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• 22:29, 15 December 2023 (diff | hist) . . (+523)‎ . . N 2023 SSMO Relay Round 4 Problems/Problem 1 ‎ (Created page with "==Problem== Martha starts out with the number <math>7</math> on her calculator, which has three buttons that multiply the current number by <math>\frac{1}{2}</math>, <math>\fr...") (current)
• 22:29, 15 December 2023 (diff | hist) . . (+2,269)‎ . . N 2023 SSMO Relay Round 3 Problems/Problem 3 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. Let <math>n = N+1.</math> A spray painter has a paint gun that paints all areas within a radius of <math>2.</math> The spray painter wal...")
• 22:29, 15 December 2023 (diff | hist) . . (+499)‎ . . N 2023 SSMO Relay Round 3 Problems/Problem 2 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. In triangle <math>ABC</math> with circumradius and inradius having lengths <math>R</math> and <math>r,</math> respectively. Given that...")
• 22:29, 15 December 2023 (diff | hist) . . (+592)‎ . . N 2023 SSMO Relay Round 3 Problems/Problem 1 ‎ (Created page with "==Problem== In triangle <math>ABC</math> with <math>AB=13,AC=14,BC=15</math>, circles <math>\omega_1,\omega_2,</math> and <math>\omega_3</math> are drawn, centered at <math>A,...") (current)
• 22:29, 15 December 2023 (diff | hist) . . (+397)‎ . . N 2023 SSMO Relay Round 2 Problems/Problem 3 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. In a committee of <math>2023</math> people, <math>N</math> are scientists and the rest are builders. In order to make a building, <math>...")
• 22:29, 15 December 2023 (diff | hist) . . (+307)‎ . . N 2023 SSMO Relay Round 2 Problems/Problem 2 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. Suppose that <math>L = \left\lfloor\sqrt{N}\right\rfloor</math> points are evenly spaced around the circle. Find the number of ways to s...")
• 22:28, 15 December 2023 (diff | hist) . . (+263)‎ . . N 2023 SSMO Relay Round 2 Problems/Problem 1 ‎ (Created page with "==Problem== Consider the cubic polynomial <math>P(x)=ax^3+bx^2+cx+d</math>, where <math>a,b,c,d</math> are single-digit integers, which has roots of approximately <cmath>x \ap...")
• 22:28, 15 December 2023 (diff | hist) . . (+180)‎ . . N 2023 SSMO Relay Round 1 Problems/Problem 3 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. Find the number of solutions to the equation <cmath>\sec^{N} (Nx) - \tan^{N}(Nx) = 1</cmath> such <math>0 \le x \le \pi</math> ==Solut...")
• 22:28, 15 December 2023 (diff | hist) . . (+195)‎ . . N 2023 SSMO Relay Round 1 Problems/Problem 2 ‎ (Created page with "==Problem== Let <math>T=</math> TNYWR. Let <math>a_0 = 3, a_1 = 1, a_2 = N</math>, and let <math>a_n = a_{n-1} - \frac{a_{n-3}}{8}</math>. Find <cmath>\sum_{i=0}^\infty a_i.</...")
• 22:28, 15 December 2023 (diff | hist) . . (+125)‎ . . N 2023 SSMO Relay Round 1 Problems/Problem 1 ‎ (Created page with "==Problem== Compute the remainder when <math>2022^{2021^{2020^{\dots}}}</math> is divided by <math>2023</math>. ==Solution==")
• 22:25, 15 December 2023 (diff | hist) . . (-270)‎ . . 2023 SSMO Team Round Problems/Problem 13 ‎
• 22:25, 15 December 2023 (diff | hist) . . (+224)‎ . . N 2023 SSMO Team Round Problems/Problem 14 ‎ (Created page with "==Problem== Find the sum of all perfect squares of the form <math>2p^3 - 5p^2q + q^2</math> where <math>p</math> and <math>q</math> are positive integers such <math>p</math> i...")
