MathDB

2016 Online Math Open Problems

Part of Online Math Open Problems

Subcontests

(30)
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2015-2016 OMO Spring #30

In triangle ABCABC, AB=330−10AB=3\sqrt{30}-\sqrt{10}, BC=12BC=12, and CA=330+10CA=3\sqrt{30}+\sqrt{10}. Let MM be the midpoint of ABAB and NN be the midpoint of ACAC. Denote ll as the line passing through the circumcenter OO and orthocenter HH of ABCABC, and let EE and FF be the feet of the perpendiculars from BB and CC to ll, respectively. Let l′l' be the reflection of ll in BCBC such that l′l' intersects lines AEAE and AFAF at PP and QQ, respectively. Let lines BPBP and CQCQ intersect at KK. XX, YY, and ZZ are the reflections of KK over the perpendicular bisectors of sides BCBC, CACA, and ABAB, respectively, and RR and SS are the midpoints of XYXY and XZXZ, respectively. If lines MRMR and NSNS intersect at TT, then the length of OTOT can be expressed in the form pq\frac{p}{q} for relatively prime positive integers pp and qq. Find 100p+q100p+q.
Proposed by Vincent Huang and James Lin

2016-2017 Fall OMO Problem 30

Let P1(x),P2(x),…,Pn(x)P_1(x),P_2(x),\ldots,P_n(x) be monic, non-constant polynomials with integer coefficients and let Q(x)Q(x) be a polynomial with integer coefficients such that x22016+x+1=P1(x)P2(x)…Pn(x)+2Q(x).x^{2^{2016}}+x+1=P_1(x)P_2(x)\ldots P_n(x)+2Q(x). Suppose that the maximum possible value of 2016n2016n can be written in the form 2b1+2b2+⋯+2bk2^{b_1}+2^{b_2}+\cdots+2^{b_k} for nonnegative integers b1<b_1< b2<b_2< ⋯<\cdots< bkb_k. Find the value of b1+b2+⋯+bkb_1+b_2+\cdots+b_k.
Proposed by Michael Ren
27
2

2015-2016 OMO Spring #27

Let ABCABC be a triangle with circumradius 22 and ∠B−∠C=15∘\angle B-\angle C=15^\circ. Denote its circumcenter as OO, orthocenter as HH, and centroid as GG. Let the reflection of HH over OO be LL, and let lines AGAG and ALAL intersect the circumcircle again at XX and YY, respectively. Define B1B_1 and C1C_1 as the points on the circumcircle of ABCABC such that BB1∥ACBB_1\parallel AC and CC1∥ABCC_1\parallel AB, and let lines XYXY and B1C1B_1C_1 intersect at ZZ. Given that OZ=25OZ=2\sqrt 5, then AZ2AZ^2 can be expressed in the form m−nm-\sqrt n for positive integers mm and nn. Find 100m+n100m+n.
Proposed by Michael Ren

2016-2017 Fall OMO Problem 27

Compute the number of monic polynomials q(x)q(x) with integer coefficients of degree 1212 such that there exists an integer polynomial p(x)p(x) satisfying q(x)p(x)=q(x2).q(x)p(x) = q(x^2).
Proposed by Yang Liu
28
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2015-2016 Spring OMO #28

Let NN be the number of polynomials P(x1,x2,…,x2016)P(x_1, x_2, \dots, x_{2016}) of degree at most 20152015 with coefficients in the set {0,1,2}\{0, 1, 2 \} such that P(a1,a2,⋯ ,a2016)≡1(mod3)P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3} for all (a1,a2,⋯ ,a2016)∈{0,1}2016.(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.
Compute the remainder when v3(N)v_3(N) is divided by 20112011, where v3(N)v_3(N) denotes the largest integer kk such that 3k∣N.3^k | N.
Proposed by Yang Liu

