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- == Problem == [[Image:AIME_1998-2.png|thumb|300px|right]]6 KB (913 words) - 16:34, 6 August 2020
- == Problem 2 == ...and <math>D</math> represent four different digits selected from <math>0,1,2,\ldots ,9.</math> If <math>(A+B)/(C+D)</math> is an integer that is as larg1,008 bytes (167 words) - 14:28, 5 July 2013
- == Problem == ...{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?797 bytes (126 words) - 15:40, 20 October 2016
- == Problem == C1=circle(O,2);4 KB (656 words) - 17:26, 20 June 2019
- ==Problem== contestants. Prove that <i>k</i>/<i>a</i> ≥ (<i>b</i> − 1)/(2<i>b</i>).451 bytes (79 words) - 23:47, 18 November 2023
- ==Problem 2== <math>\cos X = \frac{AC ^ {2} + AD ^ {2} - DC ^ {2}}{ 2 \cdot AC \cdot AD }</math>3 KB (554 words) - 08:31, 2 July 2020
- == Problem == == Solution 2==651 bytes (98 words) - 16:14, 15 June 2022
- == Problem == ...<math>b</math>, <math>(36a + b)(36b + a)</math> cannot be a power of <math>2</math>.2 KB (297 words) - 12:50, 22 July 2021
- == Problem ==564 bytes (96 words) - 15:39, 13 December 2023
Page text matches
- ...cluding Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete be ...er), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}10 KB (1,497 words) - 11:42, 10 March 2024
- ...ministered by the [[Mathematical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! ...blem 1-5</u>: 3<br><u>Problem 6-10</u>: 4<br><u>Problem 10-12</u>: 5<br><u>Problem 12-15</u>: 6}}8 KB (1,057 words) - 12:02, 25 February 2024
- This is a problem where constructive counting is not the simplest way to proceed. This next e === Example 2 ===12 KB (1,896 words) - 23:55, 27 December 2023
- ...m is the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...ion|recursively]] as <math>F_1 = F_2 = 1</math> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n \geq 3</math>. This is the simplest nontrivial example6 KB (957 words) - 23:49, 7 March 2024
- ...mplex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...from this addition, we are not only able to find the solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial.5 KB (860 words) - 15:36, 10 December 2023
- ...in the AoPSWiki, please consider adding them. Also, if you notice that a problem in the Wiki differs from the original wording, feel free to correct it. Fi | colspan="2"| [[1999 AIME | AIME]]3 KB (391 words) - 16:00, 21 February 2024
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1997 I Problems/Problem 1|Problem 1]]856 bytes (98 words) - 14:53, 3 July 2009
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1999 AIME Problems/Problem 1|Problem 1]]1 KB (118 words) - 08:41, 7 September 2011
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1998 AIME Problems]]1 KB (114 words) - 08:39, 7 September 2011
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1997 AIME Problems/Problem 1|Problem 1]]1 KB (114 words) - 08:39, 7 September 2011
- == Problem 1 == [[1997 AIME Problems/Problem 1|Solution]]7 KB (1,098 words) - 17:08, 25 June 2020
- {{AIME Problems|year=1998}} == Problem 1 ==7 KB (1,084 words) - 02:01, 28 November 2023
- == Problem 1 == [[1999 AIME Problems/Problem 1|Solution]]7 KB (1,094 words) - 13:39, 16 August 2020
