MathDB

2018 Online Math Open Problems

Part of Online Math Open Problems

Subcontests

(30)
27
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2017-2018 Spring OMO Problem 27

Let n=22018n=2^{2018} and let S={1,2,…,n}S=\{1,2,\ldots,n\}. For subsets S1,S2,…,Sn⊆SS_1,S_2,\ldots,S_n\subseteq S, we call an ordered pair (i,j)(i,j) murine if and only if {i,j}\{i,j\} is a subset of at least one of Si,SjS_i, S_j. Then, a sequence of subsets (S1,…,Sn)(S_1,\ldots, S_n) of SS is called tasty if and only if:
1) For all ii, i∈Sii\in S_i. 2) For all ii, ⋃j∈SiSj=Si\displaystyle\bigcup_{j\in S_i} S_j=S_i. 3) There do not exist pairwise distinct integers a1,a2,…,aka_1,a_2,\ldots,a_k with k≥3k\ge 3 such that for each ii, (ai,ai+1)(a_i, a_{i+1}) is murine, where indices are taken modulo kk. 4) nn divides 1+∣S1∣+∣S2∣+…+∣Sn∣1+|S_1|+|S_2|+\ldots+|S_n|.
Find the largest integer xx such that 2x2^x divides the number of tasty sequences (S1,…,Sn)(S_1,\ldots, S_n).
Proposed by Vincent Huang and Brandon Wang

2018-2019 Fall OMO Problem 27

Let p=216+1p=2^{16}+1 be a prime. Let NN be the number of ordered tuples (A,B,C,D,E,F)(A,B,C,D,E,F) of integers between 00 and p−1p-1, inclusive, such that there exist integers x,y,zx,y,z not all divisible by pp with pp dividing all three of Ax+Ez+Fy,By+Dz+Fx,Cz+Dy+ExAx+Ez+Fy, By+Dz+Fx, Cz+Dy+Ex. Compute the remainder when NN is divided by 10610^6.
Proposed by Vincent Huang
21
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2017-2018 Spring OMO Problem 21

Let ⨁\bigoplus and ⨂\bigotimes be two binary boolean operators, i.e. functions that send {True,False}×{True,False}\{\text{True}, \text{False}\}\times \{\text{True}, \text{False}\} to {True,False}\{\text{True}, \text{False}\}. Find the number of such pairs (⨁,⨂)(\bigoplus, \bigotimes) such that ⨁\bigoplus and ⨂\bigotimes distribute over each other, that is, for any three boolean values a,b,ca, b, c, the following four equations hold:
1) c⨂(a⨁b)=(c⨂a)⨁(c⨂b);c \bigotimes (a \bigoplus b) = (c \bigotimes a) \bigoplus (c \bigotimes b); 2) (a⨁b)⨂c=(a⨂c)⨁(b⨂c);(a \bigoplus b) \bigotimes c = (a \bigotimes c) \bigoplus (b \bigotimes c); 3) c⨁(a⨂b)=(c⨁a)⨂(c⨁b);c \bigoplus (a \bigotimes b) = (c \bigoplus a) \bigotimes (c \bigoplus b); 4) (a⨂b)⨁c=(a⨁c)⨂(b⨁c).(a \bigotimes b) \bigoplus c = (a \bigoplus c) \bigotimes (b \bigoplus c).
Proposed by Yannick Yao

2018-2019 Fall OMO Problem 21

Suppose that a sequence a0,a1,…a_0, a_1, \ldots of real numbers is defined by a0=1a_0=1 and an={an−1a0+an−3a2+⋯+a0an−1if n oddan−1a1+an−3a3+⋯+a1an−1if n evena_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases} for n≥1n\geq1. There is a positive real number rr such that a0+a1r+a2r2+a3r3+⋯=54.a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}. If rr can be written in the form ab−cd\frac{a\sqrt{b}-c}{d} for positive integers a,b,c,da,b,c,d such that bb is not divisible by the square of any prime and gcd⁡(a,c,d)=1,\gcd (a,c,d)=1, then compute a+b+c+da+b+c+d.
Proposed by Tristan Shin
15
2

