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• 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}|...")
• 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...")
• 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...")
• 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>...")
• 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...")
• 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...")
• 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...")
• 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...")
• 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...")
• 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>...")
• 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 <...")
• 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...")
• 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 ‎
• 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)

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