- == Problem == ..., a_2, a_3, \ldots</math> is chosen so that <math>a_n = a_{n - 1} - a_{n - 2}</math> for each <math>n \ge 3</math>. What is the sum of the first 2001 te2 KB (410 words) - 13:37, 1 May 2022
- == Problem == ...have to leave the vertex, so every vertex reached is equivalent to adding 2 more segments. So the degree of each vertex must be even, with the exceptio9 KB (1,671 words) - 22:10, 15 March 2024
- == Problem == ...angular box has half the volume of an <math>(m + 2)\times(n + 2)\times(p + 2)</math> rectangular box, where <math>m, n,</math> and <math>p</math> are in2 KB (390 words) - 21:05, 29 May 2023
- == Problem == ...the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given that <math>S_8 = - 176 - 64i</math> and <math> S_2 KB (384 words) - 19:02, 20 October 2023
- == Problem == ...ivisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>?5 KB (876 words) - 20:27, 9 June 2022
- == Problem == ...that it is of the same length of a face diagonal, making it <math>20\sqrt{2}</math>.7 KB (1,084 words) - 11:48, 13 August 2023
- == Problem == <cmath>x^2 + (r-100)^2 = (r+100)^2 \Longrightarrow x = 20\sqrt{r}</cmath>3 KB (496 words) - 13:02, 5 August 2019
- == Problem == label(rotate(45)*"$M_1-M_2\le m$",((m+60)/2,(60-m)/2),NW,fontsize(9));4 KB (624 words) - 18:34, 18 February 2018
- == Problem == | 0 || 1 || 2 || 3 || 4 || 5 || 62 KB (354 words) - 19:37, 24 September 2023
- == Problem == Note that this is an algebraic bijection, we have simplified the problem and essentially removed the odd condition, so now we can finish with plain5 KB (684 words) - 11:41, 13 August 2023
- == Problem == <math>0 = RC^2 - 112\cdot847</math>2 KB (254 words) - 19:38, 4 July 2013
- == Problem == Given that <math>A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,</math> find <math>|A_{19} + A_{20} + \cdots + A_{98}|.</math>1 KB (225 words) - 02:20, 16 September 2017
- == Problem == Nine tiles are numbered <math>1, 2, 3, \cdots, 9,</math> respectively. Each of three players randomly selects5 KB (917 words) - 02:37, 12 December 2022
- == Problem == The graph of <math> y^2 + 2xy + 40|x|= 400</math> partitions the plane into several regions. What1 KB (198 words) - 20:13, 23 February 2018
- == Problem == [[Image:AIME_1998-2.png|thumb|300px|right]]6 KB (913 words) - 16:34, 6 August 2020
- == Problem == ...<math>k</math> has only 2s and 3s in its prime factorization, or <math>k = 2^a3^b</math>.2 KB (289 words) - 22:50, 23 April 2024
- == Problem == ...e [[relatively prime]] positive integers, the [[coefficient]]s of <math>x^{2}</math> and <math>x^{3}</math> are equal. Find <math>a + b</math>.679 bytes (98 words) - 00:51, 2 November 2023
- == Problem == ...on <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math7 KB (1,131 words) - 14:49, 6 April 2023
- * [[1959 IMO Problems/Problem 1 | Problem 1]] proposed by Poland * [[1959 IMO Problems/Problem 2 | Problem 2]] proposed by Constantin Ionescu-Tiu, Romania35 KB (4,009 words) - 20:25, 21 February 2024
- ...u>Problem 1</u>: 6<br><u>Problem 2</u>: 6.5<br><u>Problem 3</u>: 7.5<br><u>Problem 4</u>: 8}} *In every competition or test there are four problem usually covering [[geometry]], [[number theory]], [[algebra]], and [[combin4 KB (578 words) - 00:12, 29 August 2018