2017-2018 Spring OMO Problem 15

Let N\mathbb{N} denote the set of positive integers. For how many positive integers k≤2018k\le 2018 do there exist a function f:N→Nf: \mathbb{N}\to \mathbb{N} such that f(f(n))=2nf(f(n))=2n for all n∈Nn\in \mathbb{N} and f(k)=2018f(k)=2018?
Proposed by James Lin

2018-2019 Fall OMO Problem 15

Iris does not know what to do with her 1-kilogram pie, so she decides to share it with her friend Rosabel. Starting with Iris, they take turns to give exactly half of total amount of pie (by mass) they possess to the other person. Since both of them prefer to have as few number of pieces of pie as possible, they use the following strategy: During each person's turn, she orders the pieces of pie that she has in a line from left to right in increasing order by mass, and starts giving the pieces of pie to the other person beginning from the left. If she encounters a piece that exceeds the remaining mass to give, she cuts it up into two pieces with her sword and gives the appropriately sized piece to the other person.
When the pie has been cut into a total of 2017 pieces, the largest piece that Iris has is mn\frac{m}{n} kilograms, and the largest piece that Rosabel has is pq\frac{p}{q} kilograms, where m,n,p,qm,n,p,q are positive integers satisfying gcd⁡(m,n)=gcd⁡(p,q)=1\gcd(m,n)=\gcd(p,q)=1. Compute the remainder when m+n+p+qm+n+p+q is divided by 2017.
Proposed by Yannick Yao
9
2

2017-2018 Spring OMO Problem 9

Let kk be a positive integer. In the coordinate plane, circle ω\omega has positive integer radius and is tangent to both axes. Suppose that ω\omega passes through (1,1000+k)(1,1000+k). Compute the smallest possible value of kk.
Proposed by Luke Robitaille

2018-2019 Fall OMO Problem 9

Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that kk is chosen with probability 2−k2^{-k} for every positive integer kk, and then it outputs kk tokens. Let NN be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann going first. Ann wins if she receives at least NN total tokens from the slot machine before Drew receives at least M=22018M=2^{2018} total tokens, and Drew wins if he receives MM tokens before Ann receives NN tokens. If each person has the same probability of winning, compute the remainder when NN is divided by 20182018.
Proposed by Brandon Wang
3
2
28
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2017-2018 Spring OMO Problem 28

In △ABC\triangle ABC, the incircle ω\omega has center II and is tangent to CA‾\overline{CA} and AB‾\overline{AB} at EE and FF respectively. The circumcircle of △BIC\triangle{BIC} meets ω\omega at PP and QQ. Lines AIAI and BCBC meet at DD, and the circumcircle of △PDQ\triangle PDQ meets BC‾\overline{BC} again at XX. Suppose that EF=PQ=16EF = PQ = 16 and PX+QX=17PX + QX = 17. Then BC2BC^2 can be expressed as mn\frac mn, where mm and nn are relatively prime positive integers. Find 100m+n100m + n.
Proposed by Ankan Bhattacharya and Michael Ren

2018-2019 Fall OMO Problem 28

Let ω\omega be a circle centered at OO with radius R=2018R=2018. For any 0<r<10090 < r < 1009, let γ\gamma be a circle of radius rr centered at a point II satisfying OI=R(R−2r)OI =\sqrt{R(R-2r)}. Choose any A,B,C∈ωA,B,C\in \omega with AC,ABAC, AB tangent to γ\gamma at E,FE,F, respectively. Suppose a circle of radius rAr_A is tangent to AB,ACAB,AC, and internally tangent to ω\omega at a point DD with rA=5rr_A=5r. Let line EFEF meet ω\omega at P1,Q1P_1,Q_1. Suppose P2,P3,Q2,Q3P_2,P_3,Q_2,Q_3 lie on ω\omega such that P1P2,P1P3,Q1Q2,Q1Q3P_1P_2,P_1P_3,Q_1Q_2,Q_1Q_3 are tangent to γ\gamma. Let P2P3,Q2Q3P_2P_3,Q_2Q_3 meet at KK, and suppose KIKI meets ADAD at a point XX. Then as rr varies from 00 to 10091009, the maximum possible value of OXOX can be expressed in the form abc\frac{a\sqrt{b}}{c}, where a,b,ca,b,c are positive integers such that bb is not divisible by the square of any prime and gcd⁡(a,c)=1\gcd (a,c)=1. Compute 10a+b+c10a+b+c.
Proposed by Vincent Huang
25
2

