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- 22:30, 15 December 2023 (diff | hist) . . (+448) . . N 2023 SSMO Tiebreaker Round Problems/Problem 3 (Created page with "==Problem== For <math>n\geq4,</math> let <math>a_n</math> be the maximum possible value of <math>P(n+1)</math> given that <math>P(x)</math> is a <math>n</math> degree monic po...") (current)
- 22:30, 15 December 2023 (diff | hist) . . (+338) . . N 2023 SSMO Tiebreaker Round Problems/Problem 2 (Created page with "==Problem== Let <math>P(x) = x^3 + 3ax^2 + 3bx + (a+b)</math> be a real polynomial with nonnegative and nonzero real roots <math>p, q, r</math>. Suppose that <cmath>(p + 1)^3...")
- 22:30, 15 December 2023 (diff | hist) . . (+276) . . N 2023 SSMO Tiebreaker Round Problems/Problem 1 (Created page with "==Problem== From the phrase "Summer Solstice", how many ways are there to make a 4 letter "word" such that the second and third letters aren't spaces and the first letter is c...") (current)
- 22:30, 15 December 2023 (diff | hist) . . (+320) . . N 2023 SSMO Relay Round 5 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR. Suppose that <math>x^2+y^2 = N.</math> Find the remainder when the expected value of the square of the maximum value <math>ax+by</math>...") (current)
- 22:30, 15 December 2023 (diff | hist) . . (+194) . . N 2023 SSMO Relay Round 5 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR. Let <math>a_n = \text{lcm} \{1, 2, \dots n\}</math> for positive integers <math>n</math>.Compute <cmath>\sum_{i=1}^N a_i \pmod {720}.</c...") (current)
- 22:30, 15 December 2023 (diff | hist) . . (+386) . . N 2023 SSMO Relay Round 5 Problems/Problem 1 (Created page with "==Problem== Let <math>S_n</math> be the set of all rational numbers of the form <math>0.\overline{a_1a_2a_3\dots a_n},</math> where <math>n</math> is an integer satisfying <ma...") (current)
- 22:29, 15 December 2023 (diff | hist) . . (+489) . . N 2023 SSMO Relay Round 4 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR. <math>N+1</math> numbers are chosen from the set <math>\{1,2,3,\dots,N+1\}</math> with replacement. If the probability that the median o...") (current)
- 22:29, 15 December 2023 (diff | hist) . . (+486) . . N 2023 SSMO Relay Round 4 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR. Let <math>n = \left\lfloor\sqrt{N}\right\rfloor.</math> Suppose that <math>N</math> points are chosen on the sides of a triangle with ar...") (current)
- 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...") (current)
- 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...") (current)
- 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>...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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.</...") (current)
- 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==") (current)
- 22:25, 15 December 2023 (diff | hist) . . (-270) . . 2023 SSMO Team Round Problems/Problem 13 (current)
- 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...") (current)
- 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...") (current)
- 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>...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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{...") (current)
- 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,...") (current)
- 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==") (current)
- 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...") (current)
- 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...") (current)
- 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>...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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) . . (+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...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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...") (current)
- 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) . . (+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) . . (+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) . . (+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) . . (+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) . . (+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)
- 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) . . (+384) . . N 2021 WSMO Team Round Problems/Problem 9 (Created page with "==Problem== In triangle <math>ABC,</math> points <math>D</math> and <math>E</math> trisect side <math>BC</math> such that <math>D</math> is closer to <math>C</math> than <math...") (current)
