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Contests
National and Regional Contests
Taiwan Contests
IMOC Shortlist
2017-IMOC
2017-IMOC
Part of
IMOC Shortlist
Subcontests
(30)
C6
1
Hide problems
Polygon with rational lengths
Consider a convex polygon in a plane such that the length of all edges and diagonals are rational. After connecting all diagonals, prove that any length of a segment is rational.
N8
1
Hide problems
x^2+y^2=n mod p, find n,p if x,y exist
Find all pairs
(
p
,
n
)
(p,n)
(
p
,
n
)
of integers so that
p
p
p
is a prime and there exists
x
,
y
≢
0
(
m
o
d
p
)
x,y\not\equiv0\pmod p
x
,
y
≡
0
(
mod
p
)
with
x
2
+
y
2
≡
n
(
m
o
d
p
)
.
x^2+y^2\equiv n\pmod p.
x
2
+
y
2
≡
n
(
mod
p
)
.
N7
1
Hide problems
chicken mcnugget theorem
For fixed coprime positive integers
a
,
b
a,b
a
,
b
, define
n
n
n
to be bad if it is not of the form
a
x
+
b
y
,
x
,
y
∈
N
∗
ax+by,\enspace x,y\in\mathbb N^*
a
x
+
b
y
,
x
,
y
∈
N
∗
Prove that there are finitely many bad positive integers. Also, find the sum of squares of them.
N6
1
Hide problems
mouse moving on plane returns to origin
A mouse walks on a plane. At time
i
i
i
, it could do nothing or turn right, then it moves
p
i
p_i
p
i
meters forward, where
p
i
p_i
p
i
is the
i
i
i
-th prime. Is it possible that the mouse moves back to the starting point?
N5
1
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f(x)+f(y)|x^2-y^2 over N
Find all functions
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
such that
f
(
x
)
+
f
(
y
)
∣
x
2
−
y
2
f(x)+f(y)\mid x^2-y^2
f
(
x
)
+
f
(
y
)
∣
x
2
−
y
2
holds for all
x
,
y
∈
N
x,y\in\mathbb N
x
,
y
∈
N
.
N4
1
Hide problems
n^(n-1)-1 square-free
Find all integers
n
n
n
such that
n
n
−
1
−
1
n^{n-1}-1
n
n
−
1
−
1
is square-free.
N3
1
Hide problems
f(mn)=f(m)f(n) and f(m+n)=min(f(m),f(n)) if f(m)≠f(n) over N->N0
Find all functions
f
:
N
→
N
0
f:\mathbb N\to\mathbb N_0
f
:
N
→
N
0
such that for all
m
,
n
∈
N
m,n\in\mathbb N
m
,
n
∈
N
, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}
N2
1
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game, filling in digits and not having a perfect power
On the blackboard, there are
K
K
K
blanks. Alice decides
N
N
N
values of blanks
(
0
−
9
)
(0-9)
(
0
−
9
)
and then Bob determines the remaining digits. Find the largest possible integer
N
N
N
such that Bob can guarantee to make the final number isn't a power of an integer.
N1
1
Hide problems
f:N->N if prod(d|n)f(d)=2^n
If
f
:
N
→
R
f:\mathbb N\to\mathbb R
f
:
N
→
R
is a function such that
∏
d
∣
n
f
(
d
)
=
2
n
\prod_{d\mid n}f(d)=2^n
d
∣
n
∏
f
(
d
)
=
2
n
holds for all
n
∈
N
n\in\mathbb N
n
∈
N
, show that
f
f
f
sends
N
\mathbb N
N
to
N
\mathbb N
N
.
C5
1
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set closed under orthocenter
We say a finite set
S
S
S
of points with
∣
S
∣
≥
3
|S|\ge3
∣
S
∣
≥
3
is good if for any three distinct elements of
S
S
S
, they are non-collinear and the orthocenter of them is also in
S
S
S
. Find all good sets.
C4
1
Hide problems
no neighbors with consecutive heights
There are
3
N
+
1
3N+1
3
N
+
1
students with different heights line up for asking questions. Prove that the teacher can drive
2
N
2N
2
N
students away such that the remain students satisfies: No one has neighbors whose heights are consecutive.
C3
1
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game in matrix, det nonzero
Alice and Bob play the following game: Initially, there is a
2016
×
2016
2016\times2016
2016
×
2016
"empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?
C2
1
Hide problems
4 pudding moving on a square form square with side 2
On a large chessboard, there are
4
4
4
puddings that form a square with size
1
1
1
. A pudding
A
A
A
could jump over a pudding
B
B
B
, or equivalently,
A
A
A
moves to the symmetric point with respect to
B
B
B
. Is it possible that after finite times of jumping, the puddings form a square with size
2
2
2
?
A6
1
Hide problems
sum sqrt(a+3b+2/c) inequality with a+b+c=3
Show that for all positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
with
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
,
∑
cyc
a
+
3
b
+
2
c
≥
3
6
.
\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.
cyc
∑
a
+
3
b
+
c
2
≥
3
6
.