2017-2018 Spring OMO Problem 25

Let mm and nn be positive integers. Fuming Zeng gives James a rectangle, such that m−1m-1 lines are drawn parallel to one pair of sides and n−1n-1 lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an m×nm\times n grid of smaller rectangles. Fuming Zeng chooses m+n−1m+n-1 of the mnmn smaller rectangles and then tells James the area of each of the smaller rectangles. Of the (mnm+n−1)\dbinom{mn}{m+n-1} possible combinations of rectangles and their areas Fuming Zeng could have given, let Cm,nC_{m,n} be the number of combinations which would allow James to determine the area of the whole rectangle. Given that A=∑m=1∞∑n=1∞Cm,n(m+nm)(m+n)m+n,A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}}, then find the greatest integer less than 1000A1000A.
Proposed by James Lin

2018-2019 Fall OMO Problem 25

Given two positive integers x,yx,y, we define z=x ⊕ yz=x\,\oplus\,y to be the bitwise XOR sum of xx and yy; that is, zz has a 11 in its binary representation at exactly the place values where x,yx,y have differing binary representations. It is known that ⊕\oplus is both associative and commutative. For example, 20⊕18=101002⊕100102=1102=620 \oplus 18 = 10100_2 \oplus 10010_2 = 110_2 = 6. Given a set S={a1,a2,…,an}S=\{a_1, a_2, \dots, a_n\} of positive integers, we let f(S)=a1⊕a2⊕a3⊕⋯⊕anf(S) = a_1 \oplus a_2 \oplus a_3\oplus \dots \oplus a_n. We also let g(S)g(S) be the number of divisors of f(S)f(S) which are at most 20182018 but greater than or equal to the largest element in SS (if SS is empty then let g(S)=2018g(S)=2018). Compute the number of 11s in the binary representation of ∑S⊆{1,2,…,2018}g(S)\displaystyle\sum_{S\subseteq \{1,2,\dots, 2018\}} g(S).
Proposed by Brandon Wang and Vincent Huang
22
2

2017-2018 Spring OMO Problem 22

Let p=9001p = 9001 be a prime number and let Z/pZ\mathbb{Z}/p\mathbb{Z} denote the additive group of integers modulo pp. Furthermore, if A,B⊂Z/pZA, B \subset \mathbb{Z}/p\mathbb{Z}, then denote A+B={a+b(modp)∣a∈A,b∈B}.A+B = \{a+b \pmod{p} | a \in A, b \in B \}. Let s1,s2,…,s8s_1, s_2, \dots, s_8 are positive integers that are at least 22. Yang the Sheep notices that no matter how he chooses sets T1,T2,…,T8⊂Z/pZT_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z} such that ∣Ti∣=si|T_i| = s_i for 1≤i≤8,1 \le i \le 8, T1+T2+⋯+T7T_1+T_2+\dots + T_7 is never equal to Z/pZ\mathbb{Z}/p\mathbb{Z}, but T1+T2+⋯+T8T_1+T_2+\dots+T_8 must always be exactly Z/pZ\mathbb{Z}/p\mathbb{Z}. What is the minimum possible value of s8s_8?
Proposed by Yang Liu

2018-2019 Fall OMO Problem 22

Let ABCABC be a triangle with AB=2AB=2 and AC=3AC=3. Let HH be the orthocenter, and let MM be the midpoint of BCBC. Let the line through HH perpendicular to line AMAM intersect line ABAB at XX and line ACAC at YY. Suppose that lines BYBY and CXCX are parallel. Then [ABC]2=a+bcd[ABC]^2=\frac{a+b\sqrt{c}}{d} for positive integers a,b,ca,b,c and dd, where gcd⁡(a,b,d)=1\gcd(a,b,d)=1 and cc is not divisible by the square of any prime. Compute 1000a+100b+10c+d1000a+100b+10c+d.
Proposed by Luke Robitaille
19
2

2017-2018 Spring OMO Problem 19

Let P(x)P(x) be a polynomial of degree at most 20182018 such that P(i)=(2018i)P(i)=\binom{2018}i for all integer ii such that 0≤i≤20180\le i\le 2018. Find the largest nonnegative integer nn such that 2n∣P(2020)2^n\mid P(2020).
Proposed by Michael Ren