- 22:16, 15 December 2023 (diff | hist) . . (+454) . . N 2021 WSMO Team Round Problems/Problem 8 (Created page with "==Problem== Isaac, Gottfried, Carl, Euclid, Albert, Srinivasa, René, Adihaya, and Euler sit around a round table (not necessarily in that order). Then, Hypatia takes a seat....") (current)
- 22:16, 15 December 2023 (diff | hist) . . (+749) . . N 2021 WSMO Team Round Problems/Problem 7 (Created page with "==Problem== A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's <math>x</math> and <math>y</math> coordinates are posit...") (current)
- 22:16, 15 December 2023 (diff | hist) . . (+812) . . N 2021 WSMO Team Round Problems/Problem 6 (Created page with "==Problem== Suppose that regular dodecagon <math>ABCDEFGHIJKL</math> has side length <math>5.</math> The area of the shaded region can be expressed as <math>a+b\sqrt{c},</math...") (current)
- 22:16, 15 December 2023 (diff | hist) . . (+25) . . 2021 WSMO Team Round Problems/Problem 5 (current)
- 22:16, 15 December 2023 (diff | hist) . . (+27) . . 2021 WSMO Team Round Problems/Problem 4 (current)
- 22:16, 15 December 2023 (diff | hist) . . (+25) . . 2021 WSMO Team Round Problems/Problem 3 (current)
- 22:12, 15 December 2023 (diff | hist) . . (+25) . . 2021 WSMO Team Round Problems/Problem 2 (current)
- 22:12, 15 December 2023 (diff | hist) . . (+26) . . 2021 WSMO Team Round Problems/Problem 1 (current)
- 22:11, 15 December 2023 (diff | hist) . . (+371) . . N 2021 WSMO Team Round Problems/Problem 5 (Created page with "==Problem== Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at <math>t+4</math> miles per hour, wh...")
- 22:08, 15 December 2023 (diff | hist) . . (+507) . . N 2021 WSMO Team Round Problems/Problem 4 (Created page with "==Problem== Consider a triangle <math>A_1B_1C_1</math> satisfying <math>A_1B_1=3,B_1C_1=3\sqrt{3},A_1C_1=6</math>. For all successive triangles <math>A_nB_nC_n</math>, we have...")
- 22:07, 15 December 2023 (diff | hist) . . (+418) . . N 2021 WSMO Team Round Problems/Problem 3 (Created page with "==Problem== Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 poun...")
- 22:07, 15 December 2023 (diff | hist) . . (+253) . . N 2021 WSMO Team Round Problems/Problem 2 (Created page with "==Problem== Bobby has some pencils. When he tries to split them into 5 equal groups, he has 2 left over. When he tries to split them into groups of 8, he has 6 left over. What...")
- 22:06, 15 December 2023 (diff | hist) . . (-1) . . 2021 WSMO Team Round Problems/Problem 1
- 22:06, 15 December 2023 (diff | hist) . . (+324) . . N 2021 WSMO Team Round Problems/Problem 1 (Created page with "==Problem== How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase lett...")
- 21:48, 15 December 2023 (diff | hist) . . (+1,211) . . N 2023 SSMO Tiebreaker Round Problems (Created page with "==Problem 1== From the phrase "Summer Solstice", how many ways are there to make a 4 letter "word" such that the second and third letters aren't spaces and the first letter i...") (current)
- 21:36, 15 December 2023 (diff | hist) . . (+1,040) . . N 2023 SSMO Relay Round 5 Problems (Created page with "==Problem 1== Let <math>S_n</math> be the set of all rational numbers of the form <math>0.\overline{a_1a_2a_3\dots a_n},</math> where <math>n</math> is an integer satisfying...") (current)
- 21:35, 15 December 2023 (diff | hist) . . (0) . . 2023 SSMO Relay Round 3 Problems (current)
- 21:35, 15 December 2023 (diff | hist) . . (+1,638) . . N 2023 SSMO Relay Round 4 Problems (Created page with "==Problem 1== 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>...") (current)
- 21:34, 15 December 2023 (diff | hist) . . (+3,500) . . N 2023 SSMO Relay Round 3 Problems (Created page with "==Problem 1== 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...")
- 21:33, 15 December 2023 (diff | hist) . . (+1,107) . . N 2023 SSMO Relay Round 2 Problems (Created page with "==Problem 1== 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...") (current)
- 21:32, 15 December 2023 (diff | hist) . . (+1) . . 2023 SSMO Relay Round 1 Problems (current)
- 21:32, 15 December 2023 (diff | hist) . . (+5) . . 2023 SSMO Relay Round 1 Problems
- 21:32, 15 December 2023 (diff | hist) . . (+634) . . N 2023 SSMO Relay Round 1 Problems (Created page with "==Problem 1== Compute the remainder when <math>2022^{2021^{2020^{\dots}}}</math> is divided by <math>2023</math>. Solution ==P...")