2016-2017 Fall OMO Problem 28

Let ABCABC be a triangle with AB=34,BC=25,AB=34,BC=25, and CA=39CA=39. Let O,H,O,H, and ω \omega be the circumcenter, orthocenter, and circumcircle of △ABC\triangle ABC, respectively. Let line AHAH meet ω\omega a second time at A1A_1 and let the reflection of HH over the perpendicular bisector of BCBC be H1H_1. Suppose the line through OO perpendicular to A1OA_1O meets ω\omega at two points QQ and RR with QQ on minor arc ACAC and RR on minor arc ABAB. Denote H\mathcal H as the hyperbola passing through A,B,C,H,H1A,B,C,H,H_1, and suppose HOHO meets H\mathcal H again at PP. Let X,YX,Y be points with XH∥AR∥YP,XP∥AQ∥YHXH \parallel AR \parallel YP, XP \parallel AQ \parallel YH. Let P1,P2P_1,P_2 be points on the tangent to H\mathcal H at PP with XP1∥OH∥YP2XP_1 \parallel OH \parallel YP_2 and let P3,P4P_3,P_4 be points on the tangent to H\mathcal H at HH with XP3∥OH∥YP4XP_3 \parallel OH \parallel YP_4. If P1P4P_1P_4 and P2P3P_2P_3 meet at NN, and ONON may be written in the form ab\frac{a}{b} where a,ba,b are positive coprime integers, find 100a+b100a+b.
Proposed by Vincent Huang
24
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2015-2016 OMO Spring #24

Bessie and her 20152015 bovine buddies work at the Organic Milk Organization, for a total of 20162016 workers. They have a hierarchy of bosses, where obviously no cow is its own boss. In other words, for some pairs of employees (A,B)(A, B), BB is the boss of AA. This relationship satisfies an obvious condition: if BB is the boss of AA and CC is the boss of BB, then CC is also a boss of AA. Business has been slow, so Bessie hires an outside organizational company to partition the company into some number of groups. To promote growth, every group is one of two forms. Either no one in the group is the boss of another in the group, or for every pair of cows in the group, one is the boss of the other. Let GG be the minimum number of groups needed in such a partition. Find the maximum value of GG over all possible company structures.
Proposed by Yang Liu

2016-2017 Fall OMO Problem 24

Let P(x,y)P(x,y) be a polynomial such that deg⁡x(P),deg⁡y(P)≤2020\deg_x(P), \deg_y(P)\le 2020 and P(i,j)=(i+ji)P(i,j)=\binom{i+j}{i} over all 202122021^2 ordered pairs (i,j)(i,j) with 0≤i,j≤20200\leq i,j\leq 2020. Find the remainder when P(4040,4040)P(4040, 4040) is divided by 20172017.
Note: deg⁡x(P)\deg_x (P) is the highest exponent of xx in a nonzero term of P(x,y)P(x,y). deg⁡y(P)\deg_y (P) is defined similarly.
Proposed by Michael Ren
29
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2015-2016 Spring OMO #29

Yang the Spinning Square Sheep is a square in the plane such that his four legs are his four vertices. Yang can do two different types of tricks:
(a) Yang can choose one of his sides, then reflect himself over the side. (b) Yang can choose one of his legs, then rotate 90∘90^\circ counterclockwise around the leg.
Yang notices that after 20162016 tricks, each leg ends up in exactly the same place the leg started out in! Let there be NN ways for Yang to perform his 20162016 tricks. What is the remainder when NN is divided by 100000100000?
Proposed by James Lin

2016-2017 Fall OMO Problem 29

Let nn be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line y=xy=x, which Yang may reach but is not permitted to go above (i.e. Yang is allowed to be located at (2016,2015)(2016,2015) and (2016,2016)(2016,2016), but not at (2016,2017)(2016,2017)). Yang is currently located at (0,0)(0,0) and wishes to reach (n,0)(n,0). Yang is permitted only to make the following moves:
(a) Yang may spring, which consists of going from a point (x,y)(x,y) to the point (x,y+1)(x,y+1). (b) Yang may stroll, which consists of going from a point (x,y)(x,y) to the point (x+1,y)(x+1,y). (c) Yang may sink, which consists of going from a point (x,y)(x,y) to the point (x,y−1)(x,y-1).
In addition, whenever Yang does a sink, he breaks his tiny little legs and may no longer do a spring at any time afterwards. Yang also expends a lot of energy doing a spring and gets bloodthirsty, so he must visit the blood river at least once afterwards to quench his bloodthirst. (So Yang may still spring while bloodthirsty, but he may not finish his journey while bloodthirsty.) Let there be ana_n different ways for which Yang can reach (n,0)(n,0), given that Yang is permitted to pass by (n,0)(n,0) in the middle of his journey. Find the 20162016th smallest positive integer nn for which an≡1(mod5)a_n\equiv 1\pmod 5.
Proposed by James Lin
21
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2015-2016 OMO Spring #21