2018-2019 Fall OMO Problem 19

Players 1,2,…,101,2,\ldots,10 are playing a game on Christmas. Santa visits each player's house according to a set of rules:
-Santa first visits player 11. After visiting player ii, Santa visits player i+1i+1, where player 1111 is the same as player 11. -Every time Santa visits someone, he gives them either a present or a piece of coal (but not both). -The absolute difference between the number of presents and pieces of coal that Santa has given out is at most 33 at every point in time. -If Santa has a choice between giving out a present and a piece of coal, he chooses with equal probability.
Let pp be the probability that player 11 gets a present before player 22 does. If p=mnp=\frac{m}{n} for relatively prime positive integers mm and nn, then compute 100m+n100m+n.
Proposed by Tristan Shin
16
2

2017-2018 Spring OMO Problem 16

In a rectangular 57×5757\times 57 grid of cells, kk of the cells are colored black. What is the smallest positive integer kk such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
Proposed by James Lin

2018-2019 Fall OMO Problem 16

Jay has a 24×2424\times 24 grid of lights, all of which are initially off. Each of the 4848 rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the 4848 rows and columns exactly once, such that after each toggle he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules which has been expended by all the lights after all 4848 toggles.
Proposed by James Lin
13
2
10
2
5
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2017-2018 Spring OMO Problem 5

A mouse has a wheel of cheese which is cut into 20182018 slices. The mouse also has a 20192019-sided die, with faces labeled 0,1,2,…,20180,1,2,\ldots, 2018, and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on kk, and the mouse has at least kk slices of cheese remaining, then the mouse eats kk slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
Proposed by Brandon Wang

2018-2019 Fall OMO Problem 5

In triangle ABCABC, AB=8,AC=9,AB=8, AC=9, and BC=10BC=10. Let MM be the midpoint of BCBC. Circle ω1\omega_1 with area A1A_1 passes through A,B,A,B, and CC. Circle ω2\omega_2 with area A2A_2 passes through A,B,A,B, and MM. Then A1A2=mn\frac{A_1}{A_2}=\frac{m}{n} for relatively prime positive integers mm and nn. Compute 100m+n100m+n.
Proposed by Luke Robitaille
7
2
4
2
1
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2017-2018 Spring OMO Problem 1

Farmer James has three types of cows on his farm. A cow with zero legs is called a <spanclass=′latex−italic′>groundbeef</span><span class='latex-italic'>ground beef</span>, a cow with one leg is called a <spanclass=′latex−italic′>steak</span><span class='latex-italic'>steak</span>, and a cow with two legs is called a <spanclass=′latex−italic′>leanbeef</span><span class='latex-italic'>lean beef</span>. Farmer James counts a total of 2020 cows and 1818 legs on his farm. How many more <spanclass=′latex−italic′>groundbeef</span><span class='latex-italic'>ground beef</span>s than <spanclass=′latex−italic′>leanbeef</span><span class='latex-italic'>lean beef</span>s does Farmer James have?
Proposed by James Lin

2018-2019 Fall OMO Problem 1

Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.
Note: Two integers are relatively prime if no positive integer other than 11 divides both numbers.
Proposed by ABCDE and Tristan Shin
30
2

2017-2018 Spring OMO Problem 30

Let p=2017p = 2017. Given a positive integer nn, an n×nn\times n matrix AA is formed with each element aija_{ij} randomly selected, with equal probability, from {0,1,…,p−1}\{0,1,\ldots,p - 1\}. Let qnq_n be probability that det⁡A≡1(modp)\det A\equiv 1\pmod{p}. Let q=lim⁡n→∞qnq=\displaystyle\lim_{n\rightarrow\infty} q_n. If d1,d2,d3,…d_1, d_2, d_3, \ldots are the digits after the decimal point in the base pp expansion of qq, then compute the remainder when ∑k=1p2dk\displaystyle\sum_{k = 1}^{p^2} d_k is divided by 10910^9.
Proposed by Ashwin Sah

