬¡ã«ïŒ$A_1$ ãéãã $A_2$ ãéãæ¹æ³ã¯ïŒåæ§ã«èã㊠$2f(2)$ éãã§ããïŒããã«ç¹°ãè¿ãããšã§ïŒ$f(4... | ãæ£æ¹åœ¢ $ABCD$ ãããïŒå蟺ã $4$ çåãããããªç¹ã«ãã£ãŠ $16~(=4\times 4)$ åã®å°æ£æ¹åœ¢ã«åãããããã«ç·ãåŒããŸãïŒããã«ïŒä»¥äžã®æäœã $3$ åè¡ããŸãïŒããã§ïŒåæç¹ã§ãå°æ£æ¹åœ¢ããšãã£ããšãïŒãã®å
éšïŒåšäžãé€ãïŒã«ãããªãç·ãåŒãããŠããªããã®ããããã®ãšããŸãïŒ
- ç·å $AC$ ãå
éšïŒåšäžãé€ãïŒãéããããªå°æ£æ¹åœ¢ãããããïŒå蟺ã®äžç¹ã«ãã£ãŠ$4~(=2\times 2)$ åã®å°æ£æ¹åœ¢ã«åãããããã«ç·ãåŒãïŒ
æçµçãªç¶æ³ã«ãããŠïŒç·ã®äžã®ã¿ãéã£ãŠ$A$ ãã $C$ ãŸã§æçã§å°éããæ¹æ³ã®ç·æ°ãæ±ããŠãã ããïŒ
<details><summary>... |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9337 | E | OMC225(E) | 500 | 20 | 52 | [
{
"content": "ã$ζ$ ã $1$ ã®åå§ $101$ ä¹æ ¹ $\\left(= \\cos \\dfrac{2\\pi}{101}+ i\\sin\\dfrac{2\\pi}{101}\\right)$ ãšããïŒãã®ãšãïŒ\r\n$$X^{100} + X^{99} + \\cdots + X + 1 = (X - ζ) (X - ζ^{2}) \\cdots (X - ζ^{100})$$\r\nãæãç«ã€ïŒãã®åŒã $(1)$ ãšããïŒãã®ãšãïŒæ±ããã¹ãå€ã¯ \r\n$$\\prod_{i = 1}^{101} (α_{i} - ζ) (α_{i} - ζ^{2}) \\cdots (α... | ã$X$ ã«é¢ãã $101$ 次æ¹çšåŒ
$$X^{101} + 2024X^{50} - 2025 = 0$$
ã®ïŒéè€åºŠã蟌ããŠïŒ$101$ åã®è€çŽ æ°è§£ã $X=α_{1}, α_{2}, \ldots , α_{101}$ ãšããŸãïŒãã®ãšãïŒ
$$\prod_{i = 1}^{101} \left (\sum_{j = 0}^{100} (α_{i})^{j} \right )$$
ã¯æ£æŽæ°å€ã«ãªãã®ã§ïŒããããã€æ£ã®çŽæ°ã®åæ°ã解çããŠãã ããïŒ |
OMC225 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc225/tasks/9208 | F | OMC225(F) | 500 | 40 | 66 | [
{
"content": "**è£é¡1.**ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$ \\ a_{2n} = 2a_{2n-2} + a_{2n-1}, \\ a_{2n+1} = a_{2n-1} + a_{2n}$ ãæç«ããïŒ\r\n\r\n**蚌æ.**ã$\\lbrace a_{n} \\rbrace$ ã®å®ãæ¹ããïŒ\r\n$$\\begin{aligned}\r\na_{2n} &= (a_{1} + a_{2} + \\cdots + a_{2n-3}) + a_{2n-2} + a_{2n-1} = 2a_{2n-2} + a_{2n-1}, \\\\\\\\\r\na_{2n+1} &= (a_{... | ãæ£æŽæ°å $\lbrace a_{n} \rbrace\_{n=1,2,\ldots}$ ã以äžã®ããã«å®ããŸãïŒ
- $a_{1} = 1$ïŒ
- $n$ ãå¶æ°ã®ãšãïŒ$a_{n} = a_{1} + a_{2} + a_{3} + \cdots + a_{n-1}$ïŒ
- $n$ ãå¥æ°ã®ãšãïŒ$a_{n} = a_{2} + a_{4} + a_{6} + \cdots + a_{n-1}$ïŒ
ããã®ãšãïŒ$\gcd(a_{n}, a_{n+4}) \geq 10^{15}$ ãã¿ããæå°ã®æ£æŽæ° $n$ ãæ±ããŠãã ããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/4432 | A | OMCB016(A) | 100 | 319 | 330 | [
{
"content": "ã$n!$ ã $10$ ã§ã¡ããã© $10$ åå²ãåããããšãæ¡ä»¶ã§ããïŒããã¯ããã« $n!$ ã $5$ ã§ã¡ããã© $10$ åå²ãåãããšèšãæããŠããïŒ$n!$ ã $5$ ã§å²ãåããåæ°ã¯å調å¢å ã§ããããšã«æ°ãã€ãããšïŒ$44!,45!,\\ldots,49!,50!$ ã¯ãããã $5$ ã§ã¡ããã© $9,10,\\ldots,10,12$ åå²ãåããããšããïŒæ±ããç·å㯠$45+\\cdots+49=\\mathbf{235}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.co... | ã$n!$ ã®åé²æ³è¡šèšã«ãããŠïŒæ«å°Ÿã« $0$ ãã¡ããã© $10$ å䞊ã¶ãããªæ£æŽæ° $n$ ã®ç·åãæ±ããŠãã ãã. |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/8193 | B | OMCB016(B) | 200 | 91 | 198 | [
{
"content": "ã$Q(x)=P(x)-(x-11)^2$ ãšãããšïŒ$Q(x)$ ããŸãæŽæ°ä¿æ°å€é
åŒã§ããïŒ$Q(3)=Q(8)=Q(13)=0$ ãã¿ããïŒãããã£ãŠïŒå æ°å®çã«ããããæŽæ°ä¿æ°å€é
åŒ $R(x)$ ãååšã㊠$Q(x)=(x-3)(x-8)(x-13)R(x)$ ãã¿ããïŒãã®ãšã $P(10)=1-42R(10)$ ãšãªãããïŒ$P(10)$ ã®åãåŸãå€ã¯ $42$ ã§å²ã£ãäœãã $1$ ã§ããæŽæ°å
šäœã§ããïŒååæ§ã¯ $R(x)$ ãå®æ°ãšããããšã§ãããïŒïŒãã£ãŠïŒè§£çãã¹ãå€ã¯\r\n$$\\sum_{k=0}^{23}(42k+1)=\\mathbf{11616}.... | ã$x$ ã®æŽæ°ä¿æ°å€é
åŒ $P(x)$ ã
$$P(3)=64, \quad P(8)=9, \quad P(13)=4$$
ãã¿ãããšãïŒ$P(10)$ ã®ãšããã $1$ ä»¥äž $1000$ 以äžã®æŽæ°å€ã®ç·åãæ±ããŠäžããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/6453 | C | OMCB016(C) | 200 | 96 | 147 | [
{
"content": "ãçŽç· $AB$ ãšçŽç· $CD$ ãšã®äº€ç¹ã $E$ ïŒçŽç· $AD$ ãšçŽç· $BC$ ãšã®äº€ç¹ã $F$ ãšããïŒ$\\cos{\\angle{BCD}}=\\dfrac{3}{5}$ ãã, $CE=5x,CF=5y$ ãšããã°ïŒ\r\n$$BC=3x,\\quad BE=4x,\\quad CD=3y,\\quad DF=4y$$\r\nãšãªãïŒæ¬¡ã«äžè§åœ¢ $ABF$ ãšäžè§åœ¢ $ADE$ ã¯çžäŒŒã§ããïŒæ¡ä»¶ããçžäŒŒæ¯ã¯ $9:7$ïŒãã£ãŠïŒ\r\n$$BF:DE=5y-3x:5x-3y=9:7$$\r\nã§ããããïŒããã解ãããšã§ $33x=31y$ ãåããïŒãããã£ãŠïŒ... | ãé¢ç©ã $4590$ ã§ããåè§åœ¢ $ABCD$ ã¯
$$\angle ABC = \angle ADC = 90^\circ,\quad AB:AD = 9:7,\quad \cos\angle BCD = \frac{3}{5}$$
ãæºãããŸãïŒãã®ãšãïŒç·å $AC$ ã®é·ãã®äºä¹ãæ±ããŠãã ããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/3945 | D | OMCB016(D) | 200 | 156 | 253 | [
{
"content": "\r\n\r\nãäžå³ã®ããã«ãã¹ç®ã«ååãã€ããïŒ\\\r\nã$6$ ã®äœçœ®ãåºæºã«ããŠèããïŒãã以å€ã®æ°ãçŽ å æ°ã $2$ ã®çŽ¯ä¹ã® $\\\\{2,4,8\\\\}$ ã® $\\bigcirc$ ã°ã«ãŒãïŒçŽ å æ°ã $3$ ã®çŽ¯ä¹ã® $\\\\{3,9\\\\}$ ã® $\\times$ ã°ã«ãŒãïŒä»ã®æ°ãšå
±éã®çŽ å æ°ãæããªã $\\\\{1,5,7\\\\}$ ã® $\\bigtriangleup$ ã°ã«ãŒãã«åãããšïŒ\r\næ¥ããããšãã§ããªã... | ã $3\times 3$ ã®ãã¹ç®ãããïŒããããã®ãã¹ã« $1$ ä»¥äž $9$ 以äžã®æŽæ°ã $1$ åãã€æžã蟌ã¿ãŸãïŒã©ã®çžç°ãªããã¹ãçžç°ãªãæ°åãæžã蟌ãŸããŠãããšãïŒæ¬¡ã®æ¡ä»¶ãæºããæžã蟌ã¿æ¹ã¯å
šéšã§äœéããããŸããïŒ
- ç·åãå
±æãããã¹ã«æžããã $2$ åã®æŽæ°ã¯ãã¹ãŠäºãã«çŽ ïŒ
ãã ãïŒå転ãå転ã«ãã£ãŠäžèŽããæžã蟌ã¿æ¹ã¯åºå¥ããŸãïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/10374 | E | OMCB016(E) | 300 | 37 | 61 | [
{
"content": "ã$0 \\leq \\theta \\lt 2\\pi$ ãšããŠããïŒäžããããæ¹çšåŒãå€åœ¢ãããš\r\n$$(x-2)(x-(\\cos2\\theta+i \\sin2\\theta))(x-(\\cos2\\theta-i \\sin2\\theta)) = 0$$\r\nãšãªãããïŒäžè¬æ§ãã \r\n$$( \\alpha, \\beta, \\gamma ) = ( 2, \\cos2\\theta+i \\sin2\\theta, \\cos2\\theta -i \\sin2\\theta )$$ \r\nãšããŠããïŒãã®ãšãïŒ\r\n$$( \\alpha^n, \\b... | ã$\theta$ ãå®æ°ãšãïŒ$x$ ã«é¢ããæ¹çšåŒ
$$x^3 - (4 \cos^2Ξ) x^2 + (4 \cos{2Ξ} + 1)x - 2 = 0$$
ã®éè€ãå«ãã $3$ ã€ã®è€çŽ æ°è§£ã $α, β, γ$ ãšããŸãïŒãããšïŒ$α^n, β^n, γ^n$ ãããããå®æ°ãšãªããããªæ£æŽæ° $n$ ãååšãïŒãã®æå°å€ã¯ $100100$ ãšãªããŸããïŒãã®ãšãïŒ$α + β + γ$ ãšããŠããåŸãå€ã®ç·åã解çããŠäžããïŒ |
OMCB016 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb016/tasks/6898 | F | OMCB016(F) | 400 | 15 | 26 | [
{
"content": "ãäžè§åœ¢ $FBQ$ ãšäžè§åœ¢ $FCQ$ ã®é¢ç©ã¯çããã®ã§ïŒä»¥äžãæç«ããïŒ\r\n$$\\frac12BF\\times BQ \\sin\\angle FBQ = \\frac12CF\\times CQ\\sin\\angle FCQ$$\r\nãŸãïŒ$\\sin\\angle FBQ = \\sin\\angle FCQ$ ã§ããããïŒä»¥äžãæç«ããïŒ\r\n$$BQ : CQ = CF : BF = AB : AC$$\r\nã§ããïŒããã«ïŒ$AB : AC = BD : CD$ ã§ããããïŒäžè§åœ¢ $ADQ$ ã®å€æ¥åã¯ïŒç·å $BC$ ã«å¯Ÿãã $AB : AC$ ã®ã¢ã... | ã$BC = 8, CA = 10, AB = 7$ ã§ããäžè§åœ¢ $ABC$ ã®å€æ¥åã $\omega$ ãšããŸãïŒãŸãïŒ$\angle BAC$ ã®äºçåç·ãšèŸº $BC$ ã®äº€ç¹ã $D$ ãšãïŒ$\omega$ äžã®ç¹ $F$ ã $AF \parallel BC$ ãæºãããŠãããšããŸãïŒããã«ïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãïŒ$FM$ ãš $\omega$ ã®äº€ç¹ã®ãã¡ $F$ ã§ãªãæ¹ã $Q$ïŒäžè§åœ¢ $ADQ$ ã®å€æ¥åãšçŽç· $FM$ ã®äº€ç¹ã®ãã¡ïŒ$Q$ ã§ãªãæ¹ã $R$ ãšããŸãïŒãã®ãšãïŒç·å $QR$ ã®é·ãã®äºä¹ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\cfrac{b}{a}$ ãšè¡šãããã®... |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/7092 | A | OMC224(A) | 200 | 286 | 309 | [
{
"content": "**ééã£ã解æ³.**ãæ¿åºŠã®é«ããã®ããé ã«ç ç³æ°Ž $C, D, E$ ãéžãã§æ··ãåãããïŒãã®ãšãïŒåŸãããç ç³æ°Žã®æ¿åºŠã¯ïŒ\r\n$$\\dfrac{500 \\times 64 + 900 \\times 40 + 1000 \\times 36}{500 + 900 + 1000} = \\dfrac{130}{3} = 43.333...$$\r\nã§ããããïŒç¹ã«è§£çãã¹ãå€ã¯ $\\bf{4333}$ ã§ããïŒ\r\n\r\n**æ£ãã解æ³.**ãäžèšãééã£ã解æ³ãã§ã®çµã¿åããã§ã®èšç®ã«ããïŒæ±ããæ倧å€ã¯ $43\\\\%$ ããã倧ããããšããããïŒ \r\nã... | ã倪éåã¯æ¿åºŠã $8\\%$ ã®ç ç³æ°Ž $A$ ã $100\rm{g}$ïŒ$24\\%$ ã®ç ç³æ°Ž $B$ ã $300\rm{g}$ïŒ$64\\%$ ã®ç ç³æ°Ž $C$ ã $500\rm{g}$ïŒ$40\\%$ ã®ç ç³æ°Ž $D$ ã $900\rm{g}$ïŒ$36\\%$ ã®ç ç³æ°Ž $E$ ã $1000\rm{g}$ ã® $5$ çš®é¡ã®ç ç³æ°ŽãçšæããŸããïŒ
ã倪éåã¯ãã®äžã® $3$ çš®é¡ã®ç ç³æ°Žãéžã³ïŒãã®å
šãŠãæ··ãåãããŠæ°ããç ç³æ°Žãäœãããšã«ããŸããïŒæ°ããç ç³æ°Žã®æ¿åºŠãšããŠèãããããã®ã®å
æ倧ã®ãã®ã¯ $x [\\%]$ ã§ãããšããŸãïŒ$100x$ ã $10^{-1}$ ã®äœã§åæšäºå
¥ããå€... |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/8687 | B | OMC224(B) | 200 | 261 | 286 | [
{
"content": "ã$N=2$ ã®ãšãã¯æããã« $f(N) = 1$ ã§ããïŒä»¥äžïŒ$N\\geq 3$ ãšããïŒ\\\r\nãå¥æ°åã®æäœã®åŸã§ $A$ 㯠$(N,N-1,\\ldots,1)$ ãå·¡åããããã®ïŒå¶æ°åã®æäœã®åŸã§ $A$ 㯠$(1,2,\\ldots,N)$ ãå·¡åããããã®ã«ãªãããïŒ$f(N)$ ã¯å¶æ°ã§ããïŒããŸïŒé£ç¶ãã $2$ åã®æäœã¯ãæ«å°Ÿã® $2$ é
ãé çªãå€ããã«å
é ã«ç§»ãããšãŸãšããããïŒãã®è¡šçŸã«ããïŒ$f(N)=\\mathrm{lcm}(N,2)$ ã§ããããšããããïŒ\\\r\nã$N$ ã®å¶å¥ã§å ŽååãããŠç·åãæ±ããããšã§ïŒæ±ããå€ã¯ $\\te... | ãé·ã $N$ ã®æ°å $A$ ãããïŒã¯ãã㯠$(1,2,\dots,N)$ ã§ãïŒ$A$ ã«å¯Ÿãã以äžã®æäœãïŒã¯ãã㊠$(1,2,\dots,N)$ ã«åã³æ»ããŸã§ç¹°ãè¿ãè¡ããŸãïŒ
- å¥æ°åç®ã®æäœã§ã¯ïŒåãã $N-1$ é
ã®äžŠã³ãååŸéã«ããïŒ
- å¶æ°åç®ã®æäœã§ã¯ïŒåŸããã $N-1$ é
ã®äžŠã³ãååŸéã«ããïŒ
è¡ãããæäœã®åæ°ã $f(N)$ ãšãããšãïŒ$f(2)+f(3)+\cdots+f(99)+f(100)$ ãæ±ããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/8724 | C | OMC224(C) | 300 | 134 | 203 | [
{
"content": "ãçµè«ããè¿°ã¹ããšïŒå
æ倪éåãåãŠãããšã¯ä»¥äžãšåå€ã§ããïŒä»¥äžïŒããã**åå©æ¡ä»¶**ãšãã¶ïŒ\r\n\r\n- $A$ ã®ã¡ããã© $1$ é
ã®ã¿ã $\\bmod\\ 3$ 㧠$2$ ã§ããïŒæ®ãã¯ãã¹ãŠ $\\bmod\\ 3$ 㧠$0$ ã§ããïŒ\r\n\r\nããŸãïŒåå©æ¡ä»¶ãæºããããŠãããšãïŒå
æ倪éåã¯ãã¹ãŠã®é
ã$\\bmod\\ 3$ 㧠$0$ ã§ãããããªç¶æ
ã«ã§ããïŒãã®åŸã¯ïŒçŽåã«åŸæ次éåãéžãã é
ãéžã³ç¶ããã°ããïŒ\\\r\nãããããµãŸãããšïŒåŸæ次éå㯠$\\bmod\\ 3$ 㧠$0$ ã§ãªãé
ãå«ãŸããç¶æ
ã§æäœãç¶ããå¿
èŠãããïŒå... | ãé·ã $11$ ã®æŽæ°å $A=(A_1,A_2,\dots,A_{11})$ ãçšããŠïŒå
æ倪éåãšåŸæ次éåãã²ãŒã ãããŸãïŒå
æ倪éåããå§ããŠïŒããããã«èš±ããã以äžã®æäœã亀äºã«è¡ãïŒå