Say a real number rr is \emph{repetitive} if there exist two distinct complex numbers z1,z2z_1,z_2 with ∣z1∣=∣z2∣=1|z_1|=|z_2|=1 and {z1,z2}≠{−i,i}\{z_1,z_2\}\neq\{-i,i\} such that z1(z13+z12+rz1+1)=z2(z23+z22+rz2+1). z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1). There exist real numbers a,ba,b such that a real number rr is \emph{repetitive} if and only if a<r≤ba < r\le b. If the value of ∣a∣+∣b∣|a|+|b| can be expressed in the form pq\frac{p}{q} for relatively prime positive integers pp and qq, find 100p+q100p+q.
Proposed by James Lin

2016-2017 Fall OMO Problem 21

Mark the Martian and Bark the Bartian live on planet Blok, in the year 20192019. Mark and Bark decide to play a game on a 10×1010 \times 10 grid of cells. First, Mark randomly generates a subset SS of {1,2,…,2019}\{1, 2, \dots, 2019\} with ∣S∣=100|S|=100. Then, Bark writes each of the 100100 integers in a different cell of the 10×1010 \times 10 grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is nn, then she stacks nn 1×1×11 \times 1 \times 1 blocks on top of one other in that cell. Let BB be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the 100100 integers into the grid of cells. Find the expected value of BB over all possible sets SS Mark could have generated.
Proposed by Yang Liu
26
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2015-2016 Spring OMO #26

Let SS be the set of all pairs (a,b)(a, b) of integers satisfying 0≤a,b≤2014.0 \le a, b \le 2014. For any pairs s1=(a1,b1),s2=(a2,b2)∈Ss_1 = (a_1, b_1), s_2 = (a_2, b_2) \in S, define s1+s2=((a1+a2)2015,(b1+b2)2015) and s1×s2=((a1a2+2b1b2)2015,(a1b2+a2b1)2015),s_1 + s_2 = ((a_1 + a_2)_{2015}, (b_1 + b_2)_{2015}) \\ \text { and } \\ s_1 \times s_2 = ((a_1a_2 + 2b_1b_2)_{2015}, (a_1b_2 + a_2b_1)_{2015}), where n2015n_{2015} denotes the remainder when an integer nn is divided by 2015.2015.
Compute the number of functions f:S→Sf : S \rightarrow S satisfying f(s1+s2)=f(s1)+f(s2) and f(s1×s2)=f(s1)×f(s2) f(s_1 + s_2) = f(s_1) + f(s_2) \text{ and } f(s_1 \times s_2) = f(s_1) \times f(s_2) for all s1,s2∈S.s_1, s_2 \in S.
Proposed by Yang Liu

2016-2017 Fall OMO Problem 26

Let ABCABC be a triangle with BC=9BC=9, CA=8CA=8, and AB=10AB=10. Let the incenter and incircle of ABCABC be II and γ\gamma, respectively, and let NN be the midpoint of major arc BCBC of the cirucmcircle of ABCABC. Line NINI meets the circumcircle of ABCABC a second time at PP. Let the line through II perpendicular to AIAI meet segments ABAB, ACAC, and APAP at C1C_1, B1B_1, and QQ, respectively. Let B2B_2 lie on segment CQCQ such that line B1B2B_1B_2 is tangent to γ\gamma, and let C2C_2 lie on segment BQBQ such that line C1C2C_1C_2 tangent to γ\gamma. The length of B2C2B_2C_2 can be expressed in the form mn\frac{m}{n} for relatively prime positive integers mm and nn. Determine 100m+n100m+n.
Proposed by Vincent Huang
25
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2015-2016 Spring OMO #25