2018-2019 Fall OMO Problem 30

Let ABCABC be an acute triangle with cos⁡B=13,cos⁡C=14\cos B =\frac{1}{3}, \cos C =\frac{1}{4}, and circumradius 7272. Let ABCABC have circumcenter OO, symmedian point KK, and nine-point center NN. Consider all non-degenerate hyperbolas H\mathcal H with perpendicular asymptotes passing through A,B,CA,B,C. Of these H\mathcal H, exactly one has the property that there exists a point P∈HP\in \mathcal H such that NPNP is tangent to H\mathcal H and P∈OKP\in OK. Let N′N' be the reflection of NN over BCBC. If AKAK meets PN′PN' at QQ, then the length of PQPQ can be expressed in the form a+bca+b\sqrt{c}, where a,b,ca,b,c are positive integers such that cc is not divisible by the square of any prime. Compute 100a+b+c100a+b+c.
Proposed by Vincent Huang
24
2

2017-2018 Spring OMO Problem 24

Find the number of ordered triples (a,b,c)(a,b,c) of integers satisfying 0≤a,b,c≤10000\le a,b,c \le 1000 for which a3+b3+c3≡3abc+1(mod1001).a^3+b^3+c^3\equiv 3abc+1\pmod{1001}.
Proposed by James Lin

2018-2019 Fall OMO Problem 24

Let p=101p = 101 and let SS be the set of pp-tuples (a1,a2,…,ap)∈Zp(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p of integers. Let NN denote the number of functions f:S→{0,1,…,p−1}f: S \to \{0, 1, \dots, p-1\} such that
[*] f(a+b)+f(a−b)≡2(f(a)+f(b))(modp)f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p} for all a,b∈Sa, b \in S, and [*] f(a)=f(b)f(a) = f(b) whenever all components of a−ba-b are divisible by pp.
Compute the number of positive integer divisors of NN. (Here addition and subtraction in Zp\mathbb{Z}^p are done component-wise.)
Proposed by Ankan Bhattacharya
18
2

2017-2018 Spring OMO Problem 18

Suppose that a,b,ca,b,c are real numbers such that a<b<ca < b < c and a3−3a+1=b3−3b+1=c3−3c+1=0a^3-3a+1=b^3-3b+1=c^3-3c+1=0. Then 1a2+b+1b2+c+1c2+a\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a} can be written as pq\frac pq for relatively prime positive integers pp and qq. Find 100p+q100p+q.
Proposed by Michael Ren

2018-2019 Fall OMO Problem 18

On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most NN passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station ii will get off at station jj for some j>ij>i (the value of jj may vary over different passengers). Call any group of four distinct stations i1,i2,j1,j2i_1, i_2, j_1, j_2 with iu<jvi_u< j_v for all u,v∈{1,2}u,v\in \{1,2\} a good group. Suppose that in any good group i1,i2,j1,j2i_1, i_2, j_1, j_2, there is a passenger who boards at station i1i_1 and de-boards at station j1j_1, or there is a passenger who boards at station i2i_2 and de-boards at station j2j_2, or both scenarios occur. Compute the minimum possible value of NN.
Proposed by Yannick Yao
12
2

2017-2018 Spring OMO Problem 12

Near the end of a game of Fish, Celia is playing against a team consisting of Alice and Betsy. Each of the three players holds two cards in their hand, and together they have the Nine, Ten, Jack, Queen, King, and Ace of Spades (this set of cards is known by all three players). Besides the two cards she already has, each of them has no information regarding the other two's hands (In particular, teammates Alice and Betsy do not know each other's cards).
It is currently Celia's turn. On a player's turn, the player must ask a player on the other team whether she has a certain card that is in the set of six cards but not in the asker's hand. If the player being asked does indeed have the card, then she must reveal the card and put it in the asker’s hand, and the asker shall ask again (but may ask a different player on the other team); otherwise, she refuses and it is now her turn. Moreover, a card may not be asked if it is known (to the asker) to be not in the asked person's hand. The game ends when all six cards belong to one team, and the team with all the cards wins. Under optimal play, the probability that Celia wins the game is pq\frac{p}{q} for relatively prime positive integers pp and qq. Find 100p+q100p+q.
Proposed by Yannick Yao