- 21:30, 15 December 2023 (diff | hist) . . (-37) . . 2023 SSMO Team Round Problems
- 21:26, 15 December 2023 (diff | hist) . . (+7,970) . . N 2023 SSMO Team Round Problems (Created page with "==Problem 1== 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> 2022 SSMO...")
- 21:21, 15 December 2023 (diff | hist) . . (+290) . . 2023 SSMO Team Round (current)
- 21:20, 15 December 2023 (diff | hist) . . (+1) . . 2023 SSMO Accuracy Round Problems
- 21:20, 15 December 2023 (diff | hist) . . (+5) . . 2023 SSMO Accuracy Round Problems
- 21:19, 15 December 2023 (diff | hist) . . (-133) . . 2023 SSMO Accuracy Round Problems
- 21:16, 15 December 2023 (diff | hist) . . (+5,770) . . N 2023 SSMO Accuracy Round Problems (Created page with "==Problem 1== 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...")
- 19:42, 15 December 2023 (diff | hist) . . (+348) . . N 2022 SSMO Tiebreaker Round Problems/Problem 2 (Created page with "==Problem== Let <math>P(x) = x^3 - 7x^2 + 9x - 2</math>. If <math>P(x) = (x - a)^3 + b(x - a) + c</math> where <math>a, b, c</math> are real numbers, then the value of <math>a...") (current)
- 19:42, 15 December 2023 (diff | hist) . . (+250) . . N 2022 SSMO Tiebreaker Round Problems/Problem 1 (Created page with "==Problem== For all positive integers <math>n,</math> let <math>S(n)</math> denote the least positive integer <math>x</math> such that <math>n+x</math> is a palindrome. Find t...") (current)
- 13:13, 14 December 2023 (diff | hist) . . (+377) . . N 2022 SSMO Relay Round 5 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR, and let <math>a_k=\text{cis}\left(\frac{k\pi}{T+1}\right)</math>. Suppose that<cmath>\sum_{k=1}^{T+1} \frac{|a_{2k}+a_{2k+2}-a_{2k-1+T}|...") (current)
- 13:13, 14 December 2023 (diff | hist) . . (+389) . . N 2022 SSMO Relay Round 5 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR, and let <math>S=\{a_1,a_2,\dots,a_{2022}\}</math> be a sequence of 2022 positive integers such that <math>a_1\le a_2\le \cdots \le a_{20...") (current)
- 13:12, 14 December 2023 (diff | hist) . . (+350) . . N 2022 SSMO Relay Round 5 Problems/Problem 1 (Created page with "==Problem== Consider an <math>8\times 8</math> chessboard with a knight in one of the center squares. The knight may move in an <math>L</math>-shaped fashion, going two square...") (current)
- 13:11, 14 December 2023 (diff | hist) . . (+255) . . N 2022 SSMO Relay Round 4 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR. If <math>f(1)=1</math>, <math>f(2)=12</math>, and<cmath>f(n+2)=12f(n+1)-20f(n)</cmath>for all positive integers <math>n</math>, find the...") (current)
- 13:11, 14 December 2023 (diff | hist) . . (+286) . . N 2022 SSMO Relay Round 4 Problems/Problem 2 (Created page with "==Problem== The roots of <math>f(x)=x^3+5x+8</math> are <math>r_1,r_2,r_3.</math> Let <math>g_n(x)</math> be a polynomial with roots <math>r_1+n, r_2+n,r_3+n.</math> If<cmath>...") (current)
- 13:11, 14 December 2023 (diff | hist) . . (+830) . . N 2022 SSMO Relay Round 4 Problems/Problem 1 (Created page with "==Problem== On any given day, there is a <math>70\%</math> chance that a robot will find a new organism, a <math>20\%</math> chance it will find an already discovered organism...") (current)
- 13:11, 14 December 2023 (diff | hist) . . (+273) . . N 2022 SSMO Relay Round 3 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR. Let <math>f(x)</math> be a polynomial of degree 10, such that <math>f(i)=i</math> for all <math>i=1,2,\dots,10</math> and <math>f(11) =T...") (current)