ã«æäœãã§ããªããªã£ãæ¹ãè² ãã§ãïŒ
- å
æ倪éåïŒ$1 \le i \le 11$ ã〠$A_i \ge 2$ ãªãæŽæ° $i$ ãä»»æã«äžã€éžã³ïŒ$A_i$ ã $2$ æžããïŒ
- åŸæ次éåïŒ$1 \le i \le 11$ ã〠$A_i \ge 1$ ãªãæŽæ° $i$ ãä»»æã«äžã€éžã³ïŒ$A_i$ ã $1$ æžããïŒ
$A_1,A_2,\ldots,A_{11}$ ããã¹ãŠ $1$ ä»¥äž $11$ 以äžã§ã... |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/6710 | D | OMC224(D) | 400 | 12 | 31 | [
{
"content": "<details>\r\n<summary>ã·ã ãœã³ç·ã«ã€ããŠ<\\/summary>\r\näžè¬ã«ïŒäžè§åœ¢ $XYZ$ ã®å€æ¥åäžã®ç¹ $W$ ããäžè§åœ¢ $XYZ$ ã®å蟺 (ã延é·ããçŽç·) ã«äžãããåç·ã®è¶³ãã¡ã¯ïŒåäžçŽç·äžã«ããïŒãã®çŽç·ã®ããšãïŒäžè§åœ¢ $XYZ$ ã«å¯Ÿãã $W$ ã®ã·ã ãœã³ç·ãšããïŒæ¬åã«ãããŠã¯ïŒ$3$ ç¹ $P, Q, R$ ã¯äžè§åœ¢ $ABC$ ã«å¯Ÿãã $D$ ã®ã·ã ãœã³ç·ããªãïŒ\r\n<\\/details>\r\n\r\nã$4$ ç¹ã®çµ $(B,D,P,R), (C,D,P,Q)$ ã¯ïŒããããç·å $BD, CD$ ãçŽåŸãšããåäžã«ååš... | ãéè§äžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ãå«ãŸãªã匧 $BC$ äžã«ç¹ $D$ ãåãïŒ$D$ ããçŽç· $BC, CA, AB$ ã«äžãããåç·ã®è¶³ããããã $P,Q,R$ ãšãããšïŒä»¥äžãæç«ããŸããïŒ
$$BD = 25,\quad CD = 29,\quad PQ = PR = 20$$
ã$\angle BAC$ ã®äºçåç·ãšèŸº $BC$ ãšã®äº€ç¹ã $E$ ãšãããšãïŒ$(BE - CE)^2$ ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/7255 | E | OMC224(E) | 400 | 100 | 202 | [
{
"content": "ã$2$ 以äžã®æ£æŽæ° $n$ ã«å¯ŸãïŒ$g(n)$ ã以äžã®ããã«å®ããïŒ\r\n- $n$ ãçŽ æ°ãªãã°ïŒ$g(n) = -n$ ãšããïŒ\r\n- $n$ ãçŽ æ°ã§ãªããªãã°ïŒ$g(n)$ 㯠$n$ ã®æ£ã®çŽæ°ã®ãã¡ $3$ çªç®ã«å°ããæ°ãšããïŒ\r\n\r\nãã®ãšã $f(n)g(n)=n$ ãæãç«ã€ïŒããŸïŒ$g(k)$ ã $3$ ã®åæ°ãšãªãã®ã¯æ¬¡ã® $3$ éãã®ç¶æ³ã«éãããïŒ\r\n\r\n- $k=3$ ã®ãšã $g(k)=-3$ ãšãªãïŒ\r\n- $6\\mid k$ ã®ãšã $g(k)=3$ ãšãªãïŒ\r\n- $2\\nmid k$ ã〠$5\\nmi... | ã$2$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒ$n$ ãå²ãåãæŽæ°ã®ãã¡ $3$ çªç®ã«å€§ãããã®ã $f(n)$ ãšããŸãïŒäŸãã°ïŒ$f(2)=-1, f(22)=2, f(224)=56$ ã§ãïŒãã®ãšãïŒ
$$\prod_{k=2}^{800}f(k)$$
ã $3$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
OMC224 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc224/tasks/9795 | F | OMC224(F) | 500 | 10 | 26 | [
{
"content": "ãäžåŒã§ $n = 1$ ã®å ŽåãèãïŒ$a_1 = 0$ ã§ããïŒãŸãïŒ$m$ ã $2$ ä»¥äž $N$ 以äžã®æŽæ°ãšãããšãïŒäžåŒã§ $n = m$ ã®å Žåãã $n = m-1$ ã®å ŽåãåŒãããšã§ïŒ\r\n$$\\sum_{d\\mid m} a_d = \\begin{cases}\r\n0 &(m \\equiv 1 \\mod 3)\\\\\\\\\r\n1 &(m \\not\\equiv 1 \\mod 3)\r\n\\end{cases}\\tag1$$\r\nãåŸãïŒ$a_1 = 0$ ãš $(1)$ ãåæã«æºãããã㪠$a_1, a_2, \\ldots, a_... | ã$N = 10^{8}$ ãšããŸãïŒ$N$ åã®å®æ° $a_1, a_2, \ldots, a_{N}$ ãïŒä»»æã® $1$ ä»¥äž $N$ 以äžã®æŽæ° $n$ ã«ã€ããŠä»¥äžã®åŒãæºãããŠããŸãïŒ
$$\bigg\lfloor \frac{2n}{3} \bigg\rfloor = \sum_{k = 1}^{n} a_k \bigg\lfloor \frac{n}{k} \bigg\rfloor$$
ã$S = \\{a_k\mid 1 \le k \le N\\}$ ãšãããšãïŒ$\sum_{a\in S}|a|$ ã解çããŠãã ããïŒããªãã¡ïŒ$a_1,\ldots, a_{N}$ ã®äžã«çŸããå®æ°ãã¹ãŠã«ã€ããŠïŒãã®çµ¶å¯Ÿ... |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/8619 | A | OMCB015(A) | 100 | 315 | 325 | [
{
"content": "ãåè§åœ¢ $ABCD$ ã®é¢ç©ã $S$ ãšãïŒ$AC$ ãš $BD$ ã®äº€ç¹ã $X$ ãšãïŒ$\\angle{AXB}=\\theta$ ãšãããšïŒ\r\n$$\r\n\\begin{aligned}\r\nS&=\\dfrac{1}{2}\\bigl(AX\\cdot BX\\sin\\theta+BX\\cdot CX\\sin\\(\\pi-\\theta)+CX\\cdot DX\\sin\\theta+DX\\cdot AX\\sin(\\pi-\\theta)\\bigr)\\\\\\\\\r\n&=\\dfrac{1}{2}(AX\\cdot BX+BX\\cdot CX... | ãåžåè§åœ¢ $ABCD$ ã«ãããŠïŒ$AC+BD=100$ ãæãç«ã€ãšãïŒåè§åœ¢ $ABCD$ ã®é¢ç©ãšããŠããããæ倧å€ãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/11333 | B | OMCB015(B) | 200 | 262 | 282 | [
{
"content": "ããŸãïŒ $z=y^2+2y+3$ ãšãããšïŒ\r\n $$\r\nz=(y+1)^2+2\r\n $$\r\nããïŒ $z\\geq 2$ ãæºããïŒ$f(x,y)=0$ ã®è§£ã¯ $x=-z\\pm \\sqrt{z^2+4}$ ãšãªãã®ã§ïŒå€§ããæ¹ã®è§£ã«ã€ããŠèãããšïŒ\r\n $$\r\nx=-z+\\sqrt{z^2+4}=\\dfrac{4}{z+\\sqrt{z^2+4}}\r\n $$\r\nãšãªãã®ã§ïŒ $z+\\sqrt{z^2+4}$ ã®æå°å€ãèããã°ããïŒ\r\nããã§ïŒ $z+\\sqrt{z^2+4}$ 㯠$z\\gt0$ ã§å調å¢å ãªé¢æ°ã§ããã®ã§ïŒ$z=... | ãå®æ° $x , y$ ã«å¯ŸããŠé¢æ° $f(x , y)$ ã以äžã®ããã«å®çŸ©ããŸãïŒ
$$
f(x , y)=x^2+2(y^2+2y+3)x-4
$$
$f(x , y)=0$ ãæºãããªãã $x,y$ ãåããšãïŒ$x$ ã®æ倧å€ãæ±ããŠãã ããïŒãã ãæ±ããå€ã¯æ£ã®æŽæ° $a , b$ ãçšã㊠$\sqrt{a}-b$ ãšè¡šããã®ã§ïŒ $ab$ ã®å€ã解çããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/3622 | C | OMCB015(C) | 200 | 186 | 265 | [
{
"content": "ã$10^{3622}-1$ 以äžã®æ£æŽæ°ã§ãã£ãŠïŒ$10^{3621}$ ã®äœã $1$ ã§ãããã®ïŒ$10^{3620}$ ã®äœã $1$ ã§ãããã®ïŒ$\\cdots$ ïŒ$10^{0}$ ã®äœã $1$ ã§ãããã®ã¯ïŒãããããã¹ãŠ $10^{3621}$ åã§ããïŒããã $2,3,\\cdots ,9$ ã§ãåæ§ã«èããããšã§ïŒ\r\n$$\r\nS=(10^{3621}\\times 3622)(1+2+\\ldots +9)+1=16299 \\underbrace{0\\cdots 0}_{3621å}1\r\n$$ \r\nãããã£ãŠïŒè§£çãã¹ãå€ã¯ $\\textbf... | ã$10^{3622}$ 以äžã®æ£æŽæ°ãã¹ãŠã«ã€ããŠïŒããããã®åæ¡ã®åã®ç·åã $S$ ãšããŸãïŒ$S$ ã®åæ¡ã®åãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/4546 | D | OMCB015(D) | 200 | 193 | 264 | [
{
"content": "ã$f(n)=180^\\circ-\\dfrac{360^\\circ}{n}$ ããïŒæ¡ä»¶ã¯æ¬¡ãšåå€ã§ããïŒ\r\n$$\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=\\frac{1}{2},\\quad 3\\leq a\\lt b\\lt c$$\r\nç¹ã« $\\dfrac{3}{a}\\gt \\dfrac{1}{2}$ ããïŒ$a=3,4,5$ ã§ããïŒ\r\n- $a=3$ ã®ãšãïŒæ¡ä»¶ã¯æ¬¡ãšåå€ãªã®ã§ $c=15,18,24,42$ ãåŸãïŒ\r\n$$(b-6)(c-6)=36,\\quad 4\\leq b\\lt c$$\r\n- $... | ã$3$ 以äžã®æŽæ° $n$ ã«å¯ŸãïŒåºŠæ°æ³ã§ã®æ£ $n$ è§åœ¢ã®äžã€ã®å
è§ã®å€§ããã $f(n)$ ã§è¡šããŸãïŒ$3\leq a\lt b\lt c$ ãªãæŽæ°ã®çµ $(a,b,c)$ ã $f(a)+f(b)+f(c)=360^\circ$ ãã¿ãããšãïŒ$c$ ãšããŠããããå€ã®ç·åãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/5163 | E | OMCB015(E) | 200 | 199 | 225 | [
{
"content": "ã蟺 $AD$ äžã« $AE = 5$ ãªãç¹ $E$ ãåããšïŒ$DE = 3$ ã§ããããäžè§åœ¢ $CDE$ ã¯æ£äžè§åœ¢ã§ããïŒåŸã£ãŠ $CE = 3$ ã§ããããïŒäžè§åœ¢ $ABC$ ãš $AEC$ ã¯äžèŸºçžçã§ååã§ããïŒåŸã£ãŠ $\\angle ABC = \\angle AEC = 120^\\circ$ ãåããã®ã§ïŒäœåŒŠå®çãã $AC = 7$ ãåããïŒãŸãïŒ\r\n$\\angle ABC + \\angle ADC = 180^\\circ$\r\nã§ããããåè§åœ¢ $ABCD$ ã¯åã«å
æ¥ããã®ã§ïŒPtolemy ã®å®çããæ±ããå€ã¯\r\n$$BD = \\fr... | ãåžåè§åœ¢ $ABCD$ ã¯ïŒ
$$AB=5,\quad BC=CD=3,\quad DA=8,\quad \angle CDA = 60^\circ$$
ãæºãããŸãïŒ$BD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããããïŒ$a+b$ ã解çããŠäžããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/8425 | F | OMCB015(F) | 200 | 172 | 210 | [
{
"content": "ãå€å¥åŒãèããããšã§ïŒæ¡ä»¶ã¯ä»¥äžã®ããã«è¡šããïŒ\r\n$$ a+b=1, \\quad a^2\\geq 4b, \\quad b^2\\geq 4a.$$\r\n$ab$ å¹³é¢äžã§ããããå³ç€ºããããšã§ïŒ$a+b=1$ ã®ããšã§ããã¯ä»¥äžãšåå€ã§ãããšãããïŒ\r\n$$ a\\leq -2-2\\sqrt{2} \\quad \\text{ãŸãã¯} \\quad a\\geq 3+2\\sqrt{2}. $$\r\nããã« $ab$ å¹³é¢äžã§ååš $a^2+b^2=k$ ãšå
±éç¹ããã€æ¡ä»¶ãèããã°ïŒ$k$ 㯠$\\\\{a,b\\\\}=\\\\{-2-2\\sqrt{2},... | ãå®æ° $a,b$ 㯠$a+b=1$ ãã¿ãããŠããŸãïŒãŸãïŒ$x$ ã«ã€ããŠã® $2$ 次æ¹çšåŒ
$$x^2+ax+b=0,\quad x^2+bx+a=0$$
ã¯ïŒããããå°ãªããšãäžã€ã®å®æ°è§£ãæã¡ãŸãïŒãã®ãšãïŒ$a^2+b^2$ ã®ãšãããæå°å€ã¯ïŒæ£ã®æŽæ° $A,B,C$
ïŒ $C$ ã¯å¹³æ¹å åãæããªãïŒãçšã㊠$A+B\sqrt{C}$ ãšè¡šãããã®ã§ïŒ$A+B+C$ ã解çããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/10781 | G | OMCB015(G) | 300 | 121 | 191 | [
{
"content": "ãé¢ã®åæ°ã $f$ ãšããïŒæãå€åŽã®éããé¢ãå«ãããšã«æ³šæããïŒãšïŒãªã€ã©ãŒã®å€é¢äœå®çããïŒ\r\n$$\r\nn=10000-f+2\r\n$$\r\nãæãç«ã€ïŒãã£ãŠ $f$ ãæ倧ã§ãããšããèããã°ããïŒåé¢ã¯ $3$ 蟺以äžæã€ããïŒæ¬¡ãæãç«ã€ïŒ\r\n$$\r\n3f \\leqq 2 \\cdot 10000\r\n$$\r\nãã£ãŠ $f$ ã®æ倧å€ã¯ $6666$ ã§ããïŒãã®ãšã $n = 3336$ ãšãªãïŒå®éïŒæ¬¡ã®ããã« $3336$ åã®ç¹ããšã£ãŠïŒå $X_k$ ãš $A,B$ ãçµã³ïŒç·å $X_iX_{i+1}(i=1,2,...,3332)$... | ã$n$ ã $2$ 以äžã®æ£æŽæ°ãšããŸãïŒå¹³é¢äžã«çžç°ãªã $n$ åã®ç¹ããšãïŒ**è¯ãç¹**ãšããŸãïŒçžç°ãªã $2$ ã€ã®è¯ãç¹ãçµã¶ç·åãïŒæ¬¡ãæãç«ã€ããã« $10000$ æ¬åŒãããšãã§ããŸããïŒ
- $10000$ æ¬ã®ç·åã®ãã¡ïŒã©ã® $2$ æ¬ã端ç¹ãé€ããŠå
±æç¹ãæããªãïŒ
$n$ ãšããŠããããæå°å€ãæ±ããŠãã ããïŒ |
OMCB015 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb015/tasks/9335 | H | OMCB015(H) | 300 | 68 | 105 | [
{
"content": "ã$v_{2}(n)v_{3}(n)$ ã¯æ¬¡ã®ããã«èšããããããïŒ\r\n\r\n- $n$ ãå²ãåã $2$ 以äžã® $2$ ã¹ã $2^a$ ãš $3$ 以äžã® $3$ ã¹ã $3^b$ ã®ã㢠$(2^a,3^b)$ ã®åæ°ïŒ\r\n\r\n$(2^a,3^b)$ ãåºå®ãããšãïŒåæ¹ã§å²ããïŒããªãã¡ $2^a3^b$ ã§å²ããïŒ$3^6+1\\leq N \\leq 3^7$ ã®åæ°ã¯ïŒ\r\n$$ \\left \\lfloor \\dfrac{3^7}{2^{a}3^{b}} \\right \\rfloor - \\left \\lfloor \\dfrac{3^6}{... | ãæ£æŽæ° $n$ ãçŽ æ° $p$ ã§å²ãåããæ倧ã®åæ°ã $v_{p}(n)$ ã§è¡šããšãïŒ
$$\displaystyle \sum_{n = 3^6+1}^{3^7} v_{2}(n)v_{3}(n)$$
ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/5029 | A | OMCB014(A) | 100 | 300 | 304 | [
{
"content": "ã$n=1$ ã¯é©ããïŒ$n=2,3,4$ ã¯é©ããïŒ$n\\ge5$ ã®ãšãïŒ$5$ çªç®ã®çŽ æ°ã $9$ ãã倧ããããšãšïŒ$n+1$ çªç®ã®çŽ æ°ã $n$ çªç®ã®çŽ æ°ãã $2$ 以äžå€§ããããšããïŒã€ãã«é©ããªãïŒãã£ãŠïŒæ±ããç·å㯠$2+3+4=\\mathbf{9}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/5029"
}
] | ã$n$ çªç®ã«å°ããçŽ æ°ã $2n-1$ ã§ãããããªïŒæ£æŽæ° $n$ ã®ç·åã解çããŠäžããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/6183 | B | OMCB014(B) | 100 | 278 | 293 | [
{
"content": "ã$a=2x+1, b=2y+1, c=2z+1$ ãšããããšã§ïŒæ¡ä»¶ãæºããçµã®åæ°ã¯ $x+y+z=49$ ãæºããé åºä»ããéè² æŽæ°ã®çµ $(x,y,z)$ ã®åæ°ã«çããããšããããïŒãã®åæ°ã¯ïŒ$49$ åã®ããŒã«ãš $2$ åã®ä»åãã䞊ã¹ãæ¹æ³ã®åæ°ã«äžèŽããã®ã§ïŒæ±ããçã㯠${}\\_{49+2}\\mathrm{C}\\_{2}=\\textbf{1275}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb014/editorial/6183"