Given a prime pp and positive integer kk, an integer nn with 0≤n<p0 \le n < p is called a (p,k)(p, k)-Hofstadterian residue if there exists an infinite sequence of integers n0,n1,n2,…n_0, n_1, n_2, \ldots such that n0≡nn_0 \equiv n and ni+1k≡ni(modp)n_{i + 1}^k \equiv n_i \pmod{p} for all integers i≥0i \ge 0. If f(p,k)f(p, k) is the number of (p,k)(p, k)-Hofstadterian residues, then compute ∑k=12016f(2017,k)\displaystyle \sum_{k = 1}^{2016} f(2017, k).
Proposed by Ashwin Sah

2016-2017 Fall OMO Problem 25

Let X1X2X3X_1X_2X_3 be a triangle with X1X2=4,X2X3=5,X3X1=7,X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7, and centroid GG. For all integers n≥3n \ge 3, define the set SnS_n to be the set of n2n^2 ordered pairs (i,j)(i,j) such that 1≤i≤n1\le i\le n and 1≤j≤n1\le j\le n. Then, for each integer n≥3n\ge 3, when given the points X1,X2,…,XnX_1, X_2, \ldots , X_{n}, randomly choose an element (i,j)∈Sn(i,j)\in S_n and define Xn+1X_{n+1} to be the midpoint of XiX_i and XjX_j. The value of
∑i=0∞(E[Xi+4G2](34)i) \sum_{i=0}^\infty \left(\mathbb{E}\left[X_{i+4}G^2\right]\left(\dfrac{3}{4}\right)^i\right)
can be expressed in the form p+qln⁡2+rln⁡3p + q \ln 2 + r \ln 3 for rational numbers p,q,rp, q, r. Let ∣p∣+∣q∣+∣r∣=mn|p| + |q| + |r| = \dfrac mn for relatively prime positive integers mm and nn. Compute 100m+n100m+n.
Note: E(x)\mathbb{E}(x) denotes the expected value of xx.
Proposed by Yang Liu
18
2

2015-2016 OMO Spring #18

Kevin is in kindergarten, so his teacher puts a 100×200100 \times 200 addition table on the board during class. The teacher first randomly generates distinct positive integers a1,a2,…,a100a_1, a_2, \dots, a_{100} in the range [1,2016][1, 2016] corresponding to the rows, and then she randomly generates distinct positive integers b1,b2,…,b200b_1, b_2, \dots, b_{200} in the range [1,2016][1, 2016] corresponding to the columns. She then fills in the addition table by writing the number ai+bja_i+b_j in the square (i,j)(i, j) for each 1≤i≤1001\le i\le 100, 1≤j≤2001\le j\le 200.
During recess, Kevin takes the addition table and draws it on the playground using chalk. Now he can play hopscotch on it! He wants to hop from (1,1)(1, 1) to (100,200)(100, 200). At each step, he can jump in one of 88 directions to a new square bordering the square he stands on a side or at a corner. Let MM be the minimum possible sum of the numbers on the squares he jumps on during his path to (100,200)(100, 200) (including both the starting and ending squares). The expected value of MM can be expressed in the form pq\frac{p}{q} for relatively prime positive integers p,qp, q. Find p+q.p + q.
Proposed by Yang Liu

2016-2017 Fall OMO Problem 18

Find the smallest positive integer kk such that there exist positive integers M,O>1M,O>1 satisfying (O⋅M⋅O)k=(O⋅M)⋅(N⋅O⋅M)⋅(N⋅O⋅M)⋅…⋅(N⋅O⋅M)⏟2016 (N⋅O⋅M)s, (O\cdot M\cdot O)^k=(O\cdot M)\cdot \underbrace{(N\cdot O\cdot M)\cdot (N\cdot O\cdot M)\cdot \ldots \cdot (N\cdot O\cdot M)}_{2016\ (N\cdot O\cdot M)\text{s}}, where N=OMN=O^M.
Proposed by James Lin and Yannick Yao
23
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2015-2016 Spring OMO #23

SS be the set of all 201722017^2 lattice points (x,y)(x,y) with x,y∈{0}∪{20,21,⋯ ,22015}x,y\in \{0\}\cup\{2^{0},2^{1},\cdots,2^{2015}\}. A subset X⊆SX\subseteq S is called BQ if it has the following properties:
(a) XX contains at least three points, no three of which are collinear. (b) One of the points in XX is (0,0)(0,0). (c) For any three distinct points A,B,C∈XA,B,C \in X, the orthocenter of △ABC\triangle ABC is in XX. (d) The convex hull of XX contains at least one horizontal line segment.
Determine the number of BQ subsets of SS.
Proposed by Vincent Huang