2018-2019 Fall OMO Problem 12

Three non-collinear lattice points A,B,CA,B,C lie on the plane 1+3x+5y+7z=01+3x+5y+7z=0. The minimal possible area of triangle ABCABC can be expressed as mn\frac{\sqrt{m}}{n} where m,nm,n are positive integers such that there does not exists a prime pp dividing nn with p2p^2 dividing mm. Compute 100m+n100m+n.
Proposed by Yannick Yao
6
2

2017-2018 Spring OMO Problem 6

Let f(x)=x2+xf(x)=x^2+x for all real xx. There exist positive integers mm and nn, and distinct nonzero real numbers yy and zz, such that f(y)=f(z)=m+nf(y)=f(z)=m+\sqrt{n} and f(1y)+f(1z)=110f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}. Compute 100m+n100m+n.
Proposed by Luke Robitaille

2018-2019 Fall OMO Problem 6

Patchouli is taking an exam with k>1k > 1 parts, numbered Part 1,2,…,k1, 2, \dots, k. It is known that for i=1,2,…,ki = 1, 2, \dots, k, Part ii contains ii multiple choice questions, each of which has (i+1)(i+1) answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to 20182018 times the probability that she gets no questions correct. Compute the number of questions that are on the exam.
Proposed by Yannick Yao
29
2

2017-2018 Spring OMO Problem 29

Let q<50q<50 be a prime number. Call a sequence of polynomials P0(x),P1(x),P2(x),...,Pq2(x)P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x) tasty if it satisfies the following conditions:
[*] PiP_i has degree ii for each ii (where we consider constant polynomials, including the 00 polynomial, to have degree 00) [*] The coefficients of PiP_i are integers between 00 and q−1q-1 for each ii. [*] For any 0≤i,j≤q20\le i,j\le q^2, the polynomial Pi(Pj(x))−Pj(Pi(x))P_i(P_j(x)) - P_j(P_i(x)) has all its coefficients divisible by qq.
As qq varies over all such prime numbers, determine the total number of tasty sequences of polynomials.
Proposed by Vincent Huang

2018-2019 Fall OMO Problem 29

For integers 0≤m,n≤22017−10 \le m,n \le 2^{2017}-1, let α(m,n)\alpha(m,n) be the number of nonnegative integers kk for which ⌊m/2k⌋\left\lfloor m/2^k \right\rfloor and ⌊n/2k⌋\left\lfloor n/2^k \right\rfloor are both odd integers. Consider a 22017×220172^{2017} \times 2^{2017} matrix MM whose (i,j)(i,j)th entry (for 1≤i,j≤220171 \le i, j \le 2^{2017}) is (−1)α(i−1,j−1). (-1)^{\alpha(i-1, j-1)}. For 1≤i,j≤220171 \le i, j \le 2^{2017}, let Mi,jM_{i,j} be the matrix with the same entries as MM except for the (i,j)(i,j)th entry, denoted by ai,ja_{i,j}, and such that det⁡Mi,j=0\det M_{i,j}=0. Suppose that AA is the 22017×220172^{2017} \times 2^{2017} matrix whose (i,j)(i,j)th entry is ai,ja_{i,j} for all 1≤i,j≤220171 \le i, j \le 2^{2017}. Compute the remainder when det⁡A\det A is divided by 20172017.
Proposed by Michael Ren and Ashwin Sah
26
2

2017-2018 Spring OMO Problem 26

Let ABCABC be a triangle with incenter II. Let PP and QQ be points such that IP⊥ACIP\perp AC, IQ⊥ABIQ\perp AB, and IA⊥PQIA\perp PQ. Assume that BPBP and CQCQ intersect at the point R≠AR\neq A on the circumcircle of ABCABC such that AR∥BCAR\parallel BC. Given that ∠B−∠C=36∘\angle B-\angle C=36^\circ, the value of cos⁡A\cos A can be expressed in the form m−np\frac{m-\sqrt n}{p} for positive integers m,n,pm,n,p and where nn is not divisible by the square of any prime. Find the value of 100m+10n+p100m+10n+p.
Proposed by Michael Ren