}... | ã$a+b+c=101$ ãæºããæ£ã®å¥æ°ã®çµ $(a, b, c)$ ã¯ããã€ååšããŸããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/9747 | C | OMCB014(C) | 100 | 282 | 291 | [
{
"content": "ãOMCãããæåã«æãæµ®ãã¹ãŠããæ°ã $n$ ãšãããšïŒOMCãããæãæµ®ãã¹ãŠããæ°ã¯æ¬¡ã®ããã«å€åããïŒ\r\n\r\n$$\\begin{aligned}\r\nn&\\longrightarrow 10n \\longrightarrow 10n+a \\longrightarrow 110n+11a \\longrightarrow 110n + 11a - b \\\\\\\\\r\n&\\longrightarrow \\dfrac{110n+11a-b}{a} \\longrightarrow \\dfrac{(110-a)n+11a-b}{a}\r\n\\end{al... | ãOMC ãããããæ£æŽæ°ãäžã€æãæµ®ãã¹ãŠããŸãïŒããã§ïŒããªãã¯æ£æŽæ° $a,b$ ãçšããŠä»¥äžã®ãããªèšç®ãããããã« OMC ããã«æ瀺ãããŸãïŒ
- æãæµ®ãã¹ãŠããæ°ã $10$ åã㊠$a$ ã足ãïŒããã $11$ åããŠãã $b$ ãåŒããŠïŒããã«ããã $a$ ã§å²ã£ãŠããæåã«æãæµ®ãã¹ãŠããæ°ãåŒãïŒ
ãã®ãšãïŒæçµçãªèšç®çµæã¯ïŒOMC ãããæåã«æãæµ®ãã¹ãŠããæ°ã«ããã $1$ ãšãªããŸãïŒ$a+b$ ã®å€ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/5732 | D | OMCB014(D) | 200 | 215 | 259 | [
{
"content": "ãæ±ããååŸã $r$ ãšããïŒãã®åã®äžå¿ã $O$ ãšããïŒãã®ãšã, $O$ ãã蟺 $CD$ ã«åç·ããããïŒãã®è¶³ã $H$ ãšãããšïŒ\r\n$$OD = r + \\frac{1}{3}, \\quad OH = \\frac{1}{2}, \\quad DH = 1 - r$$\r\nã§ããããïŒäžè§åœ¢ $DOH$ ã«ã€ããŠäžå¹³æ¹ã®å®çãã\r\n$$\\left(r + \\frac{1}{3} \\right)^2 = (1 - r)^2 + \\frac{1}{4}$$\r\nããã解ãããšã§ $r = \\dfrac{41}{96}$ ãåŸããïŒè§£çãã¹ãå€ã¯ $\... | ãäžèŸºã®é·ãã $1$ ã®æ£æ¹åœ¢ $ABCD$ ã«ã€ããŠïŒç¹ $A, D$ ãããããäžå¿ãšãïŒååŸããšãã« $\dfrac{1}{3}$ ã® $2$ ã€ã®åãèããŸãïŒãã® $2$ ã€ã®åã«å€æ¥ãïŒèŸº $BC$ ãšãæ¥ããåã®ååŸãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/4550 | E | OMCB014(E) | 200 | 251 | 269 | [
{
"content": "$$(\\text{äžåŒ}) \\iff2024\\leq\\dfrac{10^n}{m}\\lt2025\\iff\\dfrac{1}{2025}\\lt\\dfrac{m}{10^n}\\leq\\dfrac{1}{2024}$$\r\nãã㧠$\\dfrac{1}{2025}=0.000493\\cdots$ ããã³ $\\dfrac{1}{2024}= 0.000494\\cdots$ ããïŒæ±ããæå°ã® $m$ 㯠$\\bf{494}$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/c... | ã$\bigg\lfloor \dfrac{10^n}{m} \bigg\rfloor=2024$ ãæºããæ£æŽæ°ã®çµ $(m,n)$ ã®ãã¡ïŒ$m$ ãæå°ã§ãããã®ã«ã€ããŠïŒ$m$ ã®å€ãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/6090 | F | OMCB014(F) | 300 | 227 | 235 | [
{
"content": "ã$a_n+a_{n+1} \\gt a_na_{n+1}$ ãšãªãã®ã¯ïŒ$a_n$ ãš $a_{n+1}$ ã®å°ãªããšãçæ¹ã $1$ ãšãªããšãã«éãããïŒåŸã£ãŠïŒ$\\lbrace a_n \\rbrace $ ã¯ä»¥äžã®ããã«ãªããšããã. \\\r\n$$a_1=a_2=1,\\quad a_3=2,\\quad a_4=3,\\quad a_{n+2}=a_na_{n+1}\\quad (n=3,4,\\ldots)$$\r\nãããã§ïŒãã£ããããæ°åã®ç¬¬ $n$ é
ã $F_{n}$ ãšè¡šãããšã«ããïŒã€ãŸãïŒæ¬¡ã®ããã«æ°å $\\\\{F_n\\\\}$ ãå®ããïŒ\r\n... | ãæ°å$\lbrace a_n \rbrace$ ã以äžã®ããã«å®ããŸã. \
$$
a_1=a_2=1,\quad a_{n+2}=\max (a_n+a_{n+1} , a_na_{n+1})\quad(n=1,2,\ldots)
$$
ãã®ãšãïŒæ£ã®çŽæ°ã $5040$ åæã€é
$a_k$ ãã¡ããã©äžã€ååšããŸãïŒ$a_k$ ã¯çŽ æ° $p,q$ ãšæ£ã®æŽæ° $a,b$ ãçšã㊠$a_k=p^a\times q^b \space$ ãšè¡šããã®ã§ïŒ$abpq$ ã解çããŠãã ãã. |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/4319 | G | OMCB014(G) | 300 | 183 | 242 | [
{
"content": "ã$N=\\overline{abc}$ ã«ã€ããŠïŒ$a\\lt b\\lt c$ ã§ãããšããŠãäžè¬æ§ã倱ããªãïŒããŸïŒ$b-a$ ãš $c-b$ ã®å°ãªããšãäžæ¹ã¯ $4$ 以äžã§ããããïŒ$\\overline{cba}-\\overline{cab}=9(b-a)$ ãš $\\overline{acb}-\\overline{abc}=9(c-b)$ ã®å°ãªããšãäžæ¹ã¯ $36$ 以äžã§ããïŒããªãã¡ïŒ$m$ ã¯ã$36$ 以äžã® $9$ ã®åæ°ããåæ°ã«ãã€ïŒ\\\r\nãããŠïŒ$m$ ã $4$ ã®åæ°ã§ãããšãããšïŒ$a,b,c\\in\\\\{2,4,6,8\\\\}$ ã§... | ãåäœã®æ°ãçžç°ãªãïŒã〠$0$ ãå«ãŸãªã $3$ æ¡ã®æ£æŽæ° $N$ ããããŸãïŒ$N$ ã®åäœã®æ°ã䞊ã¹æ¿ããŠåŸããã $6$ åã®æ£æŽæ°ïŒ$N$ ãå«ãïŒã¯ãã¹ãŠ $m$ ã®åæ°ã§ããïŒãã®ãšãïŒ$m$ ãšããŠããåŸãæ£æŽæ°ã®ç·åãæ±ããŠãã ããïŒ |
OMCB014 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb014/tasks/7159 | H | OMCB014(H) | 300 | 104 | 146 | [
{
"content": "ãåè§åœ¢ $ABCD, BCDE, CDEA$ ã¯ããããçèå°åœ¢ã§ããïŒãã£ãŠïŒ\r\n$$\\angle ACD = \\angle CAE = \\angle CBE = \\angle BED = \\angle BAD = \\angle ADC$$\r\nã§ãããã $AC = AD$ ã§ããïŒãŸãïŒ$AC = BD, AD = CE$ ãæãç«ã€ã®ã§ïŒ\r\n$$AC = AD = BD = CE$$\r\nã§ããïŒãã®é·ãã $x$ ãšããïŒãã®ãšãïŒåè§åœ¢ $ABCD$ ã«å¯ŸããŠPtolemyã®å®çãé©çšããããšã§\r\n$4 + 3x = x^2$\r\nãåããã®ã§ïŒ... | ãå $\Gamma$ ã«å
æ¥ããåžäºè§åœ¢ $ABCDE$ ãïŒä»¥äžã®æ¡ä»¶ãæºãããŠããŸãïŒ
$$AB=CD=EA=2,\quad BC=DE=3$$
$\Gamma$ ã®é¢ç©ãæ±ããŠãã ããïŒãã ãïŒæ±ããé¢ç©ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}\pi$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/6554 | A | OMCE005(A) | 400 | 119 | 138 | [
{
"content": "ã$100$ ã®åçŽæ° $d$ ã® $f\\_{100}(100)$ ãžã®å¯äžãèããïŒ$f\\_{100}(100)$ ã«ãããŠïŒ$d$ 㯠\r\n$$a\\_{100}=100, \\quad a\\_{0}=d, \\quad a\\_{i}|a\\_{i+1}\\ (i=0,1,\\dots,99)$$ \r\nãæºããæŽæ°å $(a\\_{0},a\\_{1},\\dots,a\\_{100})$ ã®åæ°åã ãå ç®ãããïŒãã®æ°åã®åæ°ã¯åçŽ å æ°ããšã«ç¬ç«ã«èããããšãã§ãïŒ$d=2\\^{p}5^\\{q}$ ãšããã°ïŒäžèšã®æ°åã®åæ°ã¯ïŒ\r\n$$\\binom{101... | ãæ£ã®æŽæ°ã«å¯ŸããŠå®çŸ©ããæ£ã®æŽæ°å€ããšãé¢æ°ã®å $\\{ f_i \\}$ ãïŒ$f_0(n) = n$ ããã³
$$ f\_i(n)= \sum\_{d\mid n}f\_{i-1}(d) \quad (i = 1, 2, 3, \ldots) $$
ã«ãã£ãŠå®ããŸãïŒãã®ãšãïŒ$f\_{100}(100)$ ãæ±ããŠãã ããïŒãã ãïŒ$\sum\limits\_{d\mid n}$ 㯠$n$ ã®ãã¹ãŠã®æ£ã®çŽæ° $d$ ã«ã€ããŠç·åããšãããšãæå³ããŸãïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/10902 | B | OMCE005(B) | 400 | 103 | 121 | [
{
"content": "ãæ±ããå€ã $S$ ãšããïŒ$f(n)$ ã¯ïŒæ£ã®æŽæ° $i$ ã§ãã£ãŠ $\\dfrac{n}{2^i}$ ã $2$ é²æ³è¡šèšãããšãã«å°æ°ç¬¬ $1$ äœã $1$ ãšãªããããªãã®ã®åæ°ïŒããªãã¡ $n$ ã $2$ é²æ³è¡šèšãããšãã«çŸãã $1$ ã®åæ°ã§ããïŒãããã£ãŠ $f(n)=f(2n)$ ãã $f(m)+f(2m)=f(m+2m)$ ãšãªããïŒ$2$ é²æ³è¡šèšã®å ç®ã§ç¹°ãäžãããçããå Žå $1$ ã®åæ°ã¯æžãã®ã§ïŒ$2$ é²æ³è¡šèšã§ $m+2m$ ã¯ç¹°ãäžãããçããªãïŒããªãã¡ $m$ ã® $2$ é²æ³è¡šèšã§ $1$ ãé£ç¶ããªãããšããããïŒéã« $m$ ã® ... | ãæ£ã®å®æ° $x$ ã«å¯Ÿã㊠$\lceil x \rfloor$ ãã $x$ ã®å°æ°ç¹ä»¥äžç¬¬äžäœãåæšäºå
¥ããŠåŸãããæŽæ°å€ããšå®çŸ©ããŸãïŒäŸãã°
$$\lceil 2 \rfloor=2, \quad \lceil 4.2 \rfloor=4, \quad \lceil 5.5 \rfloor=6$$
ãšãªããŸãïŒæ£ã®æŽæ° $n$ ã«å¯ŸããŠæŽæ° $f(n)$ ã
$$f(n)=\sum_{i=1}^{\infty} \left( \left\lceil \dfrac n{2^i} \right\rfloor-\left\lfloor \dfrac n{2^i} \right\rfloor \right)$$
ã§å®ã... |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/7261 | C | OMCE005(C) | 500 | 28 | 39 | [
{
"content": "ã$BC=x,CA=y,AB=z$ ãšãïŒçŽç· $DP$ ãš $BC$ ã®äº€ç¹ã $X$ïŒ$DQ$ ãš $CA$ ã®äº€ç¹ã $Y$ïŒ$DR$ ãš $AB$ ã®äº€ç¹ã $Z$ ãšãïŒåé¢äœ $DXYZ$ ã®äœç©ã $V_3$ ãšããïŒãŸãïŒç©ºéå
ã®ç¹ $W$ ãšçŽç· $t$ ã®è·é¢ã $d(W,t)$ ã§è¡šãïŒåã€ã®äžè§åœ¢ $ABC, DCB, CDA, BAD$ ã¯ååã§ããããšã«æ³šæããïŒ\\\r\nã$AX:BX = AD:BD = x : y$ ãªã©ãæç«ããããšããïŒ\r\n$$\\begin{aligned}\r\n\\frac{V_3}{V_1}\r\n&=\\frac{\\t... | ãéè§äžè§åœ¢ $ABC$ ã® $3$ ã€ã®åæ¥åã®ååŸã¯ãããã $24,25,26$ ã§ãïŒç©ºéå
ã«ç¹ $D$ ã
$$AD=BC,\quad BD=AC,\quad CD=AB$$
ãæºããããã«åãïŒäžè§åœ¢ $DBC, DCA, DAB$ ã®è§ $D$ ã«å¯Ÿããåå¿ããããã $P,Q,R$ ãšããŸãïŒåé¢äœ $DABC,DPQR$ ã®äœç©ããããã $V_1,V_2$ ãšãããšãïŒ$\dfrac {V_2} {V_1}$ ã®å€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãçšã㊠$\dfrac b a$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ãæ±ããŠäžããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/3208 | D | OMCE005(D) | 700 | 29 | 58 | [
{
"content": "ã以éã¯ïŒäžã€ç®ã®æ¡ä»¶ãæºãããã®ïŒã€ãŸãïŒ`ã` ãš `ã` ã®éã« `()` ãšããé£ç¶æååããªããã®ã ããæååãšåŒã¶ããšã«ããïŒ\r\n\r\nãäžè¬ã«ïŒ`(` ãš `)` ãš `ã` ããããã $n$ åãã€ããå ŽåãèããïŒ$xy$ å¹³é¢äžã« $A(0,0)$, $B(0,1)$, $X(n,n)$, $Y(n,n+1)$ ããšãïŒ\r\n$x$ 軞æ£æ¹åãžã®é·ã $1$ ã®ç§»åã ã$\\to$ã ïŒ$y$ 軞æ£æ¹åãžã®é·ã $1$ ã®ç§»åã ã $\\uparrow$ ãã§è¡šãããšãšããïŒä»»æã®æååã«é¢ããŠïŒ\r\n- `ã` ãš `)` ã ããæãåºããŠå·Šããé ã« ... | ãçããåæ°ã® `(` ãš `)` ãããªãæååã§ãã£ãŠïŒé£ç¶ããéšåæåå `()` ãã²ãšã€éžãã§æ¶ãããšãç¹°ãè¿ãããšã§ç©ºæååã«ã§ããæååã**æ£ããæ¬åŒ§å**ãšãã³ïŒãã®ãšãåæã«æ¶ãã `(` ãš `)` ã**察å¿ããæ¬åŒ§**ãšåŒã¶ããšã«ããŸãïŒããã¯ã©ã®æ£ããæ¬åŒ§åã«å¯ŸããŠãäžæã«å®ãŸããŸãïŒïŒ\
ã`(` ãš `)` ãš `ã` ã $10$ åãã€äžŠã¹ãŠ $30$ æåã®æåå $S$ ãäœãæ¹æ³ã®ãã¡ïŒ
- $S$ ãã `ã` ããã¹ãŠæ¶å»ãããšæ£ããæ¬åŒ§å $S^\prime$ ãåŸãããïŒ
- $S^\prime$ ã®ãã¹ãŠã®å¯Ÿå¿ããæ¬åŒ§ã¯ïŒ$S$ ã«ãããŠéã«å°ãªããšã $1$ ã€ã® `ã` ... |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/9693 | E | OMCE005(E) | 700 | 10 | 20 | [
{
"content": "ãéè² æŽæ° $t$ ã«å¯Ÿã㊠\r\n$$f_t(x)=\\displaystyle\\sum_{i=1}^{128}\\bigg( a_i^t ~ \\prod_{j=1}^{127}\\dfrac{x-a_{i+j}}{a_i-a_{i+j}}\\bigg)$$ \r\nãšãããšïŒ$f_t(x)$ 㯠$127$ 次以äžã§ããïŒ$1\\leq k\\leq 128$ ã®æ£æŽæ° $k$ ã«å¯Ÿã㊠$f_t(a_k)=a_k^t$ ãšãªãã®ã§ïŒ$f_t(x)$ 㯠$x^t$ ã $\\displaystyle\\prod_{i=1}^{128}(x-a_i)$ ã§å²ã£ãäœãã§ããããšãå... | ã$1\leq k \leq 128$ ãæºããæŽæ° $k$ ã«å¯Ÿã㊠$a_k=\cos{\dfrac {2k}{257}\pi}$ ãšãïŒ$129\leq k \leq 256$ ãæºããæŽæ° $k$ ã«å¯Ÿã㊠$a_k=a_{k-128}$ ãšããŸãïŒãã®ãšãïŒ
$$\large\sum_{i=1}^{128} \normalsize\dfrac{a_i^{130}}{\small\displaystyle\prod_{j=1}^{127}\normalsize(a_i-a_{i+j})}$$
ã®å€ã¯äºãã«çŽ ãªæ£ã®æŽæ° $m,n$ ãçšã㊠$-\dfrac mn$ ãšè¡šãããã®ã§ïŒ$m+n$ ã®å€ãæ±ããŠäžããïŒ |
OMCE005 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce005/tasks/7423 | F | OMCE005(F) | 700 | 6 | 12 | [