- 13:11, 14 December 2023 (diff | hist) . . (+292) . . N 2022 SSMO Relay Round 3 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR. In cyclic quadrilateral <math>ABCD,</math> <math>\angle{BAD}=60^{\circ},</math> and <math>BC=CD=T.</math> If <math>AB</math> is a positi...") (current)
- 13:10, 14 December 2023 (diff | hist) . . (+413) . . N 2022 SSMO Relay Round 3 Problems/Problem 1 (Created page with "==Problem== Let <math>f:\mathbb Z\rightarrow\mathbb Z</math> be a function such that <math>f(0)=0</math> and <math>f\left(|x^2-4|\right)=0</math> if <math>f(x)=0</math>. Moreo...") (current)
- 13:10, 14 December 2023 (diff | hist) . . (+470) . . N 2022 SSMO Relay Round 2 Problems/Problem 3 (Created page with "==Problem== Let <math>T =</math> TNYWR. Let <math>a+b = \lfloor \sqrt{T} \rfloor</math>. If <math>a^5 + b^5 = 15</math>, then <math>ab</math> has two possible values. The abso...") (current)
- 13:10, 14 December 2023 (diff | hist) . . (+382) . . N 2022 SSMO Relay Round 2 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR. Suppose that the monic quadratic <math>f(x)</math> is tangent to the function <math>g(x)=|x+2|-T</math> at two points, when graphed on t...") (current)
- 13:10, 14 December 2023 (diff | hist) . . (+648) . . N 2022 SSMO Relay Round 2 Problems/Problem 1 (Created page with "==Problem== Let <math>P</math> be a randomly selected point on a circle, and let <math>A</math> be a randomly selected point inside the same circle. A dilation centered at <ma...") (current)
- 13:09, 14 December 2023 (diff | hist) . . (+291) . . N 2022 SSMO Relay Round 1 Problems/Problem 3 (Created page with "==Problem== Let <math>T=</math> TNYWR. Now, let <math>ABC</math> a triangle such that <math>AB=T,</math> <math>AC=100</math>, and <math>\angle{ABC}=36^{\circ}.</math> Find the...") (current)
- 13:09, 14 December 2023 (diff | hist) . . (+564) . . N 2022 SSMO Relay Round 1 Problems/Problem 2 (Created page with "==Problem== Let <math>T=</math> TNYWR. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math>...") (current)
- 13:09, 14 December 2023 (diff | hist) . . (+227) . . N 2022 SSMO Relay Round 1 Problems/Problem 1 (Created page with "==Problem== Suppose <math>a, b, c</math> are distinct digits where <math>a \not= 0</math> such that <math>\left(\overline{abc}\right)^2 = \overline{bad00}</math> where <math>d...") (current)
- 13:09, 14 December 2023 (diff | hist) . . (-68) . . 2022 SSMO Team Round Problems/Problem 5 (current)
- 13:08, 14 December 2023 (diff | hist) . . (+18) . . 2022 SSMO Team Round Problems/Problem 3 (current)
- 13:08, 14 December 2023 (diff | hist) . . (+8) . . 2022 SSMO Team Round Problems/Problem 14 (current)
- 13:07, 14 December 2023 (diff | hist) . . (+859) . . N 2022 SSMO Team Round Problems/Problem 15 (Created page with "==Problem== Consider two externally tangent circles <math>\omega_1</math> and <math>\omega_2</math> with centers <math>O_1</math> and <math>O_2</math>. Suppose that <math>\ome...") (current)
- 13:07, 14 December 2023 (diff | hist) . . (+782) . . N 2022 SSMO Team Round Problems/Problem 14 (Created page with "==Problem== On a hot summer day, three little piggies decide to play with water balloons. The three piggies travel to a 200-floor parking garage each armed with exactly one wa...")