2018-2019 Fall OMO Problem 26

Let p=2027p = 2027 be the smallest prime greater than 20182018, and let P(X)=X2031+X2030+X2029−X5−10X4−10X3+2018X2P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2. Let GF(p)\mathrm{GF}(p) be the integers modulo pp, and let GF(p)(X)\mathrm{GF}(p)(X) be the set of rational functions with coefficients in GF(p)\mathrm{GF}(p) (so that all coefficients are taken modulo pp). That is, GF(p)(X)\mathrm{GF}(p)(X) is the set of fractions P(X)Q(X)\frac{P(X)}{Q(X)} of polynomials with coefficients in GF(p)\mathrm{GF}(p), where Q(X)Q(X) is not the zero polynomial. Let D ⁣:GF(p)(X)→GF(p)(X)D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X) be a function satisfying D(fg)=D(f)⋅g−f⋅D(g)g2 D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2} for any f,g∈GF(p)(X)f,g\in \mathrm{GF}(p)(X) with g≠0g\neq 0, and such that for any nonconstant polynomial ff, D(f)D(f) is a polynomial with degree less than that of ff. If the number of possible values of D(P(X))D(P(X)) can be written as aba^b, where aa, bb are positive integers with aa minimized, compute abab.
Proposed by Brandon Wang
23
2

2017-2018 Spring OMO Problem 23

Let ABCABC be a triangle with BC=13,CA=11,AB=10BC=13, CA=11, AB=10. Let A1A_1 be the midpoint of BCBC. A variable line ℓ\ell passes through A1A_1 and meets AC,ABAC,AB at B1,C1B_1,C_1. Let B2,C2B_2,C_2 be points with B2B=B2C,B2C1⊥AB,C2B=C2C,C2B1⊥ACB_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC, and define P=BB2∩CC2P=BB_2\cap CC_2. Suppose the circles of diameters BB2,CC2BB_2, CC_2 meet at a point Q≠A1Q\neq A_1. Given that QQ lies on the same side of line BCBC as AA, the minimum possible value of PBPC+QBQC\dfrac{PB}{PC}+\dfrac{QB}{QC} can be expressed in the form abc\dfrac{a\sqrt{b}}{c} for positive integers a,b,ca,b,c with gcd⁡(a,c)=1\gcd (a,c)=1 and bb squarefree. Determine a+b+ca+b+c.
Proposed by Vincent Huang

2018-2019 Fall OMO Problem 23

Consider all ordered pairs (a,b)(a, b) of positive integers such that a2+b2+2ab\frac{a^2 + b^2 + 2}{ab} is an integer and a≤ba\le b. We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus P1=(1,1),P2=(1,3)P_1 = (1, 1), P_2 = (1, 3), and so on. If P2020=(x,y),P_{2020} = (x, y), then compute the remainder when x+yx + y is divided by 20172017.
Proposed by Ashwin Sah
20
2

2017-2018 Spring OMO Problem 20

Let ABCABC be a triangle with AB=7,BC=5,AB = 7, BC = 5, and CA=6CA = 6. Let DD be a variable point on segment BCBC, and let the perpendicular bisector of ADAD meet segments AC,ABAC, AB at E,F,E, F, respectively. It is given that there is a point PP inside △ABC\triangle ABC such that APPC=AEEC\frac{AP}{PC} = \frac{AE}{EC} and APPB=AFFB\frac{AP}{PB} = \frac{AF}{FB}. The length of the path traced by PP as DD varies along segment BCBC can be expressed as mnsin⁡−1(17)\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right), where mm and nn are relatively prime positive integers, and angles are measured in radians. Compute 100m+n100m + n.
Proposed by Edward Wan

2018-2019 Fall OMO Problem 20

For positive integers k,nk,n with k≤nk\leq n, we say that a kk-tuple (a1,a2,…,ak)\left(a_1,a_2,\ldots,a_k\right) of positive integers is tasty if
[*] there exists a kk-element subset SS of [n][n] and a bijection f:[k]→Sf:[k]\to S with ax≤f(x)a_x\leq f\left(x\right) for each x∈[k]x\in [k], [*] ax=aya_x=a_y for some distinct x,y∈[k]x,y\in [k], and [*] ai≤aja_i\leq a_j for any i<ji < j.
For some positive integer nn, there are more than 20182018 tasty tuples as kk ranges through 2,3,…,n2,3,\ldots,n. Compute the least possible number of tasty tuples there can be.
Note: For a positive integer mm, [m][m] is taken to denote the set {1,2,…,m}\left\{1,2,\ldots,m\right\}.
Proposed by Vincent Huang and Tristan Shin
17
2