{
"content": "ãäžè¬ã«ïŒ$AP = x, DP = y$ ã®å ŽåãèããïŒ\\\r\nãçŽç· $DP$ ãšäžè§åœ¢ $ABC$ ã®å
æ¥åã®äº€ç¹ã®ãã¡ $D$ ã§ãªãæ¹ã $Q$ ãšãïŒäžè§åœ¢ $ABC$ ã®å
æ¥åã® $Q$ ã§ã®æ¥ç·ãšèŸº $AB, AC$ ã®äº€ç¹ã $R, S$ ãšããïŒ\\\r\n$$\\begin{aligned}\r\n\\angle BRS\r\n&= 180^\\circ - 2\\angle FQR\\\\\\\\\r\n&= 180^\\circ - 2\\angle FDQ\\\\\\\\\r\n&= 180^\\circ - 2(90^\\circ - \\angle E... | ã$AB\neq AC$ ãªãäžè§åœ¢ $ABC$ ã®å
æ¥åãšèŸº $BC, CA, AB$ ã®æ¥ç¹ããããã $D,E,F$ ãšãïŒ$D$ ããçŽç· $EF$ ã«äžãããåç·ã®è¶³ã $P$ ãšãããšïŒ$AP \perp BC$ ãšãªããŸããïŒããã«ïŒ
$$AP = 321,\quad DP = 500$$
ã§ãããšãïŒäžè§åœ¢ $ABC$ ã®é¢ç©ãæ±ããŠãã ããïŒãã ãïŒæ±ããçãã¯äºãã«çŽ ãªæ£ã®æŽæ° $a,b$ ãšå¹³æ¹å åãæããªãæ£ã®æŽæ° $c$ ãçšã㊠$\dfrac{b\sqrt c}{a}$ ãšè¡šãããã®ã§ïŒ$a+b+c$ ã解çããŠãã ããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/7069 | A | OMC223(A) | 300 | 155 | 252 | [
{
"content": "ã$30^6 = 2^6 \\cdot 3^6 \\cdot 5^6$ ã $3$ ã€ã®æ£æŽæ°ã®ç©ã§è¡šãæ¹æ³ïŒããç®ã®é åºã¯åºå¥ããªãïŒã®ãã¡ïŒ$3$ æ°ãçžç°ãªããã®ã®åæ°ã $M$ïŒ$2$ æ°ã®ã¿ãçãããã®ã®åæ°ã $N$ ãšããïŒãã®ãšã $abc = 30^6$ ãªãæ£æŽæ°ã®çµ $(a, b, c)$ ã®ç·æ°ã¯ïŒæåã®å¯Ÿç§°æ§ã $a = b = c$ ã®ã±ãŒã¹ãèæ
®ãããš $6M + 3N + 1$ ãšè¡šãããšãã§ããïŒãã®ãããªçµã®åæ°ãå®éã«èšç®ãããšïŒ$a, b, c$ ã®çŽ å æ°å解ã«å²ãæ¯ããã $2, 3, 5$ ã®éè€åºŠã®æ±ºãæ¹ãåã
${}\\_{8}\\mathr... | ã$3$ ã€ã®ïŒæ£ãšã¯éããªãïŒæŽæ°ã®çµ $(x, y, z)$ ã§ãã£ãŠïŒ$x \lt y \lt z$ ã〠$xyz = 30^6$ ãã¿ãããã®ã¯ããã€ãããŸããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8509 | B | OMC223(B) | 300 | 136 | 206 | [
{
"content": "ãäžè¬ã« $n\\geq 3$ ã«å¯ŸããŠæ£ $2n$ è§åœ¢ $P_1P_2 \\ldots P_{2n}$ ãèãïŒãã®å€æ¥åã $\\Omega$ ãšããïŒä»¥äžãæºãããããªçµ $(i_1, i_2, i_3, i_4, i_5)$ ã®åæ°ãæ±ããã°ããïŒ\r\n- äºè§åœ¢ $P_{i_1}P_{i_2}P_{i_3}P_{i_4}P_{i_5}$ ã® $5$ ã€ã®å¯Ÿè§ç·ã®ãã¡å°ãªããšã $1$ ã€ã¯å $\\Omega$ ã®çŽåŸã§ããïŒ\r\n\r\nããã§çŽåŸãšãªãåŸã察è§ç·ã¯é«ã
$2$ ã€ã§ããããšã«æ³šæããïŒäºè§åœ¢ã®å¯Ÿè§ç·ãšãªãããçŽåŸã¯\r\n$$P_1P_{n+1}ïŒP_2... | ãæ£ $1110$ è§åœ¢ $P_1P_2 \cdots P_{1110}$ ã«å¯ŸãïŒ
$$1 \leq i_1 \lt i_2 \lt i_3 \lt i_4 \lt i_5 \leq 1110$$
ãªãæŽæ°ã®çµ $(i_1, i_2, i_3, i_4, i_5)$ ã§ãã£ãŠïŒä»¥äžãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- äºè§åœ¢ $P_{i_1}P_{i_2}P_{i_3}P_{i_4}P_{i_5}$ ã®å
è§ã®ãã¡ïŒå°ãªããšã $1$ ã€ã¯çŽè§ã§ããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8513 | C | OMC223(C) | 300 | 116 | 150 | [
{
"content": "ãä»»æã®å®æ° $x$ ã§\r\n$$g(x) = -f(-x)$$\r\nãæãç«ã¡ïŒãŸãïŒä»»æã® $0$ ãé€ãå®æ° $x$ ã§\r\n$$h(x) = x^3f \\left (\\frac{1}{x} \\right )$$\r\nãæãç«ã€ïŒããã«åé¡ã®æ¡ä»¶\r\n$$g \\left (\\frac{11}{10} \\right ) = h(1110) = 0$$\r\nã¯\r\n$$f \\left (- \\frac{11}{10} \\right ) = f \\left (\\frac{1}{1110} \\right ) = 0$$\r\nãšåå€ã§ããïŒ$- \\df... | ã$a, b$ ãå®æ°ãšãïŒå®æ°ã«å¯ŸããŠå®çŸ©ãããé¢æ° $f, g, h $ ããããã
$$
\begin{aligned}
f(x) &= ax^3 + x + b, \\\\
g(x) &= ax^3 + x - b, \\\\
h(x) &= bx^3 + x^2 + a\\\\
\end{aligned}
$$
ã§å®ããŸãïŒ
$$g \left (\frac{11}{10} \right ) = h(1110) = 0$$
ãæãç«ã€ãšãïŒ$f(x) = 0$ ãæºããæ倧ã®å®æ° $x$ ãæ±ããŠãã ããïŒãã ãæ±ããæ倧å€ã¯äºãã«çŽ ãªæ£æŽæ° $p ,q$ ã«ãã£ãŠ $\dfrac{p}{q}$ ãšè¡š... |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/10404 | D | OMC223(D) | 400 | 72 | 134 | [
{
"content": "ãäžããããçåŒãå€åœ¢ãããš\r\n$$pm^2 = q^n(q^n + 32)(q^n - 32)$$\r\nãšãªãïŒ$q^n$ ã¯å¥æ°ããïŒ$q^n - 32, q^n, q^n + 32$ ã¯ã©ã® $2$ ã€ãéžãã§ãäºãã«çŽ ã§ããïŒ$pm^2$ ãçŽ å æ°å解ãããšãã«ã¹ããå¥æ°ãšãªãçŽ æ°ã $p$ ãã $1$ ã€ã§ããããšããïŒ$q^n - 32, q^n, q^n + 32$ ã®ãã¡ã¡ããã© $2$ ã€ãå¹³æ¹æ°ãšãªãå¿
èŠãããïŒããã§è£é¡ãäžããïŒ\r\n\r\n---\r\n**è£é¡ïŒ**\\\r\nãå¥æ°ã®å¹³æ¹æ° $A, B$ ãš $3$ 以äžã®æŽæ° $k$ ã®é㧠$A ... | ãæ£æŽæ° $m, n$ ãš $3$ 以äžã®çŽ æ° $p, q$ ã
$$pm^2 + 1024q^n = q^{3n}$$
ãã¿ãããŠããŸãïŒ$p$ ã®å€ãšããŠããåŸããã®ã®**ç·ç©**ã解çããŠäžããïŒ |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/10609 | E | OMC223(E) | 400 | 22 | 55 | [
{
"content": "ã$c_1, c_2, c_3, c_4, c_5$ ãåé¡ã®æ¡ä»¶ãã¿ãããšãïŒ\r\n$$\\max \\\\{c_1, c_2, c_3, c_4, c_5\\\\} \\leq 40000$$\r\nã§ããã®ã§ïŒ\r\n$$\\sum_{n = 1}^{40000} na_n = \\sum_{i = 1}^5 \\sum_{k = 1}^{c_i} k$$\r\nãšè¡šãããšãã§ããïŒãã ã $\\displaystyle \\sum_{k = 1}^0 k = 0$ ãšããïŒïŒãããã£ãŠïŒ\r\n$$\\sum_{i = 1}^5 (2i - 1)x_i = 40000 \\tag... | $$5 \geq a_1 \geq a_2 \geq \cdots \geq a_{40000} \geq 0$$
ãªãæŽæ°ã®çµ $(a_1, ..., a_{40000})$ ãããïŒéè² æŽæ° $c_1, c_2, c_3, c_4, c_5$ ã次ã®ããã«å®ããŸãïŒ
- å $i \in \\{1, 2, 3, 4, 5\\}$ ã«å¯ŸãïŒ$1 \leq n \leq 40000$ ãªãæŽæ° $n$ ã®ãã¡ $a_n \geq i$ ãã¿ãããã®ã®åæ°ã $c_i$ ãšããïŒ
ãããš $1, 2, 3, 4, 5$ ã®äžŠã¹æ¿ã $m_1, m_2, m_3, m_4, m_5$ ã§ãã£ãŠ
$$c_{m_1} + 3c_... |
OMC223 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc223/tasks/8800 | F | OMC223(F) | 500 | 7 | 15 | [
{
"content": "ã$2$ çŽç· $AB, CD$ ã®äº€ç¹ã $F$ ãšãããšãïŒ$AE = BC$ ã§ããããšãšååšè§ã®å®çãã $\\angle ADE = \\angle FDB$ ã§ããïŒåè§åœ¢ $ABDE$ ãåã«å
æ¥ããããšãã $\\angle AED = \\angle FBD$ ãªã®ã§ïŒ $\\triangle AED \\sim \\triangle FBD$ ã§ããïŒæ¡ä»¶ $BD : DE = 11 : 10$ ãã $BF = 11$ ã§ããïŒæ£ã®å®æ° $x, y$ ãçšããŠ\r\n$$DF = 11xïŒDB = 11yïŒDA = 10xïŒDE = 10y$$\r\nãšè¡šãããšãã§ã... | ãåžäºè§åœ¢ $ABCDE$ ãåã«å
æ¥ããŠããïŒããã«ä»¥äžã®æ¡ä»¶ããã¹ãŠã¿ãããŠããŸãïŒ
$$AB = 11ïŒAE = BC = 10ïŒCD = 11 \sqrt{10}ïŒBD : DE = 11 : 10$$
ããã§ç·å $BD$ äžã«ç¹ $P$ ããšãïŒ$2$ ã€ã®ç·å $PE, DA$ ã®äº€ç¹ã $Q$ ãšãããšããïŒ$PQ = QE$ ãæãç«ã¡ãŸããïŒãã®ãšãïŒ$\dfrac{DQ}{DA}$ ã®å€ãæ±ããŠãã ããïŒ\
ããã ãïŒæ倧å
¬çŽæ°ã $1$ ã§ãã $3$ ã€ã®æ£æŽæ° $a, b, c$ ãšå¹³æ¹å åããããªãæ£æŽæ° $d$ ã«ãã£ãŠ $\dfrac{DQ}{DA} = \dfrac{a \sqrt{d} -... |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/7592 | A | OMCB013(A) | 100 | 337 | 345 | [
{
"content": "ãéè² æŽæ° $n$ ã $4$ ã§å²ã£ãäœããš $5$ ã§å²ã£ãäœãããšãã« $r$ ã§ãããšãããšïŒ$n-r$ 㯠$4$ ã®åæ°ã〠$5$ ã®åæ°ã§ããããïŒ$20$ ã®åæ°ã§ããïŒãŸãïŒ$r$ ãšããŠèããããå€ã¯ $0, 1, 2, 3$ ã§ããïŒ\\\r\nãéã«ïŒ$n=20m+r$ïŒ$m$ ã¯æŽæ°ïŒ$r=0,1,2,3$ïŒãšè¡šããã $n$ 㯠$4$ ã§å²ã£ãäœããš $5$ ã§å²ã£ãäœããçããïŒ\\\r\nãåŸã£ãŠïŒ$20$ ã§å²ã£ãŠ $0,1,2,3$ äœã $100$ æªæºã®éè² æŽæ°ã®æ°ãæ±ããã°ããïŒãã㯠$\\mathbf{20}$ åããïŒ",
"te... | ã次ã®æ¡ä»¶ãæºããæŽæ° $n$ ã¯ããã€ãããŸããïŒ
- $0\leq n\leq 99$
- $n$ ã $4$ ã§å²ã£ãäœããš $n$ ã $5$ ã§å²ã£ãäœãã¯çããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/9711 | B | OMCB013(B) | 100 | 271 | 324 | [
{
"content": "ã$A$ ããã $x,y,z$ ã®é ã«æããšããïŒãã®ãšãïŒ$B$ ãã㯠$y,z,x$ ãŸã㯠$z,x,y$ ã®é ã«æãå¿
èŠãããïŒåè
ã®å Žå㯠$C$ ãã㯠$x,y,z$ ãŸã㯠$z,x,y$ ãšæãããšã«ãªãïŒåŸè
ã®å Žå㯠$C$ ãã㯠$x,y,z$ ãšæãããšã«ãªãïŒ$A$ ããã®æãæ¹ã¯ $6$ éãããããïŒä»¥äžã«ããæ±ããå Žåã®æ°ã¯ $6 \\times 3= \\mathbf{18} $ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb01... | ã$A$ ããïŒ$B$ ããïŒ$C$ ããã® $3$ 人ãã«ã©ãªã±ã«æ¥ãŠããïŒ$A,B,C,A,B,C,A,B,C$ ã®é ã« $1$ æ²ãã€æããŸãïŒæ¬¡ã®æ¡ä»¶ãæºãããããªæ²é ãšããŠããããã®ã¯äœéããããŸããïŒ
- $3$ 人ãšããåã代ããä»°ãã°å°ãããèã®å
ãã® $3$ æ²ãäžåºŠãã€æãïŒ
- $2$ 人以äžãé£ç¶ããŠåãæ²ãæãããšã¯ãªãïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/4694 | C | OMCB013(C) | 200 | 167 | 249 | [
{
"content": "ã$N$ ã®çŽæ°ã®åæ°ã $n$ ãšãïŒ$N$ ã®ä»»æã® $\\sqrt{N}$ 以äžã®çŽæ° $d$ ã«ã€ã㊠$T_d = \\\\{d, N\\/d\\\\}$ ãšããïŒãã®ãšã, çžç°ãªã $2$ æ° $a, b$ ã®ç©ã $N$ ã§ããããšãšïŒãã $\\sqrt{N}$ 以äžã® $N$ ã®çŽæ° $d$ ãååšã㊠$T_d = \\\\{a, b\\\\}$ ãšãªãããšã¯åå€ã§ããïŒ$\\sqrt{N}$ 以äžã® $N$ ã®çŽæ°ã®åæ°ã¯ $\\lceil n\\/2\\rceil$ åã§ããããïŒé³©ã®å·£åçãã $k$ ãšããŠããåŸãæå°å€ã¯ $\\lceil n\\/2\\rc... | ãæ£æŽæ° $N$ ã«å¯ŸããŠïŒä»¥äžã®æ¡ä»¶ãæºããæ£æŽæ° $k$ ã®æå°å€ã $102$ ã§ããïŒ
- $k$ 㯠$N$ ã®æ£ã®çŽæ°ã®åæ°ä»¥äžã§ããïŒ
- çžç°ãªã $k$ åã® $N$ ã®æ£ã®çŽæ°ãã©ã®ããã«ãšã£ãŠãïŒãããã®äžã«ç©ã $N$ ã§ãããããªçžç°ãªã $2$ æ°ãååšããïŒ
$v_2(N)$ ã®å€ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ\
ããã ãïŒæ£æŽæ° $M$ ã«ã€ããŠïŒ$M$ ã $2$ ã§å²ãåããæ倧ã®åæ°ã $v_2(M)$ ãšè¡šããŸãïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/6862 | D | OMCB013(D) | 300 | 172 | 199 | [
{
"content": "ã$\\bmod\\ 5$ ã§èãããšïŒä»»æã®æŽæ° $n$ ã«ã€ããŠïŒ$n^4$ 㯠$0$ ã $1$ ãšçããïŒãŸãïŒ$38671875$ 㯠$5$ ã®åæ°ãªã®ã§ïŒ$x,y,z,w$ ã¯å
šãŠ $5$ ã®åæ°ã§ããïŒäž¡èŸºã $5^4$ ã§å²ããšïŒ\r\n$$(x\\/5)^4 + (y\\/5)^4 + (z\\/5)^4 + (w\\/5)^4 = 61875$$\r\nã§ããïŒå³èŸºã¯åã³ $5$ ã®åæ°ãšãªãã®ã§ïŒåæ§ã®è°è«ãããããšã§ $x\\/5,y\\/5,z\\/5,w\\/5$ ã¯å
šãŠ $5$ ã®åæ°ã§ããïŒåŸã£ãŠïŒ$x = 25a, y=25b, z=25c, w=25... | ã$$ x^4+y^4+z^4+w^4=38671875$$ãæºããæ£æŽæ°ã®çµ $(x, y, z, w)$ å
šãŠã«ã€ããŠïŒ$x+y+z+w$ ã®ç·åã解çããŠãã ããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/7608 | E | OMCB013(E) | 300 | 74 | 119 | [
{
"content": "ã$a_2 = 2^8$ ã§ããïŒãŸãïŒä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸãïŒ\r\n$$a_{n+2} = \\frac{4a_{n+1}^4}{\\prod_{k=1}^{n+1}a_k} = \\frac{4a_{n+1}^4}{4a_{n}^4} = \\frac{a_{n+1}^4}{a_n^4}$$\r\nãæãç«ã€ïŒãã£ãŠïŒ$b_n = \\log_2a_n$ ãšãããšïŒæ°å $\\\\{b_n\\\\}$ 㯠$b_1 = 2, b_2 = 8$ ãã€ä»»æã®æ£ã®æŽæ° $n$ ã«ã€ããŠ\r\n$$b_{n+2} = 4b_{n+1} - 4b_{n}$$\r\nãã¿ããïŒããã解ã... | ãæ£ã®æŽæ°ãããªãæ°å $\\{a_n\\}$ 㯠$a_1=4$ ããã³ïŒä»»æã®æ£ã®æŽæ° $n$ ã«å¯ŸããŠä»¥äžãæºãããŸã.