• 22:25, 15 December 2023 (diff | hist) . . (+799)‎ . . N 2023 SSMO Team Round Problems/Problem 13 ‎ (Created page with "==Problem== Let <math>D(n)</math> denote the product of all divisors of <math>n</math> Let <math>P(i,j)</math> denote the set of all integers that are both a multiple of <math...")
• 22:25, 15 December 2023 (diff | hist) . . (+653)‎ . . N 2023 SSMO Team Round Problems/Problem 12 ‎ (Created page with "==Problem== Let <math>T</math> be the set of rationals of the form <math>\frac{a}{2^b}</math> for nonnegative <math>a</math> and <math>b</math>. Define the function <math>f \c...")
• 22:25, 15 December 2023 (diff | hist) . . (+390)‎ . . N 2023 SSMO Team Round Problems/Problem 11 ‎ (Created page with "==Problem== Let <math>ABCD</math> be a cyclic quadrilateral such that <math>AC</math> is the diameter. Let <math>P</math> be the orthocenter of <math>ABD</math>. Define <math>...")
• 22:25, 15 December 2023 (diff | hist) . . (+464)‎ . . N 2023 SSMO Team Round Problems/Problem 10 ‎ (Created page with "==Problem== There exists a lane of infinite cars. Each car has a <math>\frac{1}{3}</math> chance of being high quality and a <math>\frac{2}{3}</math> chance of being low quali...")
• 22:25, 15 December 2023 (diff | hist) . . (+468)‎ . . N 2023 SSMO Team Round Problems/Problem 8 ‎ (Created page with "==Problem== Three rabbits run away from the origin at the same speed and constant velocity such that the angle between any two rabbits' directions is <math>120^\circ</math>. A...")
• 22:25, 15 December 2023 (diff | hist) . . (+369)‎ . . N 2023 SSMO Team Round Problems/Problem 9 ‎ (Created page with "==Problem== Let <math>B</math>, <math>K</math>, and <math>R</math> be the total number of possible moves for a bishop, knight, or rook from any position of a <math>9</math> by...")
• 22:25, 15 December 2023 (diff | hist) . . (+379)‎ . . N 2023 SSMO Team Round Problems/Problem 7 ‎ (Created page with "==Problem== Let <math>S = \{1, 2, 3, 4, \cdots, 23\}</math> and let there be randomly chosen sets <math>A, B, C</math> where <math>A, B, C \subseteq S</math>. The probability...")
• 22:25, 15 December 2023 (diff | hist) . . (+534)‎ . . N 2023 SSMO Team Round Problems/Problem 5 ‎ (Created page with "==Problem== Joshy is playing a game with a dartboard that has two sections. If Joshy hits the first section, he gets <math>20</math> points, and if he hits the second section,...")
• 22:25, 15 December 2023 (diff | hist) . . (+353)‎ . . N 2023 SSMO Team Round Problems/Problem 6 ‎ (Created page with "==Problem== Suppose that <math>a,b,c</math> are positive reals satisfying<cmath>(a^3+4)(b^3+6)(c^3+8) = 8(a+b+c)^3.</cmath> Find the sum of all possible values of <math>\frac{...")
• 22:25, 15 December 2023 (diff | hist) . . (+132)‎ . . N 2023 SSMO Team Round Problems/Problem 4 ‎ (Created page with "==Problem== Find the sum of values for prime <math>p</math> such that <math>p \mid (2023^{p^2}+(p-1)!+2^{p^4}).</math> ==Solution==")
• 22:25, 15 December 2023 (diff | hist) . . (+588)‎ . . N 2023 SSMO Team Round Problems/Problem 3 ‎ (Created page with "==Problem== Let <math>ABC</math> be a triangle such that <math>AB=4\sqrt{2}, BC=5\sqrt{2},</math> and <math>AC=\sqrt{82}.</math> Let <math>\omega</math> be the circumcircle of...")