- 13:07, 14 December 2023 (diff | hist) . . (+607) . . N 2022 SSMO Team Round Problems/Problem 13 (Created page with "==Problem== In regular hexagon <math>ABCDEF</math> with side length 1, an electron starts at point <math>A</math>. When the electron hits an edge, it reflects off of it, with...") (current)
- 13:07, 14 December 2023 (diff | hist) . . (+743) . . N 2022 SSMO Team Round Problems/Problem 12 (Created page with "==Problem== Regular pentagon <math>ABCDE</math> is inscribed in circle <math>\omega_1</math> with radius <math>5\sqrt{5}</math>. Circle <math>\omega_2</math> is the reflection...") (current)
- 13:07, 14 December 2023 (diff | hist) . . (+186) . . N 2022 SSMO Team Round Problems/Problem 11 (Created page with "==Problem== Find the number of solutions to <math>a^3+36ab+64=27b^3,</math> where <math>a</math> is an integer, <math>b</math> is a real number, and <math>|b|\le 50</math>. =...") (current)
- 13:06, 14 December 2023 (diff | hist) . . (+151) . . N 2022 SSMO Team Round Problems/Problem 10 (Created page with "==Problem== If <math>p, q, r</math> are the roots of the polynomial <math>x^3-2x^2-4</math>, find<cmath>(p^3+qr)(q^3+pr)(r^3+pq).</cmath> ==Solution==")
- 13:06, 14 December 2023 (diff | hist) . . (+213) . . N 2022 SSMO Team Round Problems/Problem 9 (Created page with "==Problem== Given real numbers <math>a,b,x,y</math> such that <align*> a^2+b^2&=1,\\ x^2+y^2&=1,\\ abxy-\frac{1}{8}&=b^2y^2, </align*> find the sum of all distinct values of <...") (current)
- 13:06, 14 December 2023 (diff | hist) . . (+636) . . N 2022 SSMO Team Round Problems/Problem 8 (Created page with "==Problem== A frog is at <math>0</math> on a number line and wants to go to <math>9</math>. On each turn, if the frog is at <math>n</math>, the frog hops to one of the numbers...") (current)
- 13:06, 14 December 2023 (diff | hist) . . (+373) . . N 2022 SSMO Team Round Problems/Problem 7 (Created page with "==Problem== Let <math>\cos(2A)</math>, <math>\cos(2B)</math>, and <math>\cos(2C)</math> be the not necessarily distinct roots of a monic cubic <math>f</math>. Given that <math...") (current)
- 13:06, 14 December 2023 (diff | hist) . . (+719) . . N 2022 SSMO Team Round Problems/Problem 6 (Created page with "==Problem== Let <math>n</math> be a positive integer, and let <math>x</math> be some variable. Define <math>P_{x,n}</math> as the maximum fraction of elements in the set of th...") (current)
- 13:06, 14 December 2023 (diff | hist) . . (+867) . . N 2022 SSMO Team Round Problems/Problem 5 (Created page with "==Problem== Consider the following rectangle <math>ABCD</math> where <math>BC=8.</math> If<cmath>CD=CT_2, 4T_2P_2=2DP_1=AC, [ADP_1C]=33, \text{ and }[ABP_2C]=34,</cmath>find t...")
- 13:05, 14 December 2023 (diff | hist) . . (+222) . . N 2022 SSMO Team Round Problems/Problem 4 (Created page with "==Problem== Let <math>a_1=2,b_0=3, a_n=\left(a_{n-1}\right)^2,</math> and <math>b_n=\left(b_{n-1}\right)^3.</math> If <math>c_n=a_n+b_n,</math> find the last two digits of <ma...") (current)
- 13:05, 14 December 2023 (diff | hist) . . (+1,011) . . N 2022 SSMO Team Round Problems/Problem 3 (Created page with "==Problem== Let <math>ABCD</math> be an isosceles trapezoid such that <math>AB\parallel CD.</math> Let <math>E</math> be a point on <math>CD</math> such that <math>AB=CE.</mat...")