2017-2018 Spring OMO Problem 17

Let SS be the set of all subsets of {2,3,…,2016}\left\{2,3,\ldots,2016\right\} with size 10071007, and for a nonempty set TT of numbers, let f(T)f(T) be the product of the elements in TT. Determine the remainder when ∑T∈S(f(T)−f(T)−1)2 \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2 is divided by 20172017. Note: For bb relatively prime to 20172017, we say that b−1b^{-1} is the unique positive integer less than 20172017 for which 20172017 divides bb−1−1bb^{-1} -1.
Proposed by Tristan Shin

2018-2019 Fall OMO Problem 17

A hyperbola in the coordinate plane passing through the points (2,5)(2,5), (7,3)(7,3), (1,1)(1,1), and (10,10)(10,10) has an asymptote of slope 2017\frac{20}{17}. The slope of its other asymptote can be expressed in the form −mn-\frac{m}{n}, where mm and nn are relatively prime positive integers. Compute 100m+n100m+n.
Proposed by Michael Ren
14
2

2017-2018 Spring OMO Problem 14

Let ABCABC be a triangle with AB=20AB=20 and AC=18AC=18. EE is on segment ACAC and FF is on segment ABAB such that AE=AF=8AE=AF=8. Let BEBE and CFCF intersect at GG. Given that AEGFAEGF is cyclic, then BC=mnBC=m\sqrt{n} for positive integers mm and nn such that nn is not divisible by the square of any prime. Compute 100m+n100m+n.
Proposed by James Lin

2018-2019 Fall OMO Problem 14

In triangle ABCABC, AB=13,BC=14,CA=15AB=13, BC=14, CA=15. Let Ω\Omega and ω\omega be the circumcircle and incircle of ABCABC respectively. Among all circles that are tangent to both Ω\Omega and ω\omega, call those that contain ω\omega inclusive and those that do not contain ω\omega exclusive. Let I\mathcal{I} and E\mathcal{E} denote the set of centers of inclusive circles and exclusive circles respectively, and let II and EE be the area of the regions enclosed by I\mathcal{I} and E\mathcal{E} respectively. The ratio IE\frac{I}{E} can be expressed as mn\sqrt{\frac{m}{n}}, where mm and nn are relatively prime positive integers. Compute 100m+n100m+n.
Proposed by Yannick Yao
11
2

2017-2018 Spring OMO Problem 11

Lunasa, Merlin, and Lyrica are performing in a concert. Each of them will perform two different solos, and each pair of them will perform a duet, for nine distinct pieces in total. Since the performances are very demanding, no one is allowed to perform in two pieces in a row. In how many different ways can the pieces be arranged in this concert?
Proposed by Yannick Yao

2018-2019 Fall OMO Problem 11

Let an ordered pair of positive integers (m,n)(m, n) be called regimented if for all nonnegative integers kk, the numbers mkm^k and nkn^k have the same number of positive integer divisors. Let NN be the smallest positive integer such that (20162016,N)\left(2016^{2016}, N\right) is regimented. Compute the largest positive integer vv such that 2v2^v divides the difference 20162016−N2016^{2016}-N.
Proposed by Ashwin Sah
8
2

2017-2018 Spring OMO Problem 8

Compute the number of ordered quadruples (a,b,c,d)(a,b,c,d) of distinct positive integers such that ((ab)(cd))=21\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21.
Proposed by Luke Robitaille

2018-2019 Fall OMO Problem 8

Let ABCABC be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let N=2017N = 2017, and define the AA-ntipodes to be the points A1,…,ANA_1,\dots, A_N to be the points on segment BCBC such that BA1=A1A2=⋯=AN−1AN=ANCBA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC, and similarly define the BB, CC-ntipodes. A line ℓA\ell_A through AA is called a qevian if it passes through an AA-ntipode, and similarly we define qevians through BB and CC. Compute the number of ordered triples (ℓA,ℓB,ℓC)(\ell_A, \ell_B, \ell_C) of concurrent qevians through AA, BB, CC, respectively.
Proposed by Brandon Wang
2
2