$$\prod_{k=1}^{n+1}a_k=4a_n^4$$
ããã®ãšã $a_{999}$ ã $2^{101} - 1$ ã§å²ã£ãäœãã $r$ ãšããŸãïŒ$r$ ã®æ£ã®çŽæ°ã®åæ°ãæ±ããŠãã ããïŒ |
OMCB013 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb013/tasks/6450 | F | OMCB013(F) | 400 | 45 | 66 | [
{
"content": "ã**è£é¡**ïŒäžè§åœ¢ $ABC$ ã«ãã㊠$AB:AC=p:q,~ BC=k$ ã§ãããšãïŒãã®é¢ç©ã®æ倧å€ã¯ $\\dfrac{pq}{2|p^2-q^2|}k^2$ ã§ããïŒ\r\n<details><summary> 蚌æ<\\/summary>\r\nã$A$ 㯠$BC$ ã«å¯Ÿãã $p:q$ ã®ã¢ããããŠã¹ã®åäžã«ããã®ã§ïŒ$A$ ãš $BC$ ã®è·é¢ã®æ倧å€ã¯ãã®åã®ååŸïŒããªãã¡ $\\dfrac{pq}{|p^2-q^2|}BC$ ã§ããïŒãããã£ãŠäžè§åœ¢ $ABC$ ã®é¢ç©ã®æ倧å€ã¯ $\\dfrac{pq}{2|p^2-q^2|}k^2$ ã§ããïŒ\r\n<\\/... | ã$k$ ã¯æ£ã®æŽæ°ãšããŸãïŒèªå·±äº€å·®ãæããªããïŒåžãšã¯éããªãåè§åœ¢ $ABCD$ ã¯æ¬¡ãæºãããŸãïŒ
$$AB:BC=119:124,\quad AD:DC=127:129,\quad AC=k$$
ãã®ãããªåè§åœ¢ $ABCD$ ã®é¢ç©ãšããŠããããæ倧å€ãæŽæ°å€ãšãªããã㪠$k$ ã®æå°å€ãæ±ããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/7889 | A | OMC222(A) | 100 | 341 | 346 | [
{
"content": "ã$12$ ã§å²ã£ãäœãã $a$ ãšããã°ïŒæ¡ä»¶ãã¿ããæ£ã®æŽæ°ã¯ $12\\times 2a+a=25a$ ãšè¡šãããšãã§ããïŒãããã£ãŠïŒæ±ããç·åã¯\r\n$$ 25\\times(1+2+\\cdots+11)=\\mathbf{1650}. $$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omc222/editorial/7889"
}
] | ãæ£ã®æŽæ°ã§ãã£ãŠïŒ$12$ ã§å²ã£ãåãïŒ$12$ ã§å²ã£ãäœãã®ã¡ããã© $2$ åãšãªããã®ããã¹ãŠæ±ãïŒãããã®ç·åã解çããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/4772 | B | OMC222(B) | 200 | 189 | 253 | [
{
"content": "ãäžè§åœ¢ $APQ$ ã¯äºç蟺äžè§åœ¢ãªã®ã§ïŒ$\\angle BAP=\\angle AQB$ ã§ããïŒãã£ãŠïŒäžè§åœ¢ $ABP$ ãš $QBA$ ã¯çžäŒŒã§ããã®ã§ïŒ\r\n$$BP:6=6:(BP+PQ)=6:(BP+5)$$\r\nãæãç«ã€ïŒããã解ãã° $BP=4$ ããããã®ã§ïŒäžè§åœ¢ $ABP$ ã«å¯ŸããäœåŒŠå®çãã $\\cos\\angle{B}=\\dfrac{9}{16},\\tan\\angle{B}=\\dfrac{5\\sqrt7}{9}$ ãåããïŒæ±ããå€ã¯\r\n$$\\Big(6\\times 6\\tan\\angle{B} \\times \\frac... | ã$AB=6,\angle{A}=90^{\circ}$ ã®çŽè§äžè§åœ¢ $ABC$ ã®èŸº $BC$ äžã« $2$ ç¹ $P,Q$ ã $B,P,Q,C$ ã®é ã«äžŠã¶ããã«åããšïŒ$$AP=PQ=5,\quad \angle{BAP}=\angle{PAQ}$$ ãæãç«ã¡ãŸããïŒäžè§åœ¢ $ABC$ ã®é¢ç©ã®äºä¹ãæ±ããŠãã ããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/10459 | C | OMC222(C) | 300 | 84 | 115 | [
{
"content": "ã$N = 119$ ãšããïŒæ±ããã¹ãå€ $A$ ã¯ïŒ\r\n$$\r\nA \r\n= \\frac{\\sin 46^\\circ + \\sin 47^\\circ + \\cdots + \\sin 164^\\circ}{\\sin 1^\\circ + \\sin 2^\\circ + \\cdots + \\sin 119^\\circ}\r\n= \\frac{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{(k+45)\\pi}{180}}{\\displaystyle \\sum_{k=1}^{N} \\sin \\frac{k... | ã以äžã®å€ã®æå°å€é
åŒã $f$ ãšãããšãïŒ$|f(10)|$ 以äžã®æ倧ã®æŽæ°ã解çããŠãã ããïŒ
$$
\frac{\sin 46^\circ + \sin 47^\circ + \sin 48^\circ + \cdots + \sin 164^\circ}{\sin 1^\circ + \sin 2^\circ + \sin 3^\circ + \cdots + \sin 119^\circ}
$$
<details><summary>æå°å€é
åŒã«ã€ããŠ<\/summary>
ãè€çŽ æ° $\alpha$ ã«ã€ããŠïŒ$\alpha$ ãæ ¹ã«ãã€æçæ°ä¿æ°å€é
åŒãååšãããšãïŒãã®ãã¡æ¬¡æ°ãæå°ã§ããïŒãã€æé«æ¬¡ã®ä¿... |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/9719 | D | OMC222(D) | 300 | 177 | 248 | [
{
"content": "ã$1!$ ãã $3! = 6$ åæã $2$ æãŸã§çšããããšã§ïŒ$1$ åãã $18$ åãŸã§ $1$ åå»ã¿ã§ã¡ããã©æãããšãã§ããïŒäžæ¹ã§ïŒ$n \\geq 4$ ã«ãããŠã¯ïŒåž°çŽçã«\r\n$$ 2\\cdot1!+2\\cdot2!+\\cdots+2\\cdot(n-1)! \\lt n!$$\r\nã瀺ãããšãã§ããããïŒ$n!$ åæªæºã®ãæã $2$ æãã€çšããéé¡ããã $n!$ ã®æ¹ã倧ãããªãïŒãããã£ãŠïŒäœ¿çšãã $n!$ åæã®ææ°ããšã«ç°ãªãåèšéé¡ã察å¿ããïŒãããã£ãŠ $n \\geq 4$ ã®ãšãïŒ$n!$ åæãŸã§ã䜿çšããŠäœãããšã®ã§ãã... | ãOMCåœã§ã¯ïŒ$n=1,2,\ldots,1000$ ããããã«å¯ŸãïŒ$n!$ åæãé貚ãšããŠæµéããŠããŸãïŒOMCåœã«ãã£ãŠããããªãã¯ïŒããããã®ãæã $2$ æãã€æã£ãŠããŸãïŒããªããã¡ããã©æãããšã®ã§ããéé¡ãšã㊠$1000$ çªç®ã«å°ããå€ãçããŠãã ããïŒ\
ããã ãïŒã¡ããã©æãããšã®ã§ããéé¡ãšããŠïŒ$0$ åã¯å«ããŸããïŒ |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/5667 | E | OMC222(E) | 400 | 14 | 26 | [
{
"content": "ãäžèŸº $2$ ã®æ£ $36$ è§åœ¢ $C_0C_1C_2\\cdots C_{35}$ ãèãïŒãã®äžå¿ã $O$ ãšããïŒããã§äžè§åœ¢ $C_0 C_1 O$ ã®é¢ç©ã $U$ ãšããã°ïŒäžèŸº $2$ ã®æ£ $36$ è§åœ¢ã®é¢ç©ã¯ $36U$ ã§ããé·æ¹åœ¢ $C_{0}C_{1}C_{18}C_{19}$ ã®é¢ç©ã¯ $4U$ ã§ããïŒãã®ãšãïŒå€è§åœ¢$A$ , $B$ ãåå²ãäžèŸº $2$ ã®æ£ $36$ è§åœ¢ã«ã¯ãããšå€è§åœ¢ $A$ ãå·Šå³ïŒ$B$ ãå³å³ã®ç¶²æãã®ããã«ãªãã®ã§ïŒ\r\n$$S=36U - 4U\\cdot 2 + 2\\cdot 2 = 28U + 4$$\r\n... | ãåžå€è§åœ¢ $A = A_{0}A_{1}\cdots A_{31}$ ãš $B = B_{0}B_{1}\cdots B_{29}$ ã¯ãããã以äžã®æ¡ä»¶ãã¿ãããŸãïŒ
- å€è§åœ¢ $A, B$ ã®èŸºã®é·ãã¯ãã¹ãŠ $2$ ã§ããïŒ
- $i$ ã $8$ ã®åæ°ã§ãããšã $\angle {A_i} = 160^\circ$ïŒ$i$ ã $8$ ã®åæ°ã§ãªããšã $\angle {A_i} = 170^\circ$ïŒ
- $j$ ã $5$ ã®åæ°ã§ãããšã $\angle {B_j} = 160^\circ$ïŒ$j$ ã $5$ ã®åæ°ã§ãªããšã $\angle {B_j} = 170^\circ$ïŒ
ãå€è§... |
OMC222 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc222/tasks/9299 | F | OMC222(F) | 500 | 51 | 132 | [
{
"content": "ãåé¡æäžã«ãã $2255222255255552$ ã $A$ ãšããïŒ$A$ ã®äž $m$ æ¡ $(1\\leq m\\leq 15)$ ãããªãæŽæ°ã¯ $2^m$ ã®åæ°ãªã®ã§ïŒ$m+1$ æ¡ä»¥äžã®æŽæ°ã§äž $m$ æ¡ã $A$ ãšäžèŽã $m+1$ æ¡ç®ãç°ãªããã®ã¯ $2$ ã§ã¡ããã© $m$ åå²ãããšãã§ããïŒ$m=0$ ã§ãæå³ãæã€ãïŒããã¯ããªãã¡å¥æ°ã§ãããšããæå³ãªã®ã§ïŒèæ
®ããªããŠããïŒïŒ\r\n$m+1$ æ¡ä»¥äž $16$ æ¡ä»¥äžã®æŽæ°ã§äž $m$ æ¡ã $A$ ãšäžèŽã $m+1$ æ¡ç®ãç°ãªãæŽæ°ã¯ $2^{16-m}-1$ åããã®ã§ïŒãããã®ç©ã $2... | ãåæ¡ã $2$ ãŸã㯠$5$ ã§ãã $16$ æ¡ä»¥äžã®æ£æŽæ°ã®ãã¡ïŒ$2$ ã§å²ãåããåæ°ãæãå€ãã®ã¯
$$2255222255255552$$
ã®ãã äžã€ã§ïŒãã㯠$2$ ã§æ倧 $17$ åå²ãåãããšãã§ããŸãïŒ\
ãåæ¡ã®æ°ã $2$ ãŸã㯠$5$ ãããªã $16$ æ¡ä»¥äžã®æ£æŽæ°ãã¹ãŠã®ç©ã $N$ ãšãããšãïŒ$N$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠãã ããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8033 | A | OMCE004(A) | 300 | 167 | 198 | [
{
"content": "ãäžæ¹ãæã£ãŠããå¥æ°ã®ã«ãŒããšããäžæ¹ãæã£ãŠããå¶æ°ã®ã«ãŒãã®ææ°ã¯çããïŒãŸãïŒ$1,2,\\ldots,8$ ã®ãã¡ $2$ æ°ãäºãã«çŽ ã§ãªãã®ã¯ïŒãšãã«å¶æ°ã§ããå ŽåãïŒ$3$ ãš $6$ ã®ã¿ã§ããïŒãŸãããã«ããïŒäžæ¹ã $3$ ãš $6$ ãåæã«æã£ãŠãããšãã«ã¯ïŒ$B$ ãã㯠$A$ ãããçŽåã«åºããæ°ãšå¶å¥ã®ç°ãªããã®ãä»»æã«åºãããšã§å¿
ãåãŠããšãããïŒ\\\r\nã$A$ ããã $3$ ãïŒ$B$ ããã $6$ ãæã£ãŠãããšããïŒã㟠$A$ ããã $3$ ã®ä»ã«å¥æ°ãæã£ãŠãããšãïŒããªãã¡ $B$ ããã $6$ ã®ä»ã«å¶æ°ãæã£ãŠãããšãïŒïŒ$... | ã$1$ ãã $8$ ãŸã§ã®æŽæ°ã®ãã¡äžã€ãæžãããã«ãŒãããããã $1$ æãã€ãããŸãïŒãããã $4$ æãã€ã«åã㊠$A$ ãããš $B$ ããã«é
ãïŒä»¥äžã®ãããªã²ãŒã ãè¡ããŸãïŒ
- $A$ ãããå
æïŒ$B$ ãããåŸæãšããŠïŒé
ãããã«ãŒããã亀äºã« $1$ æãã€åºããŠããïŒäžåºŠåºããã«ãŒãã¯åã³åºããªãïŒ
- çžæãçŽåã«åºããã«ãŒãã«æžãããæŽæ°ãšäºãã«çŽ ãªæŽæ°ãæžãããã«ãŒããåºãããšãã§ããïŒãã®ã«ãŒã«ã«åŸã£ãŠåºããã«ãŒããç¡ããªã£ãããã®æç¹ã§ã²ãŒã ãçµäºããïŒ
- æåŸã«ã«ãŒããåºãã人ã®åã¡ãšããïŒç¹ã«ïŒãã¹ãŠã®ã«ãŒããåºãåã£ãã $B$ ããã®åã¡ã§ããïŒïŒ
ãããšïŒäž¡è
ãæ... |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8291 | B | OMCE004(B) | 500 | 49 | 89 | [
{
"content": "ããŸãïŒåããŒã«ãç¬ç«ã«ãããã $\\dfrac{1}{3}$ ã®ç¢ºçã§èµ€ïŒéïŒçœã«å¡ã£ãŠãããšèããããšã§ïŒæ±ããã¹ã㯠$RBW$ ã®æåŸ
å€ãšèšãæããããšãã§ããïŒãŸãïŒå¡ãæ¹ãåºå®ãããšãã® $RBW$ ã®å€ã¯ïŒé£ãåãããŒã«å士ã®é åºä»ãã $3$ ã€ã®çµ $(A_1,A_2,A_3)$ ã§ãã£ãŠïŒ$A_1, A_2, A_3$ ã«å«ãŸãã $2$ ã€ã®ããŒã«ãããããèµ€ãšèµ€ïŒéãšéïŒçœãšçœã§å¡ãããŠãããããªãã®ã®æ°ã«çããïŒ\r\n$ \\\\\\ $ãããã§, $(A_1, A_2, A_3)$ ãéžãã æ, $A_1, A_2, A_3$ ã«å«ãŸãã $2$ ã€ã®ããŒã«... | ãååšäžã« $2024$ åã®äºãã«åºå¥ã§ããããŒã«ãããé åºã§äžŠãã§ããŸãïŒãããã®ããŒã«ãããããèµ€è²ïŒéè²ïŒçœè²ã®ãããã $1$ è²ã§å¡ã£ãŠãããŸãïŒãã®ãšãïŒé£ãåã $2$ ã€ã®ããŒã«ã®çµã§ãã£ãŠïŒäž¡æ¹ãšãèµ€è²ã§å¡ããããã®ã®æ°ã $R$ïŒäž¡æ¹ãšãéè²ã§å¡ããããã®ã®æ°ã $B$ïŒäž¡æ¹ãšãçœè²ã§å¡ããããã®ã®æ°ã $W$ ãšããŸãïŒããŒã«ãå¡ãæ¹æ³ã¯å
šéšã§ $3^{2024}$ éããããŸããïŒãããå
šãŠã«å¯Ÿãã $RBW$ ã®çžå å¹³åãæ±ããŠäžããïŒãã ãïŒçãã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\cfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠäžãã. |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/10686 | C | OMCE004(C) | 500 | 26 | 57 | [
{
"content": "ã$\\angle ACB = \\theta$ ãšããïŒãããšïŒ$4$ ç¹ $A, X, D, C$ ã®å
±åãã $\\angle AXD = 180^\\circ - \\theta$ ãåŸãïŒ$\\angle AXC = \\angle ADC = 90^\\circ$ ãã $\\angle DXC = 90^\\circ - \\theta$ ãåŸãããïŒãŸãïŒ$4$ ç¹ $A, O, X, B$ ã®å
±åãã $\\angle BXE = \\angle BAO = 90^\\circ - \\theta$ ãåããã®ã§ïŒ\r\n$$\\angle BXC = 360^\\cir... | ãéè§äžè§åœ¢ $ABC$ ã®å€å¿ã $O$ïŒ$A$ ãã蟺 $BC$ ã«äžãããåç·ã®è¶³ã $D$ ãšããŸãïŒäžè§åœ¢ $AOB$ ã®å€æ¥åãšäžè§åœ¢ $ADC$ ã®å€æ¥åã¯äžè§åœ¢ $ABC$ ã®å
éšã®ç¹ $X (\neq A)$ ã§äº€ãããŸããïŒããã«çŽç· $OX$ ãšçŽç· $BC$ ã®äº€ç¹ã $E$ ãšãããšïŒ$4$ ç¹ $B, E, D, C$ ã¯ãã®é ã«äžŠã³ïŒ
$$BE = 4, \quad ED = 3, \quad DC = 2$$
ãæç«ããŸããïŒãã®ãšã $AX^{2}$ ã¯äºãã«çŽ ãªæ£æŽæ° $a,b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã®å€ã解çããŠãã ããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8808 | D | OMCE004(D) | 600 | 30 | 79 | [
{
"content": "ã$$\\sum_{k = 0}^{1012} 2024^ {1012-k} x^{2k} = \\dfrac{x^{2026} - 2024^{1013}}{x^{2} - 2024}$$ \r\nã§ããããïŒ$Ï = \\cos \\dfrac{2\\pi}{2026} + i \\sin \\dfrac{2\\pi}{2026}$ ãšãããšïŒæ¹çšåŒãæ〠$2024$ åã®è€çŽ æ°è§£ã¯ \r\n$$ \\pm \\sqrt{2024} à Ï^{k} \\quad (1 \\leq k \\leq 1012) $$\r\n\r\nãšè¡šãããããšãåããïŒããã§ïŒ$Ï^{k}$ ãå®æ°ã«... | ã$x$ ã«é¢ãã $2024$ 次æ¹çšåŒ
$$ \sum_{k = 0}^{1012} 2024^ {1012-k} x^{2k} = 0 $$
ã¯çžç°ãªã $2024$ åã®è€çŽ æ°è§£ãæã€ã®ã§ïŒãããå
šãŠãèŠçŽ ãšããŠæã€éåã $S$ ãšããŸãïŒãã®ãšãïŒ$S$ ã®**空ã§ãªã**éšåéåã§ãã£ãŠïŒèŠçŽ ãšããŠå«ãŸããè€çŽ æ°ãå
šãŠæãåããããšæŽæ°ãšãªããããªãã®ã®åæ°ãçŽ æ° $1009$ ã§å²ã£ãäœãã解çããŠäžããïŒ |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8146 | E | OMCE004(E) | 700 | 6 | 14 | [
{
"content": "ã$YZ$ ã®äžç¹ã $L^\\prime$ ãšãããšïŒ$AY = AZ$ ãã $\\angle AL^\\prime Y = \\angle AL^\\prime Z = 90^\\circ$ ã§ãã.ããŸãïŒ\r\n$$\\angle YAL^\\prime = \\angle ZAL^\\prime = \\dfrac{\\angle YAZ}{2} = \\angle BAC$$ \r\nã§ããããïŒ\r\n$$\\triangle YAL^\\prime \\sim \\triangle BAH_B, \\quad \\triangle ZAL^\\prime \\sim \\... | ãåå¿ã $H$ ã§ããäžè§åœ¢ $ABC$ ãããïŒ$A$, $B$, $C$ ããçŽç· $BC$, $CA$, $AB$ ã«äžãããåç·ã®è¶³ããããã $H_A, H_B, H_C$ ãšããŸãïŒç·å $H_AC$ äžïŒäž¡ç«¯ç¹ãé€ãïŒã«ç¹ $X$ ãåãïŒçŽç· $AB, AC$ ã«é¢ã㊠$X$ ãšå¯Ÿç§°ãªç¹ããããã $Y, Z$ ãšãããšïŒ$3$ çŽç· $YZ , AH , H_BH_C$ ã $1$ ç¹ $L$ ã§äº€ãããŸããïŒããã«ç·å $XY$ ã®äžç¹ã $M$ ãšãïŒäžè§åœ¢ $ALM$ ã®å€æ¥åãšäžè§åœ¢ $ABC$ ã®å€æ¥åã® $A$ ã§ãªãæ¹ã®äº€ç¹ã $P$ ãšãããšïŒ