2016-2017 Fall OMO Problem 23

Let N\mathbb N denote the set of positive integers. Let f:N→Nf: \mathbb N \to \mathbb N be a function such that the following conditions hold:
(a) For any n∈Nn\in \mathbb N, we have f(n)∣n2016f(n) | n^{2016}. (b) For any a,b,c∈Na,b,c\in \mathbb N satisfying a2+b2=c2a^2+b^2=c^2, we have f(a)f(b)=f(c)f(a)f(b)=f(c).
Over all possible functions ff, determine the number of distinct values that can be achieved by f(2014)+f(2)−f(2016)f(2014)+f(2)-f(2016).
Proposed by Vincent Huang
22
2

2015-2016 Spring OMO #22

Let ABCABC be a triangle with AB=5AB=5, BC=7BC=7, CA=8CA=8, and circumcircle ω\omega. Let PP be a point inside ABCABC such that PA:PB:PC=2:3:6PA:PB:PC=2:3:6. Let rays AP→\overrightarrow{AP}, BP→\overrightarrow{BP}, and CP→\overrightarrow{CP} intersect ω\omega again at XX, YY, and ZZ, respectively. The area of XYZXYZ can be expressed in the form pqr\dfrac{p\sqrt q}{r} where pp and rr are relatively prime positive integers and qq is a positive integer not divisible by the square of any prime. What is p+q+rp+q+r?
Proposed by James Lin

2016-2017 Fall OMO Problem 22

Let ABCABC be a triangle with AB=3AB=3 and AC=4AC=4. It is given that there does not exist a point DD, different from AA and not lying on line BCBC, such that the Euler line of ABCABC coincides with the Euler line of DBCDBC. The square of the product of all possible lengths of BCBC can be expressed in the form m+npm+n\sqrt p, where mm, nn, and pp are positive integers and pp is not divisible by the square of any prime. Find 100m+10n+p100m+10n+p.
Note: For this problem, consider every line passing through the center of an equilateral triangle to be an Euler line of the equilateral triangle. Hence, if DD is chosen such that DBCDBC is an equilateral triangle and the Euler line of ABCABC passes through the center of DBCDBC, then consider the Euler line of ABCABC to coincide with "the" Euler line of DBCDBC.
Proposed by Michael Ren
20
2
15
2

2015-2016 Spring OMO #15

Let a,b,c,da,b,c,d be four real numbers such that a+b+c+d=20a+b+c+d=20 and ab+bc+cd+da=16ab+bc+cd+da=16. Find the maximum possible value of abc+bcd+cda+dababc+bcd+cda+dab.
Proposed by Yannick Yao

2016-2017 Fall OMO Problem 15

Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $1,$2,$5\$ 1, \$ 2,\$ 5, and $10\$ 10 bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $1,000,000\$1{,}000{,}000, and whoever makes the last move wins the $1,000,000\$1{,}000{,}000 in the pile (if the pile is worth more than $1,000,000\$1{,}000{,}000 after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
Proposed by Yannick Yao
17
2

2015-2016 Spring OMO #17

A set S⊆NS \subseteq \mathbb{N} satisfies the following conditions: (a) If x,y∈Sx, y \in S (not necessarily distinct), then x+y∈Sx + y \in S. (b) If xx is an integer and 2x∈S2x \in S, then x∈Sx \in S.
Find the number of pairs of integers (a,b)(a, b) with 1≤a,b≤501 \le a, b\le 50 such that if a,b∈Sa, b \in S then S=N.S = \mathbb{N}.
Proposed by Yang Liu

2016-2017 Fall OMO Problem 17

Let nn be a positive integer. SS is a set of points such that the points in SS are arranged in a regular 20162016-simplex grid, with an edge of the simplex having nn points in SS. (For example, the 22-dimensional analog would have n(n+1)2\dfrac{n(n+1)}{2} points arranged in an equilateral triangle grid). Each point in SS is labeled with a real number such that the following conditions hold:
(a) Not all the points in SS are labeled with 00.
(b) If ℓ\ell is a line that is parallel to an edge of the simplex and that passes through at least one point in SS, then the labels of all the points in SS that are on ℓ\ell add to 00.
(c) The labels of the points in SS are symmetric along any such line ℓ\ell.
Find the smallest positive integer nn such that this is possible.
Note: A regular 20162016-simplex has 20172017 vertices in 20162016-dimensional space such that the distances between every pair of vertices are equal.
Proposed by James Lin
19
2