• 22:25, 15 December 2023 (diff | hist) . . (+352)‎ . . N 2023 SSMO Team Round Problems/Problem 2 ‎ (Created page with "==Problem== A plane and a car start both move northward. The car moves northbound at 60 miles per hour. The plane moves northeast and increases in altitude at an angle of <mat...")
• 22:25, 15 December 2023 (diff | hist) . . (+172)‎ . . N 2023 SSMO Team Round Problems/Problem 1 ‎ (Created page with "==Problem== Let <math>(a, b, c, d)</math> be a permutation of <math>(2, 0, 2, 3)</math>. Find the largest possible value of <math>a^b + b^c + c^d + d^a</math> ==Solution==")
• 22:24, 15 December 2023 (diff | hist) . . (+1,656)‎ . . N 2023 SSMO Team Round Problems/Problem 15 ‎ (Created page with "==Problem== Consider a piece of paper in the shape of a regular pentagon with sidelength <math>2.</math> We fold it in half. We then fold it such that the vertices of the long...")
• 22:22, 15 December 2023 (diff | hist) . . (+1,087)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 9 ‎ (Created page with "==Problem== Consider a <math>2023 \times 2023</math> grid called <math>A = P_{2023}</math>. We take one of the four smaller <math>2022 \times 2022</math> grids located in <mat...")
• 22:22, 15 December 2023 (diff | hist) . . (+741)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 10 ‎ (Created page with "==Problem== Let <math>\triangle ABC</math> be a triangle such <math>AB = 13</math>, <math>BC = 14</math>, <math>CA = 15</math>. Let the incircle of <math>\triangle ABC</math>...")
• 22:21, 15 December 2023 (diff | hist) . . (+539)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 7 ‎ (Created page with "==Problem== Concentric circles <math>\omega</math> and <math>\omega_1</math> are drawn, with radii <math>3</math> and <math>5,</math> respectively. Chords <math>AB</math> and...")
• 22:21, 15 December 2023 (diff | hist) . . (+679)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 8 ‎ (Created page with "==Problem== There is a quadrilateral <math>ABCD</math> inscribed in a circle <math>\omega</math> with center <math>O</math>. In quadrilateral <math>ABCD</math>, diagonal <math...")
• 22:21, 15 December 2023 (diff | hist) . . (+387)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 5 ‎ (Created page with "==Problem== Define the <math>\textit{relationship}</math> between two numbers <math>a</math> and <math>b</math> to be <math>\frac{\sigma(ab)}{\sigma(a)\sigma(b)}</math> where...")
• 22:21, 15 December 2023 (diff | hist) . . (+211)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 6 ‎ (Created page with "==Problem== Let the roots of <math>P(x) = x^3 - 2023x^2 + 2023^{2023}</math> be <math>p, q, r</math>. Find <cmath>\frac{p^2 + q^2}{p + q} + \frac{q^2 + r^2}{q + r} + \frac{r^2...")
• 22:21, 15 December 2023 (diff | hist) . . (+264)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 3 ‎ (Created page with "==Problem== Suppose that <math>a, b, c</math> are real numbers such <cmath> \begin{align*} a + b - c &= 4 \\ a^2 + b^2 + c^2 &= 14 \\ a^3 + b^3 - c^3 &= 34 \\ \end...")
• 22:21, 15 December 2023 (diff | hist) . . (+483)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 4 ‎ (Created page with "==Problem== In square <math>ABCD,</math> point <math>E</math> is selected on diagonal <math>AC.</math> Let <math>F</math> be the intersection of the circumcircles of triangles...")
• 22:21, 15 December 2023 (diff | hist) . . (+487)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 1 ‎ (Created page with "==Problem== Mr. Sammy proposes a Hamburger Proclamation, which has <math>500</math> lines, divided into paragraphs of <math>5</math> lines each. It takes him <math>23</math> s...")