$$ AP = 5, \quad PM = \s... |
OMCE004 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce004/tasks/8673 | F | OMCE004(F) | 700 | 12 | 38 | [
{
"content": "ã以äžïŒããŒãåºãã人ã $P$ïŒã°ãŒãåºãã人ã $G$ïŒãã§ããåºãã人ã $C$ ãšè¡šãïŒãŸãïŒäžè¬æ§ã倱ããã«ïŒæ®ã $3$ 人ã«ãªã£ã段é㧠$P, G, C$ ããã®é ã«æèšåãã«äžŠãã§ãããšããïŒãŸãïŒ$k$ åæŠãçµãã£ãçŽåŸã« $P$ ãš $G$ ããã®é ã«æèšåãã«ãªãããã«é£ãåã£ãŠãããšãïŒãã®äºäººã®éã«ã㊠$k$ åæŠã§è±èœãã人ã«ã€ããŠèããïŒ$P$ ã®æèšåãã«é£ã«ãã人ã§è±èœãã人ããããšãããšïŒãã®äººã¯äž¡é£ã®ã©ã¡ãã«ãåã£ãŠããªãããïŒ$P$ ã $G$ ã ãšåããïŒãŸãïŒ$P$ ã ãšãããšïŒãã®æèšåãã«é£ã®äººã $C$ ãšç¢ºå®ãïŒãã®äººã $k$ å... | ãOMC æã«ã¯ $1200$ 人ã®äœäººãããŸãïŒãããã®äœäººå
šå¡ã§ä»¥äžã®ã«ãŒã«ã«åŸãããããã倧äŒãè¡ãããšã«ããŸããïŒ
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šå¡ãã©ã®æãåºããäºåã«æ±ºããŠããïŒãã®æã¯å€§äŒãçµäºãããŸã§å€ããããšã¯ãªãïŒ
- äžåæŠã¯ $1200$ 人å
šå¡ã§è¡ãïŒå
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- ããããã®åè² ã«ãããŠïŒåã£ãå Žå㯠$1$ ç¹ãïŒè² ããå Žå㯠$-1$ ç¹ãç²åŸãïŒãããã®å Žåã¯åŸç¹ã¯ç²åŸããªãïŒ(ã€ãŸãïŒåäœäººã®åŸç¹ã¯ $-2$ ç¹ä»¥äž $2$ ç¹ä»¥äžã®æŽæ°å€ãåãåŸãïŒ)
- å
šãŠã®ããããããçµäºããã®ã¡ïŒåŸç¹ã $0$ ç¹**æªæº**... |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/7601 | A | OMCB012(A) | 100 | 358 | 359 | [
{
"content": "ãå¹³æ¹æ°ãæ£æŽæ° $n$ ãçšã㊠$n^2$ ãšè¡šããšïŒ$n^2-1=(n+1)(n-1)$ ãçŽ æ°ã§ãããã $n-1=1$ïŒåŸã£ãŠåé¡ã®æ¡ä»¶ã«åœãŠã¯ãŸãå¹³æ¹æ°ã¯ $n^2=\\mathbf{4}$ ã®ã¿ïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/7601"
}
] | ãçŽ æ°ã« $1$ ã足ããŠåŸãããå¹³æ¹æ°ãšããŠãããããã®ã®ç·åãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/8383 | B | OMCB012(B) | 100 | 344 | 347 | [
{
"content": "ã$xy-2x-3y+6=(x-3)(y-2)$ ã§ããïŒ$x-3,y-2$ ã¯ã©ã¡ããæ¡ä»¶äžã«ãããŠåžžã«éè² ã§ããããïŒæ倧å€ã¯ $(x,y)=(19,21)$ ã®ãšã㧠$304$ïŒæå°å€ã¯ $(x,y)=(16,t)$ ã®ãšã㧠$13t-26$ ã§ããïŒãã®å·®ã¯ $330-13t$ ã§ããããïŒ$330-13t=135$ ã解ã㊠$t=\\mathbf{15}$ ãåŸãïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/8383"
}
] | ãå®æ° $t$ 㯠$2\leq t\leq 21$ ããã³æ¬¡ãã¿ãããŸããïŒ$t$ ã®å€ã解çããŠãã ããïŒ
- å®æ° $x,y$ ã $16\leq x\leq 19,t\leq y\leq 21$ ãæºããããã«åããšãïŒ
$$xy-2x-3y+6$$
ã®æ倧å€ãšæå°å€ã®å·®ã¯ $135$ ã§ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/3194 | C | OMCB012(C) | 200 | 289 | 320 | [
{
"content": "ãäžè¬æ§ã倱ãã $x \\leq y \\leq z$ ãšããŠèãããš $x^3+3x\\leq 33$ ã§ããïŒãããã $x=1,2$ ã§ããïŒ\\\r\nã$x=1$ ã®ãšãïŒ$yz+y+z=32$ ãã\r\n$$(y+1)(z+1)=33$$\r\nã§ããïŒ$y+1\\geq 2$ ãã $y+1=3,z+1=11$ ãšãªãã»ããªãïŒãã£ãŠ $(x,y,z)=(1,2,10)$ïŒ\\\r\n ã$x=2$ ã®ãšãïŒ$2yz+y+z=31$ ãã\r\n$$(2y+1)(2z+1)=63$$\r\nã§ããïŒ$2y+1\\geq 2x+1=5$ ãã\r\n$(2y+1,2z+1)... | ã$xyz+x+y+z=33$ ãã¿ããæ£ã®æŽæ°ã®çµ $(x,y,z)$ ãã¹ãŠã«ã€ããŠïŒ
$x+y+z$ ã®ç·åãçããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/6768 | D | OMCB012(D) | 200 | 221 | 274 | [
{
"content": "ã$f$ ãå¶æ°æ¬¡ã®é
ãããªãããšãã $f$ ã¯å¶é¢æ°ãªã®ã§, $f(-1)=f(-2)=f(-3)=f(-4)=5$ ã§ãã. åŸã£ãŠ, \r\n$$f(x)=(x-4)(x-3)(x-2)(x-1)(x+1)(x+2)(x+3)(x+4)+5$$\r\nãåãããã, $f(5)=\\mathbf{72581}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb012/editorial/6768"
},
{
"content": "ãæ¡ä»¶ãã $f(x... | ã以äžãå
šãŠæºãã $8$ 次ã®å®æ°ä¿æ°å€é
åŒ $f$ ã¯äžæã«ååšããã®ã§ïŒ$f(5)$ ã解çããŠãã ãã.
- å¥æ°æ¬¡ã®é
ã®ä¿æ°ã¯å
šãŠ $0$ ã§ãã.
- $8$ 次ã®é
ã®ä¿æ°ã¯ $1$ ã§ãã.
- $f(1)=f(2)=f(3)=f(4)=5$. |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/3865 | E | OMCB012(E) | 200 | 243 | 288 | [
{
"content": "ãåæ¡ã®æ°ã $a\\lt b$ ã® $2$ çš®é¡ãããªãè¯ãæ°ã¯ ${}_5 \\mathrm{C}_1+{}_5 \\mathrm{C}_2+{}_5 \\mathrm{C}_3+{}_5 \\mathrm{C}_4=30$ åããïŒ$2^5-2$ ãšèããŠãããïŒïŒãã㧠$\\overline{aaaab}$ ã«ã¯ $\\overline{bbbba}$ ãïŒ$\\overline{aaabb}$ ã«ã¯ $\\overline{bbbaa}$ ã察å¿ãããèŠé 㧠$2$ ã€ãã€ãã¢ã«ããããšã§ïŒ$30$ åã®ç·å㯠$11111(a+b)\\times 15$ ã§ããïŒ\\\r\n... | ãåé²æ³è¡šèšã§åæ¡ã®æ°ã $0$ 以å€ã®ã¡ããã© $2$ çš®é¡ã®æ°ãããªãæ°ã**è¯ãæ°**ãšåŒã³ãŸãïŒäŸãã° $377$ ã $9494$ ã¯è¯ãæ°ã§ããïŒ$888$ ã $2022$ ã¯è¯ãæ°ã§ã¯ãããŸããïŒã¡ããã© $5$ æ¡ã®è¯ãæ°ã®ç·åãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/8432 | F | OMCB012(F) | 200 | 149 | 206 | [
{
"content": "ã$A$ ãã $BC$ ã«äžãããåç·ã®è¶³ã $J$ ãšãããšïŒ$BP = PJ$ ããã³å¹³è¡ç·ã®æ¯ã®æ§è³ªãã\r\n$$AE:EC=JQ:QC=1:5$$\r\nã§ããïŒãŸã $B$ ãã $AC$ ã«äžãããåç·ã®è¶³ã $K$ ãšãããšïŒ$AE = EK$ ãã\r\n$$AE:EK:KC=1:1:4$$\r\nãåŸãïŒ$â³AJC \\sim â³BKC$ ãªã®ã§\r\n$$AC:6 = AC : JC=BC:KC=10:\\dfrac{2}{3}AC$$ ãã $AC^2=\\bf{90}$ \r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "htt... | ãéè§äžè§åœ¢ $ABC$ ã«ãããŠèŸº $AB$ ã®äžç¹ã $D$ ãšãïŒ$D$ ããçŽç· $AC$ ã«ããããåç·ã®è¶³ã $E$ ãšããŸãïŒ$D,E$ ããçŽç· $BC$ ã«ããããåç·ã®è¶³ããããã $P,Q$ ãšãããšãïŒ$$BP=2, \quad PQ=3, \quad QC=5$$
ãæãç«ã¡ãŸããïŒ$AC^2$ ãæ±ããŠãã ããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/7177 | G | OMCB012(G) | 300 | 125 | 205 | [
{
"content": "ãéšå¡æ°ã $555, 666, 777, 567$ 人ã®éšæŽ»ããããã $A, B, C, D$ ã§è¡šãïŒã¡ããã© $2$ ã€ã®éšæŽ»ã«å
¥ã£ãŠããã®ã $x$ 人ã§ãããšãïŒãã® $x$ 人ã®ãã¡åéšæŽ»ã«å
¥ã£ãŠããã®ã $a,b,c,d$ 人ãããšããïŒ$a+b+c+d=2x$ ã§ããããšã«æ³šæããïŒããã $x$ 人以å€ã®ååžã«ã€ããŠæ³šç®ããããšã§ïŒã\r\n$$\\max(555-a, 666 - b, 777 - c, 567 - d) \\leq 1000 - x$$\r\nãæãç«ã€ããšãåããïŒããããç¹ã«\r\n$$2565-2x = (555-a)+(666-b)+(777... | ãOMCåŠåã«ã¯ $1000$ 人ã®çåŸãåšç±ããŠããŸãïŒOMCåŠåã«ã¯ $4$ ã€ã®éšæŽ»ãããïŒéšå¡æ°ã¯ãããã $555$ 人ïŒ$666$ 人ïŒ$777$ 人ïŒ$567$ 人ã§ãïŒã¡ããã© $2$ ã€ã®éšæŽ»ã«æå±ããŠããçåŸã®äººæ°ãšããŠããããæ倧å€ã解çããŠãã ããïŒãã ãïŒã©ã®éšæŽ»ã«ãæå±ããŠããªãçåŸãïŒ$3$ ã€ä»¥äžã®éšæŽ»ã«æå±ããŠããçåŸãããŠãããŸããŸããïŒ |
OMCB012 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb012/tasks/4575 | H | OMCB012(H) | 300 | 52 | 85 | [
{
"content": "ãåããåã $C$ ,ãã®äžå¿ã $P$ ãšãïŒ$A(0,1),B(0,-1)$ ãšããïŒãŸãïŒ$A,B$ ãããããäžå¿ãšããååŸ $2$ ã®åããããã $C_1,C_2$ ãšããïŒãã®ãšãïŒ$PA=PB=\\dfrac{1}{\\cos \\alpha}$ ãšãªãïŒäžå¿ã®è·é¢ãšååŸã®é¢ä¿ããå $C$ ã¯å $C_1$ ãšå $C_2$ ã«å
æ¥ããªããåãããšããããïŒããããïŒæ±ããééé åã¯ïŒ$x\\geq 0$ ã®ç¯å²ã§ã¯å $C_1$ ãšå $C_2$ ã®å
±ééšåã§ããïŒ$x\\leq 0$ ã®ç¯å²ã§ã¯ $\\alpha=0$ ã®ãšãã®å $C$ ã®ååã§ããããšããããïŒ... | ã$0\leq \alpha\leq \dfrac{\pi}{3}$ ã®ç¯å²ã§å®æ° $\alpha$ ãåãããšãïŒ $(\tan \alpha,0)$ ãäžå¿ã«æã€ååŸ $2-\dfrac{1}{\cos \alpha}$ ã®ååšãééããé åã®é¢ç©ãæ±ããŠãã ããïŒ\
ããã ãïŒæ±ããçãã¯ïŒæ£æŽæ° $a,b,c$ ãçšããŠïŒ$\dfrac{a}{b}\pi-\sqrt{c}$ ãšè¡šããã®ã§ïŒãã ã $a,b$ ã¯äºãã«çŽ ïŒïŒ$a+b+c$ ã®å€ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/10741 | A | OMC221(A) | 100 | 337 | 364 | [
{
"content": "ãããããã®ã¢ã«ãã¡ãããã§æžãæããã®ã¡ã®æ°åãè¡šãããšãšããïŒ$a$ ã $3$ ã€ïŒ$r$ ã $2$ ã€ãããã®ä»ã®ã¢ã«ãã¡ããã㯠$1$ ã€ãã€ããããšããïŒ\r\n$$2a+r+(0+1+\\dots+9)=54$$ \r\nããªãã¡ $2a+r=9$ ã§ããïŒãŸãïŒ$a\\neq r$ ã§ããããïŒãããæºããã®ã¯\r\n$$(a,r)=(0,9),(1,7),(2,5),(4,1)$$\r\nã§ããïŒããããã«å¯ŸããŠæ®ã $8$ ã€ã®ã¢ã«ãã¡ãããã«ä»£å
¥ããæ¹æ³ã¯ $8!$ éãããã®ã§ïŒæ±ããå Žåã®æ°ã¯ $4\\times8!=\\mathbf{161280}$ éãã§... | ãé»æ¿ã« $10$ çš®é¡ïŒ$13$ åã®ã¢ã«ãã¡ããã
$$m,a,r,t,h,s,a,k,u,r,a,n,o$$
ãæžãããŠããŸã ($a$ ã $3$ ã€ïŒ$r$ ã $2$ ã€ïŒä»ã®ã¢ã«ãã¡ããã㯠$1$ ã€ãã€æžãããŠããŸã) ïŒä»¥äžãã¿ããããã«ããããã®ã¢ã«ãã¡ãããã $0$ ä»¥äž $9$ 以äžã®æŽæ°ã«æžãæããæ¹æ³ã¯ããã€ãããŸããïŒ
- åãã¢ã«ãã¡ãããã¯åãæ°åã«ïŒç°ãªãã¢ã«ãã¡ãããã¯ç°ãªãæ°åã«æžãæããïŒ
- å
šãŠã®ã¢ã«ãã¡ããããæžãæããã®ã¡ïŒé»æ¿ã«æžãããæ°åã®å㯠$54$ ã§ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/7797 | B | OMC221(B) | 200 | 198 | 272 | [
{
"content": "**解æ³1.**ãå¹³è¡å蟺圢ã®èŸºã®é·ãã $x, y$ ãšãããšïŒäœåŒŠå®çãã\r\n$$x^2+y^2-2xy\\cos89^\\circ=199^2,\\quad x^2+y^2+2xy\\cos89^\\circ=201^2$$\r\nã§ããããïŒ\r\n$$xy=\\dfrac{201^2-199^2}{4\\cos89^\\circ}=\\dfrac{200}{\\cos89^\\circ}$$\r\nãåŸãïŒãã£ãŠïŒ\r\n$$S=xy\\sin89^\\circ=200\\tan89^\\circ$$\r\nãšè¡šããïŒãŸãïŒé è§ã $2^\\circ$ïŒåºèŸºã®é·ãã $1$ ... | ãå
è§ã®ã²ãšã€ã $89^\circ$ ã§ãã£ãŠïŒ$2$ æ¬ã®å¯Ÿè§ç·ã®é·ãããããã $199$ ãš $201$ ã§ããå¹³è¡å蟺圢ã®é¢ç©ã $S$ïŒäžèŸºã®é·ãã $1$ ã§ããæ£ $180$ è§åœ¢ã®é¢ç©ã $T$ ãšãããšãïŒ$\dfrac{S}{T}$ ãæ±ããŠãã ããïŒãã ãïŒæ±ããå€ã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šãããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/9871 | C | OMC221(C) | 400 | 115 | 191 | [
{
"content": "ã$X_n=\\\\{1,1,2,2,\\dots,n,n\\\\}$ ãšããïŒ$f$ ã®å®çŸ©ããã®éšåéåã«æ¡åŒµããïŒããã§ïŒ$f(\\emptyset)=1$ ãšããïŒãã®ãšãïŒ$n=2,3,\\dots$ ã«ã€ããŠïŒ$X_{n-1}$ ã®éšåéå $V$ ãš $\\\\{n,n\\\\}$ ã®éšåéåã®åéåã«ã€ããŠïŒä»¥äžãæç«ããïŒ\r\n$$\r\n\\begin{aligned}\r\nf(V\\cup \\emptyset)&=f(V),\\\\\\\\\r\nf(V\\cup \\\\{n\\\\})&=\r\n\\begin{cases}\r\nf(V)&(Vã®èŠçŽ æ°ãå¥... | ãå€ééå $\\{1,1,2,2,3,3,4,4,5,5\\}$ ã®ç©ºã§ãªãéšåéå $U$ ã«ã€ããŠïŒãã®èŠçŽ ãæé ã«äžŠã¹ããšãå¥æ°çªç®ã«ããããã®ã®ç·ç©ã $f(U)$ ãšããŸãïŒäŸãã° $U=\\{1,2,2,4,5,5\\}$ ã®ãšãïŒ$f(U)=1\times2\times5$ ã§ãïŒ$U$ ãšããŠèãããããã®ã¯ $3^{5}-1$ éããããŸããïŒããã§ïŒåãæ°ã¯åºå¥ããªããã®ãšããŸãïŒïŒããããã¹ãŠã«å¯Ÿãã $f(U)$ ã®ç·åãæ±ããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/5046 | D | OMC221(D) | 400 | 10 | 47 | [
{
"content": "ã$\\angle DAP+\\angle PRD=180^\\circ$ ãã $A, P, R, D$ ã¯å
±åã§ããïŒååšè§ã®å®çãã $\\angle RAP=\\angle RDP=\\angle RPD=\\angle RAD$ ã§ããããïŒ$R$ 㯠$AC$ äžã«ããïŒ\\\r\nã$PD$ ãš $AC$ ã®äº€ç¹ã $S$ ãšããïŒ$\\triangle BRS$ ã $AC$ ã«é¢ããŠç·å¯Ÿç§°ç§»åããå³åœ¢ã $\\triangle DRS$ ã§ããïŒãã£ãŠïŒ$\\angle RBS=\\angle RDS=\\angle RPS$ ãšãªããã $B, R, S, P$ ã¯å
±å... | ãäžèŸºã®é·ãã $1$ ã®ã²ã圢 $ABCD$ ãããïŒèŸº $AB$ äžã«ç¹ $P$ïŒèŸº $CD$ äžã«ç¹ $Q$ïŒç·å $BQ$ äžã«ç¹ $R$ ããšããŸãïŒããŸïŒ
$$AP:DQ=10:9,\quad PR=RD,\quad \angle ABC=\angle PRD$$
ãæãç«ã€ãšãïŒç·å $AC$ ã®é·ããšããŠããããæ倧å€ãæ±ããŠãã ããïŒååšãä¿èšŒãããŸãïŒïŒãã ãïŒæ±ããå€ã¯æ£æŽæ° $a, b, c$ïŒ$a, c$ ã¯äºãã«çŽ ã§ããïŒ$b$ ã¯å¹³æ¹å åããããªãïŒãçšã㊠$\dfrac{a \sqrt b}{c}$ ãšè¡šãããã®ã§ïŒ$a+b+c$ ã解çããŠãã ããïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/5047 | E | OMC221(E) | 400 | 58 | 110 | [
{