- ...ficulty=4 - 6|breakdown=<u>Problem 1</u>: 4<br><u>Problem 2/3</u>: 5<br><u>Problem 4</u>: 6}} *In every competition or test there are four problem usually covering [[geometry]], [[number theory]], [[algebra]], and [[combin3 KB (472 words) - 18:49, 2 February 2019
- * [[1997 JBMO Problems/Problem 1 | Problem 1]] proposed by Bulgaria * [[1997 JBMO Problems/Problem 2 | Problem 2]] proposed by Cyprus6 KB (524 words) - 17:17, 17 August 2018
- * [[1984 BMO Problems/Problem 1 | Problem 1]] * [[1984 BMO Problems/Problem 2 | Problem 2]]4 KB (371 words) - 16:41, 1 January 2024
- ...ate|[[2007 AMC 12A Problems|2007 AMC 12A #22]] and [[2007 AMC 10A Problems/Problem 25|2007 AMC 10A #25]]}} == Problem ==15 KB (2,558 words) - 19:33, 4 February 2024
- == Problem I1 == Calculate: <math>\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}</math>11 KB (1,738 words) - 19:25, 10 March 2015
- ==Problem== ...0 thousands and multiply it by 2 and subtract 5, since we are leaving out 1998 and 1999.1 KB (202 words) - 14:39, 20 April 2014
- == Problem 1 == <math>1 - 2 + 3 -4 + \cdots - 98 + 99 = </math>13 KB (1,945 words) - 18:28, 19 June 2023
- ...contains the full set of test problems. The rest contain each individual problem and its solution. ** [[1999 USAMO Problems/Problem 1 | Problem 1]]808 bytes (95 words) - 15:27, 29 April 2020
- == Problem == ...that each <math>d_i</math> can be any of the ten decimal digits <math>0,1,2, \ldots, 9</math>, the number of different memorable telephone numbers is2 KB (330 words) - 10:14, 10 August 2016
- == Problem == \qquad\mathrm{(D)}\ 21 KB (138 words) - 10:41, 22 April 2016
- == Problem == : (b) <math>f(0) = 2</math>.656 bytes (94 words) - 22:16, 28 March 2024
- == Problem == pair D=(0,0), C=(0,24*3^0.5), A=(46,0), E=(72,0), B=(46+13/2,13*3^.5/2);3 KB (391 words) - 14:30, 5 July 2013
- == Problem == [[Image:1998 AHSME num. 27.png]]3 KB (403 words) - 23:00, 10 August 2020
- ** [[1989 APMO Problems/Problem 1 | Problem 1]] ** [[1989 APMO Problems/Problem 2 | Problem 2]]7 KB (626 words) - 21:52, 19 August 2023
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1998 APMO Problems|Problems]]549 bytes (60 words) - 09:45, 30 May 2012
- ...contains the full set of test problems. The rest contain each individual problem and its solution. ** [[1999 AHSME Problems/Problem 1|Problem 1]]2 KB (178 words) - 13:28, 20 February 2020
- |year = 1998 == Problem 1 ==15 KB (2,222 words) - 10:40, 11 August 2020
- == Problem == ...t angle <math>CAD</math> is twice angle <math>DAB</math>. If <math>AC/AD = 2/3</math>, then <math>CD/BD = m/n</math>, where <math>m</math> and <math>n</4 KB (662 words) - 00:51, 3 October 2023
- == Problem == ...8,1),N); MP("I",(.17/2,.25),(0,0));MP("I",(.17/2,.75),(0,0));MP("I",(1-.17/2,.75),(0,0));MP("II",(.5-.17,.4),(0,0));MP("II",(.5-.17,.6),(0,0));MP("II",(2 KB (347 words) - 11:28, 2 December 2019
- == Problem == <center><math>\left[[60,30,90],[2,1,3],[10,5,15]\right]?</math></center>714 bytes (95 words) - 14:28, 5 July 2013
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1998 AHSME Problems|Entire Exam]]2 KB (173 words) - 03:44, 29 September 2014
- == Problem == <math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }3 \qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ }7 </m638 bytes (84 words) - 14:29, 5 July 2013
- ==Problem== ...)}\ \dfrac{1}{2} \qquad \text{(D)}\ \dfrac{1}{3} \qquad \text{(E)}\ \dfrac{2}{3}</math>2 KB (242 words) - 19:53, 31 October 2016