- 13:05, 14 December 2023 (diff | hist) . . (+897) . . N 2022 SSMO Team Round Problems/Problem 2 (Created page with "==Problem== Consider <math>8</math> marbles in a line, where the color of each marble is either black or white and is randomly chosen. Define the period of a lineup of 8 marbl...") (current)
- 13:05, 14 December 2023 (diff | hist) . . (+506) . . N 2022 SSMO Team Round Problems/Problem 1 (Created page with "==Problem== In triangle <math>ABC</math>, circumcircle <math>\omega</math> is drawn. Let <math>I</math> be the incenter of <math>\triangle{ABC}</math>. Let <math>H_A</math> be...") (current)
- 13:04, 14 December 2023 (diff | hist) . . (0) . . 2022 SSMO Accuracy Round Problems (current)
- 13:04, 14 December 2023 (diff | hist) . . (+328) . . N 2022 SSMO Accuracy Round Problems/Problem 10 (Created page with "==Problem== Let<cmath>S=\sum^{\infty}_{k=5}2^{-10k}\dbinom{2k}{10}.</cmath>Then the value of <math>S</math> can be expressed as <math>\frac mn</math>, where <math>m</math> and...") (current)
- 13:04, 14 December 2023 (diff | hist) . . (+475) . . N 2022 SSMO Accuracy Round Problems/Problem 9 (Created page with "==Problem== The graph <math>\Gamma</math> has <math>52</math> vertices labeled <math>A, A', B, B', \dots, Z, Z'</math>, such that <math>A</math> is not connected to <math>A'</...") (current)
- 13:03, 14 December 2023 (diff | hist) . . (+501) . . N 2022 SSMO Accuracy Round Problems/Problem 8 (Created page with "==Problem== Let <math>ABCD</math> be a trapezoid with <math>AB \parallel CD</math>. Suppose that <math>AD=1</math>, <math>DC=4</math>, <math>CB=2</math>, and <math>AB<CD</math...") (current)
- 13:03, 14 December 2023 (diff | hist) . . (+522) . . N 2022 SSMO Accuracy Round Problems/Problem 7 (Created page with "==Problem== After a robber drives in a car for <math>t</math> (not necessarily integral) minutes, the car goes at <math>120-t</math> miles per hour. Whenever the car's speed d...") (current)
- 13:03, 14 December 2023 (diff | hist) . . (+727) . . N 2022 SSMO Accuracy Round Problems/Problem 6 (Created page with "==Problem== Consider an unfair <math>6</math>-sided die labeled from <math>1</math> to <math>6</math>, such that the probability of rolling a number <math>m</math> is directly...") (current)
- 13:03, 14 December 2023 (diff | hist) . . (+271) . . N 2022 SSMO Accuracy Round Problems/Problem 5 (Created page with "==Problem== Find the number of ordered pairs <math>(a, b)</math>, where <math>1\leq a, b\leq 9</math>, for which the largest integer <math>n</math> that satisfies<cmath>(a-b)^...") (current)
- 13:03, 14 December 2023 (diff | hist) . . (+362) . . N 2022 SSMO Accuracy Round Problems/Problem 4 (Created page with "==Problem== A monic polynomial <math>f</math> has real roots <math>r,s,t.</math> A monic polynomial <math>g</math> has roots <math>r^3,s^3,t^3.</math> Given that the minimum p...") (current)
- 13:02, 14 December 2023 (diff | hist) . . (+501) . . N 2022 SSMO Accuracy Round Problems/Problem 3 (Created page with "==Problem== Let <math>A=(6,3,2), B=(2,-9,-6),</math> and <math>O=(0,0,0).</math> Suppose that <math>D</math> is a point in space such that <math>OD</math> bisects <math>\angle...") (current)
- 13:02, 14 December 2023 (diff | hist) . . (+486) . . N 2022 SSMO Accuracy Round Problems/Problem 2 (Created page with "==Problem== Consider a cone with radius <math>5</math> and height <math>12</math>, and a point <math>P</math> in the same plane as the base of the cone, but a distance of <mat...") (current)
- 13:02, 14 December 2023 (diff | hist) . . (+284) . . N 2022 SSMO Accuracy Round Problems/Problem 1 (Created page with "==Problem== Consider a bijective function (meaning each element in the domain maps to a distinct element in the range) <math>f:S\rightarrow S,</math> where <math>S=\{1, 2, 3,...") (current)