2015-2016 Spring OMO #19

Let Z≥0\mathbb{Z}_{\ge 0} denote the set of nonnegative integers.
Define a function f:Z≥0→Zf:\mathbb{Z}_{\ge 0} \to\mathbb{Z} with f(0)=1f\left(0\right)=1 and f(n)=512⌊n/10⌋f(⌊n/10⌋) f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) for all n≥1n \ge 1. Determine the number of nonnegative integers nn such that the hexadecimal (base 1616) representation of f(n)f\left(n\right) contains no more than 25002500 digits.
Proposed by Tristan Shin

2016-2017 Fall OMO Problem 19

Let SS be the set of all polynomials Q(x,y,z)Q(x,y,z) with coefficients in {0,1}\{0,1\} such that there exists a homogeneous polynomial P(x,y,z)P(x,y,z) of degree 20162016 with integer coefficients and a polynomial R(x,y,z)R(x,y,z) with integer coefficients so that P(x,y,z)Q(x,y,z)=P(yz,zx,xy)+2R(x,y,z)P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z) and P(1,1,1)P(1,1,1) is odd. Determine the size of SS.
Note: A homogeneous polynomial of degree dd consists solely of terms of degree dd.
Proposed by Vincent Huang
14
2

2015-2016 Spring OMO #14

Let ABCABC be a triangle with BC=20BC=20 and CA=16CA=16, and let II be its incenter. If the altitude from AA to BCBC, the perpendicular bisector of ACAC, and the line through II perpendicular to ABAB intersect at a common point, then the length ABAB can be written as m+nm+\sqrt{n} for positive integers mm and nn. What is 100m+n100m+n?
Proposed by Tristan Shin

2016-2017 Fall OMO Problem 14

In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of 9090, Michael M had an average test score of 9191, and Michael R had an average test score of 9292.
(b) Michael K took more tests than Michael M, who in turn took more tests than Michael R.
(c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests)
What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?
Proposed by James Lin
12
2
16
2

2015-2016 Spring OMO #16

Jay is given a permutation {p1,p2,…,p8}\{p_1, p_2,\ldots, p_8\} of {1,2,…,8}\{1, 2,\ldots, 8\}. He may take two dividers and split the permutation into three non-empty sets, and he concatenates each set into a single integer. In other words, if Jay chooses a,ba,b with 1≤a<b<81\le a< b< 8, he will get the three integers p1p2…pa‾\overline{p_1p_2\ldots p_a}, pa+1pa+2…pb‾\overline{p_{a+1}p_{a+2}\ldots p_{b}}, and pb+1pb+2…p8‾\overline{p_{b+1}p_{b+2}\ldots p_8}. Jay then sums the three integers into a sum N=p1p2…pa‾+pa+1pa+2…pb‾+pb+1pb+2…p8‾N=\overline{p_1p_2\ldots p_a}+\overline{p_{a+1}p_{a+2}\ldots p_b}+\overline{p_{b+1}p_{b+2}\ldots p_8}. Find the smallest positive integer MM such that no matter what permutation Jay is given, he may choose two dividers such that N≤MN\le M.
Proposed by James Lin

2016-2017 Fall OMO Problem 16

For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day 00. On each day afterwards, Emilia attempts to use her magic to turn each sapling into a perfectly-shaped apple tree, and for each sapling she succeeds in turning it into a perfectly-shaped apple tree that day with a probability of 12\frac{1}{2}. (Once a sapling is turned into a perfectly-shaped apple tree, it will stay a perfectly-shaped apple tree.) The expected number of days it will take Emilia to obtain six perfectly-shaped apple trees is mn\frac{m}{n} for relatively prime positive integers mm and nn. Find 100m+n100m+n.
Proposed by Yannick Yao
11
2
13
2