- == Problem == ...= \min(x_n,y_n)</math>. The last non-zero digit is the last digit of <math>2^{x_n-z_n} 5^{y_n-z_n} r_n</math>.3 KB (486 words) - 10:37, 11 August 2020
- == Problem == A piece of graph paper is folded once so that <math>(0,2)</math> is matched with <math>(4,0)</math>, and <math>(7,3)</math> is match4 KB (722 words) - 19:47, 28 October 2022
- == Problem == The graphs of <math>x^2 + y^2 = 4 + 12x + 6y</math> and <math>x^2 + y^2 = k + 4x + 12y</math> intersect when <math>k</math> satisfies <math>a \le k1 KB (193 words) - 09:12, 2 December 2018
- == Problem == ...{d^2}h \qquad \mathrm{(D) \ } \frac {h^2}d \qquad \mathrm{(E) \ } \frac{d^2}{h-d}</math>1 KB (220 words) - 14:29, 5 July 2013
- == Problem == <math>\textrm{(A)}\ 2 \qquad \textrm{(B)}\ 3 \qquad \textrm{(C)}\ 4 \qquad \textrm{(D)}\ 5 \qquad2 KB (328 words) - 16:40, 29 September 2018
- == Problem == <math> \mathrm{(A) \ }0 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 8</m2 KB (368 words) - 11:08, 4 February 2017
- == Problem == <math>\qquad \mathrm{(D) \ }A^2 - M^2 + C^2 = 0 \qquad \mathrm{(E) \ } 2A + 2M = 3C </math>1 KB (187 words) - 14:29, 5 July 2013
- == Problem == We have <math>144 = 2^4 3^2</math>.1 KB (195 words) - 14:29, 5 July 2013
- == Problem == <math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }4 \qquad \mathrm{(D) \ }5 \qquad \mathrm{(E) \ }6 </m929 bytes (132 words) - 14:29, 5 July 2013
- ==Problem== <math> \mathrm{(A) \ } -2004 \qquad \mathrm{(B) \ } -2 \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } 4003 \qquad \mathrm{(E) \ }2 KB (364 words) - 11:41, 13 October 2021
- == Problem == ...ider the sequence of numbers: <math>4,7,1,8,9,7,6,\dots</math> For <math>n>2</math>, the <math>n</math>-th term of the sequence is the units digit of th2 KB (274 words) - 11:06, 18 July 2023
- ...test], [http://www.oma.org.ar/internacional/resultados_generales-may4.htm 1998 test results] ===Problem 1===4 KB (668 words) - 14:52, 17 August 2020
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1998 USAMO Problems|Entire Test]]773 bytes (91 words) - 18:11, 17 September 2012
- == Problem == ...cdots a_n</math> be real numbers in the interval <math>\left(0,\frac {\pi}{2}\right)</math> such that2 KB (322 words) - 13:31, 23 August 2023
- == Problem== If <math>2^{1998}-2^{1997}-2^{1996}+2^{1995} = k \cdot 2^{1995},</math> what is the value of <math>k</math>?878 bytes (110 words) - 14:28, 5 July 2013
- ==Problem 1== <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text13 KB (1,880 words) - 13:35, 19 February 2020
- ...contains the full set of test problems. The rest contain each individual problem and its solution. ** [[1997 AJHSME Problems/Problem 1]]2 KB (141 words) - 23:42, 8 October 2014
- ==Problem 1== [[1997 AJHSME Problems/Problem 1|Solution]]12 KB (1,702 words) - 12:35, 6 November 2022
- ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[1998 AJHSME Problems|Entire Test]]2 KB (141 words) - 23:43, 8 October 2014
- |year = 1998 ==Problem 1==14 KB (1,920 words) - 19:31, 31 January 2024
- Problems of the [[1998 USAMO | 1998]] [[USAMO]]. ===Problem 1===3 KB (486 words) - 06:11, 24 November 2020
- == Problem == Suppose that the set <math>\{1,2,\cdots, 1998\}</math> has been partitioned into disjoint pairs <math>\{a_i,b_i\}</math>1 KB (184 words) - 10:56, 30 January 2021