- 13:01, 14 December 2023 (diff | hist) . . (+335) . . N 2021 WSMO Team Round/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)
- 13:01, 14 December 2023 (diff | hist) . . (+323) . . N 2021 WSMO Team Round/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)
- 13:01, 14 December 2023 (diff | hist) . . (+418) . . N 2021 WSMO Team Round/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)
- 13:01, 14 December 2023 (diff | hist) . . (+357) . . N 2021 WSMO Team Round/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)
- 13:01, 14 December 2023 (diff | hist) . . (+2) . . 2021 WSMO Team Round/Problem 11 (current)
- 13:00, 14 December 2023 (diff | hist) . . (+161) . . N 2021 WSMO Team Round/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>) ==Solution")
- 13:00, 14 December 2023 (diff | hist) . . (+359) . . N 2021 WSMO Team Round/Problem 9 (Created page with "==Problem== In triangle <math>ABC,</math> points <math>D</math> and <math>E</math> trisect side <math>BC</math> such that <math>D</math> is closer to <math>C</math> than <math...") (current)
- 13:00, 14 December 2023 (diff | hist) . . (+724) . . N 2021 WSMO Team Round/Problem 7 (Created page with "==Problem== A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's <math>x</math> and <math>y</math> coordinates are posit...") (current)
- 13:00, 14 December 2023 (diff | hist) . . (+371) . . N 2021 WSMO Team Round/Problem 5 (Created page with "==Problem== Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at <math>t+4</math> miles per hour, wh...") (current)
- 12:59, 14 December 2023 (diff | hist) . . (+418) . . N 2021 WSMO Team Round/Problem 3 (Created page with "==Problem== Farmer Sam has <math>n</math> dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 poun...") (current)
- 12:58, 14 December 2023 (diff | hist) . . (+468) . . N 2021 WSMO Accuracy Round Problems/Problem 8 (Created page with "==Problem== 20 unit spheres are stacked in a triangular pyramid formation, such that the first layer has 1 sphere, the second layer has 3 spheres, the third layer has 6 sphere...") (current)
- 12:56, 14 December 2023 (diff | hist) . . (+40) . . 2023 SSMO Team Round
- 12:56, 14 December 2023 (diff | hist) . . (+40) . . 2023 SSMO Accuracy Round (current)
- 12:56, 14 December 2023 (diff | hist) . . (+40) . . 2023 SSMO Speed Round (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Relay Round 4 (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Relay Round 3 (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Relay Round 5 (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Relay Round 2 (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Relay Round 1 (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Team Round (current)
- 12:55, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Accuracy Round (current)
- 12:54, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Speed Round (current)
- 12:54, 14 December 2023 (diff | hist) . . (+25) . . SMO Problem Archive (current)
- 12:53, 14 December 2023 (diff | hist) . . (+40) . . 2021 WSMO Speed Round (current)
- 12:53, 14 December 2023 (diff | hist) . . (+40) . . 2021 WSMO Team Round (current)
- 12:53, 14 December 2023 (diff | hist) . . (+40) . . 2021 WSMO Accuracy Round (current)
- 12:52, 14 December 2023 (diff | hist) . . (+40) . . 2022 SSMO Tiebreaker Round (current)
- 12:52, 14 December 2023 (diff | hist) . . (+481) . . N 2023 SSMO Tiebreaker Round (Created page with "'''2023 SSMO Tiebreaker Round''' problems and solutions. The first link contains the full set of test problems. The second link contains the answer key. The rest contain each...") (current)
- 12:52, 14 December 2023 (diff | hist) . . (-53) . . SMO Problem Archive
- 12:51, 14 December 2023 (diff | hist) . . (-2) . . SMO Problem Archive
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