"content": "ã$X^2$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ãïŒæŽæ° $Y$ ãçšã $Y^2$ ãšããïŒãŸãæ¡ä»¶ãé ã« (a),(b),(c) ãšããïŒ$X\\geq4$ ãš (b),(c) ããïŒããæ£æŽæ° $n$ ãååšããŠ\r\n$$(X+Y)(X-Y)=2^{n+3}\\times5^n\\quad\\cdots(1),\\qquad10^n\\gt Y^2\\quad\\cdots(2)$$\r\nãæãç«ã€ïŒãã®ãšã $n+1$ 㯠$X^2$ ã®æ¡æ°ãšäžèŽããïŒïŒ\r\nç°¡åãªè°è«ã«ããïŒ(1) ãã¿ãã $(X,Y)$ ã®çµã¯ãã $i,j\\ (0\\leq i\\leq n+1,0... | ã次ã®æ¡ä»¶ããã¹ãŠæºãã $4$ 以äžã®æŽæ° $X$ ãšããŠããåŸãå€ã®ãã¡ $6$ çªç®ã«å°ãããã®ãæ±ããŠãã ããïŒãã ãïŒæ¡ä»¶ã¯ãã¹ãŠåé²æ³è¡šèšã§èããŸãïŒ
- $X^2$ ã® $1$ ã®äœã¯ $0$ ã§ãªãïŒ
- $X^2$ ã®æé«äœã®æ°ã¯ $8$ ã§ããïŒ
- $X^2$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ã¯å¹³æ¹æ°ã§ããïŒ
ãäŸãã°ïŒ$10201$ ã®æé«äœã®æ°ãåãé€ããŠã§ããæ°ã¯ $201$ ã§ãïŒãŸãïŒ$0.301 \lt\log_{10}2 \lt 0.302$ ãæãç«ã¡ãŸãïŒ |
OMC221 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc221/tasks/10100 | F | OMC221(F) | 600 | 11 | 34 | [
{
"content": "ã$P$ ã®ç§»åã«ä»¥äžã®ããã«ååãã€ããïŒ\r\n- æäœ $X$ : ç¹ $P$ ã $x$ æ¹åã« $1$ ã ã移åããã\r\n- æäœ $Y$ : ç¹ $P$ ã $y$ æ¹åã« $1$ ã ã移åããã\r\n- æäœ $Z$ : ç¹ $P$ ã $z$ æ¹åã« $1$ ã ã移åããã\r\n\r\nã$P$ ã®ç§»åæ¹æ³ã«å¯Ÿã㊠$X,Y,Z$ ãããªãæååã察å¿ãããããšãèããïŒããšãã°ïŒæäœ $X$ $\\rightarrow$ æäœ $Y$ $\\rightarrow$ æäœ $Z$ $\\rightarrow$ $\\cdots$ã«å¯ŸããŠã¯ïŒ$XYZ\\cdots$ ... | ã座æšç©ºéå
ã®ç¹ $P$ ãã¯ããåç¹ $(0,0,0)$ ã«ãããŸãïŒ$P$ ã $x, y, z$ ã®ããããã®æ£æ¹åã« $1$ ã ã移åãããæäœãèš $900$ åè¡ãããšã§ïŒç¹ $(400,400,100)$ ã«ç§»åãããããšãèããŸãïŒ\
ã$k=0,1,\dots,100$ ã«å¯ŸããŠïŒå蟺ã $x$ 軞ãŸã㯠$y$ 軞ã«å¹³è¡ãªé·æ¹åœ¢ã§ãã£ãŠïŒ$P$ ã®éã£ã $901$ åã®æ Œåç¹ã®ãã¡å¹³é¢ $z=k$ å
ã«å«ãŸãããã®å
šãŠããã®å
éšãŸãã¯åšäžã«å«ããã®ã®é¢ç©ã®æå°å€ã $S_k$ ãšããŸãïŒãã ãïŒ$z=k$ å
ã§ã® $P$ ã®éã£ãæ Œåç¹ãåäžçŽç·äžã«äžŠã¶ïŒãŸã㯠$1$ ç¹ã®ã¿ã§ãããšãïŒ$S_k=0$ ãšã... |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/7038 | A | OMCB011(A) | 100 | 337 | 360 | [
{
"content": "ã$MN \\parallel BC$ ã§ããïŒ$â MBC=108^\\circ \\div 2 =54^\\circ$ ã§ãããã $â BMN=180^\\circ - 54^\\circ=\\mathbf{126}^\\circ$ ã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/7038"
}
] | ãæ£äºè§åœ¢ $ABCDE$ ã«ã€ããŠïŒç·å $AC$ ã®äžç¹ïŒç·å $BD$ ã®äžç¹ããããã $M,N$ ãšããŸãïŒ$â BMN$ ã®å€§ããã床æ°æ³ã§è§£çããŠäžããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/4459 | B | OMCB011(B) | 100 | 329 | 354 | [
{
"content": "ã$(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)$ ã®å±éãèããã°ïŒ$\\dfrac{6!}{x}$ ãæŽæ°ã«ãªã $x$ ãæ±ããã°ããïŒãããæºãã $x$ 㯠$6!=2^4 \\times 3^2 \\times 5$ ã®æ£ã®çŽæ°ã§ããããïŒæ±ããå€ã¯$$(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2)(5^0+5^1)=\\mathbf{2418}.$$",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/4459... | ã$\dfrac{(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)}{x}$ ãæŽæ°ãšãªããããªæ£æŽæ° $x$ ã®ç·åãæ±ããŠãã ããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/11081 | C | OMCB011(C) | 100 | 314 | 326 | [
{
"content": "ã$A$ ã®æ¬¡ã«éãç¹ã $P$ïŒãã®æ¬¡ã«éãç¹ã $Q$ ãšããïŒ$(P,Q)$ ã®å®ãæ¹ã¯ $3 \\times 2 = 6$ éãããïŒ$Q$ ãã $G$ ãžãšåããæ¹æ³ã¯ã¡ããã© $3$ éãååšããïŒå¯Ÿç§°æ§ãããããšã«æ³šæïŒïŒãããã£ãŠçã㯠$6 \\times 3 = \\mathbf{18}$ éãã§ããïŒ\r\n\r\n------\r\n**äŸïŒ**ããšãã° $(P,Q) = (B,C)$ ã®ãšãïŒ$C$ ãã $G$ ãžè¡ãæ¹æ³ã¯\r\n- $C \\to G$\r\n- $C \\to D \\to H \\to G$\r\n- $C \\to D \\to H ... | ãç«æ¹äœ $ABCD - EFGH$ ã«ãããŠïŒç·å $AG$ ã¯ç«æ¹äœã®äœå¯Ÿè§ç·ïŒå
éšãéãæãé·ã察è§ç·ïŒã§ãïŒé ç¹ $A$ ãã蟺äžã®ã¿ãéã£ãŠé ç¹ $G$ ãŸã§éäžã§æ¥ãéãæ»ããã«ç§»åããæ¹æ³ã®ãã¡ïŒåãé ç¹ã $2$ å以äžééããªããã®ã¯ããã€ãããŸããïŒ\
ããã ãïŒã¹ã¿ãŒãå°ç¹ã® $A$ ããŽãŒã«å°ç¹ã® $G$ ãåºçºïŒå°çããæç¹ã§ééãããšã¿ãªããŸãïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/5182 | D | OMCB011(D) | 200 | 195 | 310 | [
{
"content": "ã$x \\lt y$ ãšããã° $x+y=13$ ãæºãã $(x,y)$ ã®çµã¯ $(1,12),(2,11),\\cdots ,(6,7)$ ã® $6$ åããïŒããããã«ã€ã㊠$(f(x),f(y))$ ã®çµã¿ãšããŠããåŸããã®ã¯ $(1,12),(2,11), \\cdots ,(12,1)$ ã® $12$ åååšããããæ±ããå€ã¯ $12^6=\\bf{2985984}$ ã§ãã.",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/5182"
... | ã$S=\lbrace1,2,3,\ldots ,12\rbrace$ ãšããŸãïŒä»¥äžã®æ¡ä»¶ãæºããé¢æ° $f\colon S\rightarrow S$ ã¯ããã€ãããŸããïŒ
- $x+y=13$ ãæºããå
šãŠã® $S$ ã®å
ã®çµ $(x,y)$ ã«ã€ã㊠$f(x)+f(y)=13$ ãæºããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/6308 | E | OMCB011(E) | 200 | 312 | 318 | [
{
"content": "$$\\overline{ABCDE}\\times A=\\overline{EEEEEE}=3\\cdot7\\cdot11\\cdot13\\cdot37\\cdot E$$\r\nãã $11\\cdot 13\\cdot 37\\mid\\overline{ABCDE}\\cdot A$ ã ãïŒ$A$ 㯠$9$ 以äžã®æ£æŽæ°ãªã®ã§ $11,13,37$ ã®ãããã§ãå²ãåããªãïŒãã£ãŠ $11\\cdot 13\\cdot 37\\mid \\overline{ABCDE}$ ã§ããïŒæ¬¡ãæºããæ£æŽæ° $k$ ãååšããïŒ$$\\overline{ABCDE}=5291k\\q... | ã$A,E$ 㯠$1$ ä»¥äž $9$ 以äžã®æŽæ°ïŒ$B,C,D$ 㯠$0$ ä»¥äž $9$ 以äžã®æŽæ°ã§ããïŒ$A,B,C,D,E$ ã¯çžç°ãªããŸãïŒ$$\overline{ABCDE}\times A=\overline{EEEEEE}$$ãæç«ããæïŒ$\overline{ABCDE}$ ã®å€ã¯äžæã«å®ãŸãã®ã§ïŒãã® $\overline{ABCDE}$ ã®å€ã解çããŠãã ããïŒ\
ããã ãïŒ$$\displaystyle\overline{a_na_{n-1}\cdots a_1a_0}=10^na_n+10^{n-1}a_{n-1}+\cdots+10a_1+a_0$$ ã§ãïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/10719 | F | OMCB011(F) | 200 | 246 | 281 | [
{
"content": "ãååŸ $5$ ã®è²çŽã¯å¿
ãèŠãããšãã§ãããïŒãã®ä»ã®è²çŽã¯èªåããã倧ããè²çŽã®äžã«ããã°èŠãããšãã§ããªãïŒãã£ãŠïŒååŸ $5$ ã®è²çŽã¯ãè¡šããè£ãã® $2$ éãïŒãã®ä»ã®è²çŽã¯ãè¡šããè£ããèŠããªããã® $3$ éãããïŒåè²çŽã®èŠãæ¹ãå®ãããšãïŒãããå®çŸãããéãæ¹ã¯ååšããïŒãããã£ãŠæ±ããèŠãæ¹ã¯ $2 \\times 3^4 = \\textbf{162}$ éãã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/contests/omcb011/editorial/10719"
... | ãæ°Žå¹³ãªæºãšïŒååŸããããã $1,2,3,4,5$ ã®åã®åœ¢ãããè²çŽã $1$ æãã€ããïŒ$i=1,2,3,4,5$ ã«å¯ŸããŠïŒååŸ $i$ ã®è²çŽã®è¡šé¢ã¯è² $2i-1$ ã§å¡ãããŠããïŒè£é¢ã¯è² $2i$ ã§å¡ãããŠããŸãïŒ
ãã ãïŒè² $1,2,...,10$ ã¯çžç°ãªããšããŸãïŒ\
ãæºã®äžã®å®ç¹ãå®ãïŒãã®äžã«èš $5$ æã®è²çŽãïŒå®ç¹ãšå
šãŠã®äžå¿ãäžèŽããããã«éããŸãïŒéããé çªïŒããã³ã©ã¡ãã®é¢ãäžã«ãããã¯ä»»æã§ãïŒéããç¶æ
ãçäžããèŠããšã $5$ æã®è²çŽã®**èŠãæ¹**ã¯äœéããããŸããïŒ\
ãéããé çªãç°ãªã£ããïŒé¢ã®åããéã£ããããªéãæ¹ã§ãã£ãŠãïŒçäžããèŠããšãã®èŠãæ¹ãåãã§ãã... |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/6700 | G | OMCB011(G) | 300 | 154 | 193 | [
{
"content": "ã蟺 $AC, BC$ ã®äžç¹ããããã $D, M$ ãšãïŒçŽç· $BH$ ãš $AC$ ã®äº€ç¹ã $E$ ãšããïŒ \r\nã$|GD|=4a, |HM|=b$ ãšããïŒéå¿ã¯åé ç¹ãšå¯ŸèŸºã®äžç¹ã $2:1$ ã«å
åããç¹ã§ãããã $BD=12a$ ã§ããïŒãŸãïŒ$\\angle{CBH}=\\angle{GBH}$ ããïŒ\r\n$$BM=BG\\times \\frac{HM}{GH} = 2ab$$\r\nã§ããïŒããã«ïŒ\r\n$$\\angle CDB = 90^\\circ - \\angle GBH = 90^\\circ - \\angle CBH = \\angl... | ã$AB=AC,\angle BAC\lt 60^\circ$ ãªãäžè§åœ¢ $ABC$ ã®éå¿ã $G$ïŒåå¿ã $H$ ãšãããšããïŒ
$$\angle{CBH}=\angle{GBH},\quad GH=4$$
ãšãªããŸããïŒ
ãäžè§åœ¢ $ABC$ ã®é¢ç©ã®äºä¹ã¯æŽæ°ãšãªããŸãïŒãã®å€ãæ±ããŠãã ããïŒ |
OMCB011 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb011/tasks/7115 | H | OMCB011(H) | 300 | 54 | 132 | [
{
"content": "$$a_n=(999+\\sqrt{999997})^n+(999-\\sqrt{999997})^n$$\r\nãšãããš $a_n$ ã¯æ£æŽæ°ã§ããïŒæ¬¡ã®æŒžååŒãæãç«ã€ïŒ\r\n$$a_{n+2}=1998a_{n+1}+1996a_n$$\r\nãã£ãŠæ¬¡ãåŸãïŒ\r\n$$a_0\\equiv 2,\\quad a_1\\equiv 8,\\quad a_{n+2}\\equiv 8a_{n+1}+6a_n\\quad\\pmod{10}$$\r\nãããçšãããš $a_1$ 以é㯠$10$ ãæ³ãšã㊠$8,6,6,4,8,8,2,4,4,6,2,2$ ãç¹°ãè¿ãããšããããïŒ\... | ã$(999+\sqrt{999997})^n$ ã $10$ é²æ³ã®å°æ°ã§è¡šãããšãã® $1$ ã®äœã $5$ ã§ãããããªïŒ$1$ ä»¥äž $100$ 以äžã®æŽæ° $n$ ã®ç·åãæ±ããŠãã ããïŒ\
ãäŸãã°ïŒå°æ° $7115.11$ ã® $1$ ã®äœã¯ $5$ ã§ãïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/5690 | A | OMC220(A) | 100 | 348 | 357 | [
{
"content": "$$\\angle ABP = \\angle PBC=30^{\\circ},\\quad AB = BC$$\r\nãæãç«ã€ããïŒäžè§åœ¢ $BPA$ ãš äžè§åœ¢ $BPC$ ã¯ååã§ããïŒãã£ãŠïŒ\r\n$$\\angle APC=2(180^\\circ - \\angle APB) = 2(180^\\circ - (180^\\circ - 12^\\circ - 30^\\circ))=\\textbf{84}^{\\circ}$$\r\nã§ããïŒ",
"text": "å
¬åŒè§£èª¬",
"url": "https://onlinemathcontest.com/... | ãæ£äžè§åœ¢ $ABC$ ã«ãããŠïŒãã®å
éšã«ç¹ $P$ ããšããšïŒ
$$\angle BAP=12^{\circ}, \quad \angle ABP=30^{\circ}$$
ãæç«ããŸããïŒãã®ãšãïŒ$\angle APC$ ã®å€§ããã床æ°æ³ã§æ±ããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/4571 | B | OMC220(B) | 300 | 248 | 294 | [
{
"content": "ã$10^6$ ã®äœãé©åã«è£ãããšã§ïŒåé¡ã¯ä»¥äžã®ããã«è¡šçŸã§ããïŒ\r\n\r\n- åæ¡ã®åã $6$ ã§ãã $7$ æ¡ä»¥äžã®æ£æŽæ°ãã¹ãŠã«ã€ããŠïŒãã®äž $6$ æ¡ã®ç·åãæ±ããïŒ\r\n\r\nãããŠïŒåæ¡ã®åã $6$ ã§ãã $7$ æ¡ä»¥äžã®æ£æŽæ°ã¯ïŒåºå¥ã®ãªã $6$ åã®çãåºå¥ã®ãã $7$ åã®ç®±ã«å
¥ããæ¹æ³ïŒç©ºã®ç®±ãèš±ãïŒã«å¯Ÿå¿ããããïŒãã®ç·æ°ã¯ ${}\\_{12}\\mathrm{C}\\_{6}$ åã§ããïŒããã ${}\\_{12}\\mathrm{C}\\_{6}$ åã®æ°ã®åæ¡ã®ç·å㯠${}\\_{12}\\mathrm{C}\\_{6}\\ti... | ã$10^6$ æªæºã®æ£æŽæ°ã®ãã¡ïŒåé²æ³è¡šèšã§åæ¡ã®åã $6$ 以äžã§ãããã®å
šãŠã«ã€ããŠïŒãã®ç·åãæ±ããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/7581 | C | OMC220(C) | 300 | 203 | 276 | [
{
"content": "ã$M$ ã®èŠçŽ ãé ç¹ã«æã€ã°ã©ãã§ãã£ãŠïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯Ÿã㊠$x$ ãã $f(x)$ ã«åããŠæå¹èŸºã匵ãããŠãããã®ãèããïŒ\r\n\r\n----\r\n**è£é¡.**ã$f$ ãæ¡ä»¶ãæºããããšã¯ïŒåé ç¹ãé ç¹æ° $3$ ã®éè·¯ã«å«ãŸããããšãšåå€ã§ããïŒ\\\r\n**蚌æ.**ããŸãå¿
èŠæ§ã瀺ãïŒ$f$ ãæ¡ä»¶ãæºãããšãïŒ$f$ ã¯å
šå°ã§ãã $M$ ã¯æééåã§ããããïŒåé ç¹ã¯éè·¯ã«å«ãŸããïŒæ¬¡ã«ïŒ$f(f(f(x))) = x$ ããåéè·¯ã®é ç¹æ°ã¯ $3$ ã®çŽæ°ã§ãããïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯Ÿã㊠$f(x) \\neq x... | ã$M = \\{1,2,\ldots,99\\}$ ãšããŸãïŒ$f:M\to M$ ã§ãã£ãŠïŒä»»æã® $M$ ã®èŠçŽ $x$ ã«å¯ŸããŠ
$$f(x) \neq x,\quad f\big(f(f(x))\big) = x$$
ãæºãããã®ã®æ°ã $N$ ãšããŸãïŒ$N$ ã $2$ ã§å²ãåããæ倧ã®åæ°ãæ±ããŠäžããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/7444 | D | OMC220(D) | 400 | 145 | 187 | [
{
"content": "ã$n=100$ ãšããïŒ\r\n$$S_1=\\sum_{k=1}^{n^2} \\bigg\\lfloor\\dfrac{k^2}{n^2}\\bigg\\rfloor,\\quad\r\nS_2=\\sum_{k=1}^{n^2}\\Big\\lfloor n\\sqrt{k}\\Big\\rfloor$$\r\nãšããïŒãã®ãšãïŒ$S_1$ 㯠$1\\le x\\le n^2$ ã®é åå
ã® $y=\\dfrac{x^2}{n^2}$ ãš $x$ 軞ã§å²ãŸããé åã«å«ãŸããæ Œåç¹ã®æ°ã«çããïŒ$S_2$ 㯠$1\\le x\\le n^2$ ã®é åå
ã® $y=n\\sqrt x$... | ã以äžã®å€ãæ±ããŠãã ãã.