- ...contains the full set of test problems. The rest contain each individual problem and its solution. ** [[1999 AMC 8 Problems/Problem 1|Problem 1]]2 KB (163 words) - 00:24, 31 March 2015
- ==Problem 1== <math>(6?3) + 4 - (2 - 1) = 5</math> To make this statement true, the question mark between the17 KB (2,394 words) - 19:51, 8 May 2023
- ==Problem== draw((2,0)--(2,-.5));5 KB (686 words) - 15:35, 29 May 2021
- ==Problem== ...e is a set <math>S</math> of <math>n</math> integers such that <math>(a-b)^2</math> divides <math>ab</math> for every distinct <math>a,b\in S</math>.1 KB (244 words) - 09:44, 20 July 2016
- == Problem 3 == <center><math>\begin{tabular}{rr}&\ \texttt{7 a 2}\\ -& \texttt{4 8 b} \\1 KB (230 words) - 19:46, 10 March 2015
- == Problem == If <math>1998</math> is written as a product of two positive integers whose difference is1 KB (157 words) - 14:28, 5 July 2013
- == Problem == <math>\frac{70}{2} = 35</math> people heard <math>20</math> minutes of the talk, for a total1 KB (189 words) - 14:29, 5 July 2013
- == Problem 1 == ...<math>w = 7, x = 5, y = 4, z = 8</math>. In rectangle E, <math>w = 9, x = 2, y = 7, z = 0</math>. These five rectangles are placed, without rotating or2 KB (346 words) - 14:28, 5 July 2013
- == Problem== ...the larger square. Thus the area of the larger square is <math>(x+y)^2=7^2=\boxed{49\text{ (A)}}</math>.1 KB (208 words) - 14:29, 5 July 2013
- == Problem == ...positive, and one negative. If this parabola is the graph of <math>y = ax^2 + bx + c,</math> which of <math>a,b,</math> and <math>c</math> must be posi1 KB (219 words) - 14:29, 5 July 2013
- == Problem== <math> \mathrm{(A) \ }\sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }\sqrt{6} \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \764 bytes (123 words) - 14:29, 5 July 2013
- == Problem == [[Image:Problem 16.png|center|###px]]2 KB (307 words) - 18:58, 11 January 2014
- == Problem 2 == ...and <math>D</math> represent four different digits selected from <math>0,1,2,\ldots ,9.</math> If <math>(A+B)/(C+D)</math> is an integer that is as larg1,008 bytes (167 words) - 14:28, 5 July 2013
- == Problem== <math> \mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \907 bytes (132 words) - 00:54, 30 January 2019
- ==Problem== {{AJHSME box|year=1998|before=First question|num-a=2}}764 bytes (106 words) - 17:27, 25 November 2022
- ==Problem== draw((1,1)--(2,0)--(4,2));1 KB (155 words) - 00:16, 31 March 2015
- == Problem == ...{c}</math>, what is the value of <math>\begin{tabular}{r|l}3&4 \\ \hline 1&2\end{tabular}</math>?797 bytes (126 words) - 15:40, 20 October 2016
- ==Problem== <math>2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + \cd1 KB (157 words) - 11:48, 20 December 2018
- ==Problem== <math>\text{(A)}\ 1 \qquad \text{(B)} \dfrac{25}{16} \qquad \text{(C)}\ 2 \qquad \text{(D)}\ \dfrac{43}{20} \qquad \text{(E)}\ \dfrac{47}{16}</math>590 bytes (62 words) - 00:09, 31 March 2015
- ==Problem== ==Solution 2==1 KB (153 words) - 22:10, 13 January 2023
- ==Problem== draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);2 KB (294 words) - 17:52, 26 October 2020
- ==Problem== ...ext{(C)}\ (199.8)^2 \qquad \text{(D)}\ (1998)^2 \qquad \text{(E)}\ (19980)^2</math>508 bytes (51 words) - 00:28, 5 July 2013