A5
1
Hide problems
FE over Z
Find all functions
f
:
Z
→
Z
f:\mathbb Z\to\mathbb Z
f
:
Z
→
Z
such that
f
(
m
f
(
n
+
1
)
)
=
f
(
m
+
1
)
f
(
n
)
+
f
(
f
(
n
)
)
+
1
f(mf(n+1))=f(m+1)f(n)+f(f(n))+1
f
(
m
f
(
n
+
1
))
=
f
(
m
+
1
)
f
(
n
)
+
f
(
f
(
n
))
+
1
for all integer pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
.
A4
1
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f(x+f(y))<=xf(y)+x over R, prove no solutions
Show that for all non-constant functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
, there are two real numbers
x
,
y
x,y
x
,
y
such that
f
(
x
+
f
(
y
)
)
>
x
f
(
y
)
+
x
.
f(x+f(y))>xf(y)+x.
f
(
x
+
f
(
y
))
>
x
f
(
y
)
+
x
.
A3
1
Hide problems
x^3+y+z=1 and cyclic, solve over R
Solve the following system of equations:
{
x
3
+
y
+
z
=
1
x
+
y
3
+
z
=
1
x
+
y
+
z
3
=
1
\begin{cases} x^3+y+z=1\\ x+y^3+z=1\\ x+y+z^3=1\end{cases}
⎩
⎨
⎧
x
3
+
y
+
z
=
1
x
+
y
3
+
z
=
1
x
+
y
+
z
3
=
1
A2
1
Hide problems
x+f(y)|f(y+f(x)) and f(x)-2017|x-2017
Find all functions
f
:
N
→
N
f:\mathbb N\to\mathbb N
f
:
N
→
N
such that \begin{align*} x+f(y)&\mid f(y+f(x))\\ f(x)-2017&\mid x-2017\end{align*}
A1
1
Hide problems
(a^n+1)(b^n+1)>=4 if a+b=2
Prove that for all
a
,
b
>
0
a,b>0
a
,
b
>
0
with
a
+
b
=
2
a+b=2
a
+
b
=
2
, we have
(
a
n
+
1
)
(
b
n
+
1
)
≥
4
\left(a^n+1\right)\left(b^n+1\right)\ge4
(
a
n
+
1
)
(
b
n
+
1
)
≥
4
for all
n
∈
N
≥
2
n\in\mathbb N_{\ge2}
n
∈
N
≥
2
.
C1
1
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Combinatorics involving an operation
On a blackboard , the 2016 numbers
1
2016
,
2
2016
,
.
.
.
2016
2016
\frac{1}{2016} , \frac{2}{2016} ,... \frac{2016}{2016}
2016
1
,
2016
2
,
...
2016
2016
are written. One can perfurm the following operation : Choose any numbers in the blackboard, say
a
a
a
and
b
b
b
and replace them by
2
a
b
−
a
−
b
+
1
2ab-a-b+1
2
ab
−
a
−
b
+
1
. After doing 2015 operation , there will only be one number
t
t
t
Onthe blackboard . Find all possible values of
t
t
t
.
C7
1
Hide problems
12 monsters.
There are
12
12
12
monsters in a plane. Each monster is capable of spraying fire in a
30
30
30
-degree cone. Prove that monsters can destroy the plane.
A7
1
Hide problems
IMOC 2017 A7
Determine all non negative integers
k
k
k
such that there is a function
f
:
N
→
N
f : \mathbb{N} \to \mathbb{N}
f
:
N
→
N
that satisfies
f
n
(
n
)
=
n
+
k
f^n(n) = n + k
f
n
(
n
)
=
n
+
k
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
G6
1
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IMOC 2017 G6 ((x + y)^2+(y + z)^2 +(z + x)^2)/(x + y + z)<= a + b + c
A point
P
P
P
lies inside
△
A
B
C
\vartriangle ABC
△
A
BC
such that the values of areas of
△
P
A
B
,
△
P
B
C
,
△
P
C
A
\vartriangle PAB, \vartriangle PBC, \vartriangle PCA
△
P
A
B
,
△
PBC
,
△
PC
A
can form a triangle. Let
B
C
=
a
,
C
A
=
b
,
A
B
=
c
,
P
A
=
x
,
P
B
=
y
,
P
C
=
z
BC = a,CA = b,AB = c, PA = x,PB = y, PC = z
BC
=
a
,
C
A
=
b
,
A
B
=
c
,
P
A
=
x
,
PB
=
y
,
PC
=
z
, prove that
(
x
+
y
)
2
+
(
y
+
z
)
2
+
(
z
+
x
)
2
x
+
y
+
z
≤
a
+
b
+
c
\frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{x + y + z} \le a + b + c
x
+
y
+
z
(
x
+
y
)
2
+
(
y
+
z
)
2
+
(
z
+
x
)
2
≤
a
+
b
+
c
G7
1
Hide problems