2015-2016 Spring OMO #13

For a positive integer nn, let f(n)f(n) be the integer formed by reversing the digits of nn (and removing any leading zeroes). For example f(14172)=27141f(14172)=27141. Define a sequence of numbers {an}n≥0\{a_n\}_{n\ge 0} by a0=1a_0=1 and for all i≥0i\ge 0, ai+1=11aia_{i+1}=11a_i or ai+1=f(ai)a_{i+1}=f(a_i) . How many possible values are there for a8a_8?
Proposed by James Lin

2016-2017 Fall OMO Problem 13

Let A1B1C1A_1B_1C_1 be a triangle with A1B1=16,B1C1=14,A_1B_1 = 16, B_1C_1 = 14, and C1A1=10C_1A_1 = 10. Given a positive integer ii and a triangle AiBiCiA_iB_iC_i with circumcenter OiO_i, define triangle Ai+1Bi+1Ci+1A_{i+1}B_{i+1}C_{i+1} in the following way:
(a) Ai+1A_{i+1} is on side BiCiB_iC_i such that CiAi+1=2BiAi+1C_iA_{i+1}=2B_iA_{i+1}. (b) Bi+1≠CiB_{i+1}\neq C_i is the intersection of line AiCiA_iC_i with the circumcircle of OiAi+1CiO_iA_{i+1}C_i. (c) Ci+1≠BiC_{i+1}\neq B_i is the intersection of line AiBiA_iB_i with the circumcircle of OiAi+1BiO_iA_{i+1}B_i.
Find (∑i=1∞[AiBiCi])2. \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2.
Note: [K][K] denotes the area of KK.
Proposed by Yang Liu
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2015-2016 Spring OMO #8

Let ABCDEFABCDEF be a regular hexagon of side length 33. Let X,Y,X, Y, and ZZ be points on segments AB,CD,AB, CD, and EFEF such that AX=CY=EZ=1AX=CY=EZ=1. The area of triangle XYZXYZ can be expressed in the form abc\dfrac{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. Find 100a+10b+c100a+10b+c.
Proposed by James Lin

2016-2017 Fall OMO Problem 8

For a positive integer nn, define the nnth triangular number TnT_n to be n(n+1)2\frac{n(n+1)}{2}, and define the nnth square number SnS_n to be n2n^2. Find the value of S62+T63S61+T62⋯S2+T3S1+T2.\sqrt{S_{62}+T_{63}\sqrt{S_{61}+T_{62}\sqrt{\cdots \sqrt{S_2+T_3\sqrt{S_1+T_2}}}}}.
Proposed by Yannick Yao
10
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2015-2016 Spring OMO #10

Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
Proposed by Yannick Yao

2016-2017 Fall OMO Problem 10

Let a1<a2<a3<a4a_1 < a_2 < a_3 < a_4 be positive integers such that the following conditions hold:
-gcd⁡(ai,aj)>1\gcd(a_i,a_j)>1 holds for all integers 1≤i<j≤41\le i < j\le 4. -gcd⁡(ai,aj,ak)=1\gcd(a_i,a_j,a_k)=1 holds for all integers 1≤i<j<k≤41\le i < j < k\le 4.
Find the smallest possible value of a4a_4.
Proposed by James Lin
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2015-2016 Spring OMO #7

Compute the number of ordered quadruples of positive integers (a,b,c,d)(a,b,c,d) such that a!⋅b!⋅c!⋅d!=24!. a!\cdot b!\cdot c!\cdot d!=24!.
Proposed by Michael Kural

2016-2017 Fall OMO Problem 7

The 20162016 players in the Gensokyo Tennis Club are playing Up and Down the River. The players first randomly form 10081008 pairs, and each pair is assigned to a tennis court (The courts are numbered from 11 to 10081008). Every day, the two players on the same court play a match against each other to determine a winner and a loser. For 2≤i≤10082\le i\le 1008, the winner on court ii will move to court i−1i-1 the next day (and the winner on court 11 does not move). Likewise, for 1≤j≤10071\le j\le 1007, the loser on court jj will move to court j+1j+1 the next day (and the loser on court 10081008 does not move). On Day 11, Reimu is playing on court 123123 and Marisa is playing on court 876876. Find the smallest positive integer value of nn for which it is possible that Reimu and Marisa play one another on Day nn.
Proposed by Yannick Yao
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