$$\sum_{k=1}^{10000} \biggl( \bigg\lfloor\frac{k^2}{10000}\bigg\rfloor+\Big\lfloor 100\sqrt{k}\Big\rfloor \biggr)$$ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/6311 | E | OMC220(E) | 400 | 59 | 104 | [
{
"content": "ãçŽç· $HX$ ãšçŽç· $AB$ ã®äº€ç¹ã $Y$ ãšãïŒç·å $CY$ ã®äžç¹ã $M$ïŒ$M$ ã«ã€ã㊠$X$ ãšå¯Ÿç§°ãªç¹ã $Z$ ãšããïŒãã®ãšãïŒ\r\n$$\\angle XHC=\\angle AHC=180^{\\circ}-\\angle ABC=\\angle YBC$$\r\nãã $H, B, Y, C$ ã¯å
±åã§ããïŒãããã£ãŠäžè§åœ¢ $FHB$ ãš $FYC$ ã¯çžäŒŒãšãªãïŒãŸãïŒäžè§åœ¢ $XHB$ ãš $XCY$ïŒ$ZYC$ ã¯ãã¹ãŠçžäŒŒã§ããããïŒåè§åœ¢ $FHXB$ ãš $FYZC$ ã¯çžäŒŒïŒãããã£ãŠïŒ$FZ=22x, CZ=2x, YZ=23x$ ... | ãäžè§åœ¢ $ABC$ ã®åå¿ã $H$ïŒ$C$ ãã蟺 $AB$ ã«äžãããåç·ã®è¶³ã $F$ ãšããŸãïŒèŸº $BC$ äžã« $\angle AHF=\angle XHF$ ãã¿ããç¹ $X$ ãååšãïŒ
$$BX=2, \quad FX=22, \quad HX=23$$
ãæãç«ã€ãšãïŒ$AH$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ãçšã㊠$\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a+b$ ã解çããŠãã ããïŒ |
OMC220 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omc220/tasks/5773 | F | OMC220(F) | 400 | 43 | 125 | [
{
"content": "ãã¹ç®ã®é ç¹ã®ãã¡ïŒäžãã $m$ çªç®ïŒå·Šãã $n$ çªç®ã®ãã®ã $(m, n)$ ã§è¡šãïŒ$1, 4$ ãæžã蟌ãŸãããã¹ãèµ€ãïŒ$2, 3$ ãæžã蟌ãŸãããã¹ãéãã¬ãïŒç°ãªãè²ã®å¢çã«ç·åãåŒãïŒãã®ãšãæ¡ä»¶ããïŒé£ãåã $2$ ãã¹ã«æžã蟌ãŸããæŽæ°ã®åã¯ïŒè²ãåãã§ããç®æã§ã¯ $5$ ã§ããïŒè²ãç°ãªãç®æã§ã¯ $5$ ã§ãªãïŒ\\\r\nãå¡ãæ¹ïŒæç·ã®åŒãæ¹ïŒã以äžã® $2$ éãã«åé¡ãã.\r\n- é·ã $2$ ã®æç·ã $2$ ã€åŒãããå Žå \\\r\né·ã $2$ ã®æç·ãšããŠãããããã®ã¯ $(1, n)$ ãš $(3, n)$ ãçµã¶ãã® ( $n... | ã$2Ã1468$ ã®ãã¹ç®ããããŸãïŒãã®ãšãïŒåãã¹ã«ä»¥äžã®æ¡ä»¶ãæºããããã« $1, 2, 3, 4$ ã®æŽæ°ãæžã蟌ãæ¹æ³ã¯äœéããããŸããïŒ
- é£ããããã¹ã«ã¯ç°ãªãæŽæ°ãæžã蟌ãïŒ
- é£ããã $2$ ãã¹ã«æžã蟌ãŸããæŽæ°ã®åã $5$ ã§ãªããããªç®æã¯ã¡ããã© $4$ ç®æååšããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/8763 | A | OMCE003(A) | 300 | 108 | 162 | [
{
"content": "ã$P, Q, R, S$ ã®é ã§å€ãå®ããããšãèãããïŒ\\\r\nããŸã $P$ ã®éžã³æ¹ã¯ $1110$ éãããïŒããããã® $P$ ã«å¯Ÿãã $Q$ ã®å€ã®åè£ã¯\r\n$$P - 1000ïŒP - 100ïŒP - 10ïŒP + 110ïŒP + 1010ïŒP + 1100$$\r\nã® $6$ ã€ã§ãããïŒãã®ãã¡ $1$ ä»¥äž $1110$ 以äžã®ç¯å²ã«å«ãŸãããã®ã¯ïŒ$P$ ã®éžã³æ¹ã«ãããïŒã¡ããã© $3$ ã€ã§ããïŒ\r\n\r\n<details><summary>çç±<\\/summary>\r\n$$\\begin{aligned}\r\nQ_1 &= P - 1... | ã$1$ ä»¥äž $1110$ 以äžã®æŽæ°ã®çµ $(P, Q, R, S)$ ã§ãã£ãŠïŒä»¥äžãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- $P - Q, Q - R, R - S$ ã¯ãããã
$$-1100, -1010, -110, 10, 100, 1000$$
ã®ããããã«çããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/8320 | B | OMCE003(B) | 400 | 62 | 91 | [
{
"content": "ãäžè§åœ¢ $ABM$ ã®å€æ¥åã $\\Omega$ ãšãããšãïŒåçŽç· $BD$ ãš $\\Omega$ ãç¹ $B$ 以å€ã«å
±æç¹ã $1$ ã€ãã€ã®ã§ïŒããã $E$ ãšããïŒãããšååšè§ã®å®çãã\r\n$$\\angle AME = \\angle MAE = 60^{\\circ}$$\r\nãããããã®ã§ïŒäžè§åœ¢ $AME$ ã¯æ£äžè§åœ¢ã§ããïŒããã§æ¬¡ã®è£é¡ãæãç«ã€ïŒ\r\n\r\n---\r\n\r\n**è£é¡ïŒ**$E$ ã¯ç·å $BD$ïŒäž¡ç«¯ãé€ãïŒäžã®ç¹ã§ããïŒ\r\n\r\n<details><summary>è£é¡ã®èšŒæ<\\/summary>\r\nã$BE \... | ãåžåè§åœ¢ $ABCD$ ãäžããããŠããïŒèŸº $BC$ ã®äžç¹ã $M$ ãšãããšããïŒ
$$AB = 1ïŒAC : AD = 16 : 19ïŒ \\\\
\angle ABD = \angle CBD = 60^{\circ}ïŒ\angle MAC = \angle ADB$$
ãæãç«ã¡ãŸããïŒãã®ãšãç·å $BD$ ã®é·ãã¯äºãã«çŽ ãªæ£æŽæ° $a, b$ ã«ãã£ãŠ $\dfrac{a}{b}$ ãšè¡šããã®ã§ïŒ$a + b$ ã®å€ã解çããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/11150 | C | OMCE003(C) | 400 | 116 | 148 | [
{
"content": "ãæ Œåç¹ã§ãã£ãŠ $x, y$ 座æšããšãã« $3$ ã®åæ°ãšãªããã®ã**çœãç¹**ãšåŒã³ïŒ$x, y$ 座æšããããã $3$ ã§å²ã£ããšãïŒäžæ¹ãäœã $1$ ã§ããäžæ¹ãäœã $2$ ãšãªããã®ã**é»ãç¹**ãšåŒã¶ïŒä»»æã® $1 \\leq n \\leq 36$ ãªãæŽæ° $n$ ã«å¯Ÿã $P_n$ ã® $x, y$ 座æšã®å㯠$n$ ãªã®ã§ïŒOMCåã®ç§»åãç¹°ãè¿ãæ¹æ³ã«ããã $P_3, P_6, ..., P_{36}$ ã® $12$ åã¯å¿
ãçœãç¹ãé»ãç¹ã§ããïŒéã«ïŒ$P_1, P_2, ..., P_{36}$ ã®ãã¡å
ã»ã©ã® $12$ å以å€ã® $24$ åã¯ïŒ... | ãã¯ããïŒOMCå㯠$xy$ å¹³é¢ã®åç¹ã«ããïŒä»¥äžã©ã¡ããã®ç§»åãåèš $36$ åç¹°ãè¿ããŸãïŒ
- $x$ 軞ã®æ£ã®æ¹åã« $1$ 移åããïŒ
- $y$ 軞ã®æ£ã®æ¹åã« $1$ 移åããïŒ
ãã㧠$1 \leq n \leq 36$ ãªãæŽæ° $n$ ã«å¯ŸãïŒç§»åã $n$ åç¹°ãè¿ããæç¹ã§OMCåãããç¹ã $P_n$ ãšè¡šããŸãïŒOMCåã移åãç¹°ãè¿ãæ¹æ³ã¯å
šéšã§ $2^{36}$ éããããŸããïŒãã®ãã¡æ¬¡ã®æ¡ä»¶ãã¿ãããã®ã¯å
šéšã§ããã€ãããŸããïŒ
- $P_1, P_2, ..., P_{36}$ ã®ãã¡ã¡ããã© $1$ ã€ã¯ïŒ$x, y$ 座æšããšãã« $3$ ã®åæ°ã§ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/10277 | D | OMCE003(D) | 400 | 68 | 82 | [
{
"content": "ãæ£æŽæ° $n$ ãšçŽ æ° $p$ ã«å¯ŸããŠïŒ $\\mathrm{ord}_p(n)$ ã $n$ ã $p$ ã§å²ãåããæ倧ã®åæ°ãšå®çŸ©ããïŒ\r\n\r\nã$c$ ãå²ãåããªãä»»æã®çŽ æ° $p$ ã«ã€ããŠïŒ1ã€ç®ã®æ¡ä»¶ããïŒ $$ \\mathrm{ord}_p(a)+\\mathrm{ord}_p(b)\\leq \\mathrm{ord}_p(a^c+b)$$\r\nãšãªãïŒãã㧠$\\mathrm{ord}_p(b)\\lt \\mathrm{ord}_p(a^c)$ ãšä»®å®ãããšïŒ $$ \\mathrm{ord}_p(a)+\\mathrm{ord}_p(b)\\le... | ã以äžã®æ¡ä»¶ãå
šãŠæºããæ£æŽæ°ã®çµ $(a,b,c)$ ã**ãŸã¶ããçµ**ãšãã³ãŸãïŒ
- $ab$ 㯠$c(a^c+b)$ ãå²ãåãïŒ
- $a$ 㯠$c^{10}$ ãå²ãåããªãïŒ
- $a,b,c$ ã¯å
šãŠ $1000$ 以äžã®æ£æŽæ°ïŒ
ãŸã¶ããçµ $(a, b, c)$ ã«ããã $c$ ãšããŠããåŸãæ倧å€ã $c_\mathrm{max}$ ãšããŸãïŒ $c=c_\mathrm{max}$ ãæºãããŸã¶ããçµ $(a,b,c)$ å
šãŠã«ã€ã㊠$abc$ ã®ç·åãæ±ããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/10300 | E | OMCE003(E) | 500 | 24 | 43 | [
{
"content": "ã$n = 10000$ ãšãïŒ$p_k = \\displaystyle\\sum_{i=1}^n a_i^k$ ãšããïŒãŸã $a_1, a_2, \\ldots, a_n$ ã® $d$ 次ã®åºæ¬å¯Ÿç§°åŒã $\\sigma_d$ ãšããïŒãã®ãšã $1 \\leq k \\leq n$ ã«ã€ã㊠$p_k$ 㯠$\\sigma_1, \\ldots, \\sigma_k$ ã®å€é
åŒã§è¡šãããããšã«æ³šæããïŒ\r\n\r\nã$p_k = 2^k + 3^k $ $(k = 1, 2, \\ldots, n-1)$ ããã³ $\\sigma_n = 2024$ ãã¿ããå€é
åŒ\r\n$... | ãè€çŽ æ° $a_1,a_2,âŠ,a_{10000}$ ã¯ä»¥äžã®æ¡ä»¶ããšãã«æºãããŸãïŒ
- $k=1,2,âŠ,9999$ ããããã«å¯ŸããŠïŒ$\displaystyle \sum_{i=1}^{10000}a_i^k=2^k+3^k$ïŒ
- $\displaystyle \prod_{i=1}^{10000}a_i=2024$ïŒ
ããã®ãšãïŒ $\displaystyle \sum_{i=1}^{10000}a_i^{10000}$ ã®å€ã¯äžæã«å®ãŸãïŒæ£æŽæ°å€ã«ãªããŸãïŒãã®å€ãçŽ æ° $4999$ ã§å²ã£ãäœããæ±ããŠãã ããïŒ |
OMCE003 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omce003/tasks/9224 | F | OMCE003(F) | 700 | 12 | 21 | [
{
"content": "ãæ£æŽæ° $n$ ã«å¯Ÿã $n$ 以äžã®æ£æŽæ°å
šäœãããªãéåã $[n]$ ãšæžãïŒ$X, Y, Z$ ãæ£æŽæ°ãšãïŒäžè¬ã« $\\Gamma$ ã次ã®ããã«å®ããïŒåé¡ã®èšå®ã§ã¯ $(X, Y, Z) = (7, 11, 13)$ ã§ããïŒïŒ\r\n\r\n- $\\Gamma$ ã $x \\in [3X], y \\in [3Y], z \\in [3Z]$ ãªãæ Œåç¹ $(x, y, z)$ ãããªãéåãšããïŒ\r\n\r\nãŸãïŒ$\\Omega$ ã®éšåéå $\\Omega_x, \\Omega_y, \\Omega_z$ ããããã次ã®ããã«å®ããïŒãªãïŒããã㯠$\... | ãäžæ¬¡å
空éã«ãããŠïŒ$x, y, z$ 座æšããã¹ãŠæŽæ°ã§ãããããªç¹ã**æ Œåç¹**ãšåŒã¶ããšã«ããŸãïŒæ¬¡ã®æ¡ä»¶ãã¿ããæ Œåç¹å
šäœãããªãéåã $\Omega$ ãšããŸãïŒ
- $x, y, z$ 座æšã®ãã¡**å°ãªããšãäºã€**ã¯ïŒ$3$ ã§å²ããš $2$ äœãæ°ã§ããïŒ
ãŸãïŒä»¥äžã® $3$ æ¡ä»¶ããã¹ãŠã¿ããæ Œåç¹å
šäœãããªãéåã $\Gamma$ ãšããŸãïŒ
- $x$ 座æšã¯ $1$ 以äžã〠$21$ 以äžã§ããïŒ
- $y$ 座æšã¯ $1$ 以äžã〠$33$ 以äžã§ããïŒ
- $z$ 座æšã¯ $1$ 以äžã〠$39$ 以äžã§ããïŒ
ããã§ïŒ$\Gamma$ ããçžç°ãªã $1000$ ... |
OMCB010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb010/tasks/4981 | A | OMCB010(A) | 100 | 331 | 346 | [
{
"content": "ã$A\\text{ - }B,B\\text{ - }C,...,E\\text{ - }F$ ã® $5$ ç®æã§é
ç¹ã $100$ ç¹äžããããïŒå®éã«é
ç¹ãäžããã®ã¯ $3$ ç®æãªã®ã§æ¡ä»¶ãæºããé
ç¹ã®çµã¿åãã㯠${}\\_{5}\\mathrm{C}\\_{3}=\\bf10$ éãã§ããïŒ\r\n\r\n\r\n----\r\n**å¥è§£.**\r\nã以äžã® $2$ éãã®å ŽåãããåŸãïŒ\r\n- $100, 200, 300, 400$ ã®ãããã $2$ ã€ã $2$ åïŒæ®ãã® $2$ ã€ã $1$ åã®å ŽåïŒ\r\n- $100, 200, 300, 400$ ã®... | ãããOMCã®ã³ã³ãã¹ãã«ã€ããŠïŒä»¥äžã®æ¡ä»¶ããšãã«æç«ããŸããïŒ
- ã³ã³ãã¹ãåé¡ã¯é ã« $A, B, C, D, E, F$ ã® $6$ åã§ããïŒé
ç¹ãäœãé ã«äžŠãã§ããïŒ
- $6$ åã«ã¯ $100$ ç¹ïŒ$200$ ç¹ïŒ$300$ ç¹ïŒ$400$ ç¹ããããã $1$ å以äžå«ã¿ïŒãŸããã以å€ã®é
ç¹ã®åé¡ã¯å«ãŸãªãïŒ
ãã®ãšãïŒ$6$ åã®é
ç¹ã®çµã¿åãããšããŠãããããã®ã¯ããã€ãããŸããïŒ |
OMCB010 | https://onlinemathcontest.com/contests/all?page=1 | https://onlinemathcontest.com/contests/omcb010/tasks/6404 | B | OMCB010(B) | 200 | 272 | 300 | [
{
"content": "ãåè§åœ¢ $ABCD$ ã¯äžçµã®å¯ŸèŸºãçããå
æ¥åè§åœ¢ãªã®ã§ç¹ã«çèå°åœ¢ã§ããïŒ$A$ ãã 蟺 $BC$ ã«äžãããåç·ã®è¶³ã $H$ ãšããïŒ$ {AD}=2a$ ãšãããšïŒ$BH = \\dfrac{1}{2}(BC - AD) = a$ ã§ããããïŒäžè§åœ¢ $ABC$ ã«å¯Ÿããäžå¹³æ¹ã®å®çããïŒ\r\n$$AH = \\sqrt{AB^2 - BH^2} = \\sqrt{4-a^2}$$\r\nãåŸãïŒåŸã£ãŠïŒåè§åœ¢ $ {ABCD}$ ã®é¢ç©ã«ã€ããŠïŒä»¥äžã®ãããªæ¹çšåŒãç«ãŠãããïŒ\r\n$$6=\\frac12\\times AH\\times(AD+BC) = 3a\\s... | ãåã«å
æ¥ããåžåè§åœ¢ $ {ABCD}$ ã«ã€ããŠïŒ
$$ {AB}= {CD}=2, \quad {BC}=2 {AD}$$
ãæãç«ã¡ãŸããïŒåè§åœ¢ $ {ABCD}$ ã®é¢ç©ã $6$ ã®ãšãïŒ$ {BC}$ ã®é·ãã® $2$ ä¹ã解çããŠãã ããïŒ |
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