• 22:21, 15 December 2023 (diff | hist) . . (+283)‎ . . N 2023 SSMO Accuracy Round Problems/Problem 2 ‎ (Created page with "==Problem== Suppose that the average of all <math>n</math>-digit palindromes is denoted by <math>P_{n}</math> and the average of all <math>n</math>-digit numbers is denoted by...")
• 22:19, 15 December 2023 (diff | hist) . . (+13)‎ . . N 2023 SSMO Tiebreaker Round Answer Key ‎ (Created page with "#1162 #10 #25") (current)
• 22:19, 15 December 2023 (diff | hist) . . (+12)‎ . . N 2023 SSMO Relay Round 5 Answer Key ‎ (Created page with "#25 #321 #22") (current)
• 22:19, 15 December 2023 (diff | hist) . . (+12)‎ . . N 2023 SSMO Relay Round 4 Answer Key ‎ (Created page with "#88 #160 #50") (current)
• 22:19, 15 December 2023 (diff | hist) . . (+12)‎ . . N 2023 SSMO Relay Round 3 Answer Key ‎ (Created page with "#325 #13 #39") (current)
• 22:19, 15 December 2023 (diff | hist) . . (+13)‎ . . N 2023 SSMO Relay Round 2 Answer Key ‎ (Created page with "#119 #1200 #2") (current)
• 22:19, 15 December 2023 (diff | hist) . . (+19)‎ . . N 2023 SSMO Relay Round 1 Answer Key ‎ (Created page with "#2022 #16176 #64704") (current)
• 22:18, 15 December 2023 (diff | hist) . . (+74)‎ . . N 2023 SSMO Team Round Answer Key ‎ (Created page with "#18 #61952 #101 #36 #1040 #0 #64 #59 #2560 #254 #191 #3073 #9021 #416 #831") (current)
• 22:18, 15 December 2023 (diff | hist) . . (+46)‎ . . N 2023 SSMO Accuracy Round Answer Key ‎ (Created page with "#1983 #49 #4 #40 #340 #8092 #506 #98 #16 #2389") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+348)‎ . . N 2021 WSMO Team Round Problems/Problem 14 ‎ (Created page with "==Problem== Suppose that <math>x</math> is a complex number such that <math>x+\frac{1}{x}=\frac{\sqrt{6}+\sqrt{2}}{2}</math> and the imaginary part of <math>x</math> is nonneg...") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+361)‎ . . N 2021 WSMO Team Round Problems/Problem 15 ‎ (Created page with "==Problem== Let <math>ABCD</math> and <math>DEFG</math> (vertices labelled clockwise) be squares that intersect exactly once and with areas <math>1011^2</math> and <math>69^2<...") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+382)‎ . . N 2021 WSMO Team Round Problems/Problem 12 ‎ (Created page with "==Problem== Choose three integers <math>x,y,z</math> randomly and independently from the nonnegative integers. The probability that the sum of the factors of <math>2^x3^y5^z</...") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+443)‎ . . N 2021 WSMO Team Round Problems/Problem 13 ‎ (Created page with "==Problem== Square <math>BCDE</math> is drawn outside of equilateral triangle <math>ABC.</math> Regular hexagon <math>DEFGHI</math> is drawn outside of square <math>BCDE.</mat...") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+196)‎ . . N 2021 WSMO Team Round Problems/Problem 10 ‎ (Created page with "==Problem== The minimum possible value of <cmath>\sqrt{m^2+n^2}+\sqrt{3m^2+3n^2-6m+12n+15}</cmath> can be expressed as <math>a.</math> Find <math>a^2.</math> ''Proposed by pi...") (current)
• 22:16, 15 December 2023 (diff | hist) . . (+190)‎ . . N 2021 WSMO Team Round Problems/Problem 11 ‎ (Created page with "==Problem== Find the remainder when <cmath>\sum_{x+y+z\leq10}\frac{(x+y+z)!}{x!y!z!}</cmath> is divided by <math>100</math>. (<math>x,y,z\geq 0</math>) ''Proposed by pinkpig'...") (current)

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