IMOC 2017 G7 (concyclic wanted, <ABD =< DCA amd circumcircle related)
Given
△
A
B
C
\vartriangle ABC
△
A
BC
with circumcenter
O
O
O
. Let
D
D
D
be a point satisfying
∠
A
B
D
=
∠
D
C
A
\angle ABD = \angle DCA
∠
A
B
D
=
∠
D
C
A
and
M
M
M
be the midpoint of
A
D
AD
A
D
. Suppose that
B
M
,
C
M
BM,CM
BM
,
CM
intersect circle
(
O
)
(O)
(
O
)
at another points
E
,
F
E, F
E
,
F
, respectively. Let
P
P
P
be a point on
E
F
EF
EF
so that
A
P
AP
A
P
is tangent to circle
(
O
)
(O)
(
O
)
. Prove that
A
,
P
,
M
,
O
A, P,M,O
A
,
P
,
M
,
O
are concyclic. https://2.bp.blogspot.com/-gSgUG6oywAU/XnSKTnH1yqI/AAAAAAAALdw/3NuPFuouCUMO_6KbydE-KIt6gCJ4OgWdACK4BGAYYCw/s320/imoc2017%2Bg7.png
G5
1
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IMOC 2017 G5 (<A=120 => E, F, Y,Z are concyclic, incenter related)
We have
△
A
B
C
\vartriangle ABC
△
A
BC
with
I
I
I
as its incenter. Let
D
D
D
be the intersection of
A
I
AI
A
I
and
B
C
BC
BC
and define
E
,
F
E, F
E
,
F
in a similar way. Furthermore, let
Y
=
C
I
∩
D
E
,
Z
=
B
I
∩
D
F
Y = CI \cap DE, Z = BI \cap DF
Y
=
C
I
∩
D
E
,
Z
=
B
I
∩
D
F
. Prove that if
∠
B
A
C
=
12
0
o
\angle BAC = 120^o
∠
B
A
C
=
12
0
o
, then
E
,
F
,
Y
,
Z
E, F, Y,Z
E
,
F
,
Y
,
Z
are concyclic. https://1.bp.blogspot.com/-5IFojUbPE3o/XnSKTlTISqI/AAAAAAAALd0/0OwKMl02KJgqPs-SDOlujdcWXM0cWJiegCK4BGAYYCw/s1600/imoc2017%2Bg5.png
G4
1
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IMOC 2017 G4 collinearity, orthocenter and circumcircle related
Given an acute
△
A
B
C
\vartriangle ABC
△
A
BC
with orthocenter
H
H
H
. Let
M
a
M_a
M
a
be the midpoint of
B
C
.
M
a
H
BC. M_aH
BC
.
M
a
H
intersects the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
at
X
a
X_a
X
a
and
A
X
a
AX_a
A
X
a
intersects
B
C
BC
BC
at
Y
a
Y_a
Y
a
. Define
Y
b
,
Y
c
Y_b, Y_c
Y
b
,
Y
c
in a similar way. Prove that
Y
a
,
Y
b
,
Y
c
Y_a, Y_b,Y_c
Y
a
,
Y
b
,
Y
c
are collinear. https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png
G3
1
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IMOC 2017 G3 (AB CX DX)^2+(CD AX BX)^2=(AD BX CX)^2 +(BC AX DX)^2
Let
A
B
C
D
ABCD
A
BC
D
be a circumscribed quadrilateral with center
O
O
O
. Assume the incenters of
△
A
O
C
,
△
B
O
D
\vartriangle AOC, \vartriangle BOD
△
A
OC
,
△
BO
D
are
I
1
,
I
2
I_1, I_2
I
1
,
I
2
, respectively. If circumcircles of
△
A
I
1
C
\vartriangle AI_1C
△
A
I
1
C
and
△
B
I
2
D
\vartriangle BI_2D
△
B
I
2
D
intersect at
X
X
X
, prove the following identity:
(
A
B
⋅
C
X
⋅
D
X
)
2
+
(
C
D
⋅
A
X
⋅
B
X
)
2
=
(
A
D
⋅
B
X
⋅
C
X
)
2
+
(
B
C
⋅
A
X
⋅
D
X
)
2
(AB \cdot CX \cdot DX)^2 + (CD\cdot AX \cdot BX)^2 = (AD\cdot BX \cdot CX)^2 + (BC \cdot AX \cdot DX)^2
(
A
B
⋅
CX
⋅
D
X
)
2
+
(
C
D
⋅
A
X
⋅
BX
)
2
=
(
A
D
⋅
BX
⋅
CX
)
2
+
(
BC
⋅
A
X
⋅
D
X
)
2
G2
1
Hide problems
IMOC 2017 G2 , (ABC) <= (DEF) . perpendiculars related
Given two acute triangles
△
A
B
C
,
△
D
E
F
\vartriangle ABC, \vartriangle DEF
△
A
BC
,
△
D
EF
. If
A
B
≥
D
E
,
B
C
≥
E
F
AB \ge DE, BC \ge EF
A
B
≥
D
E
,
BC
≥
EF
and
C
A
≥
F
D
CA \ge FD
C
A
≥
F
D
, show that the area of
△
A
B
C
\vartriangle ABC
△
A
BC
is not less than the area of
△
D
E
F
\vartriangle DEF
△
D
EF
G1
1
Hide problems
IMOC 2017 G1 (MT // angle bisector, BP = CQ)
Given
△
A
B
C
\vartriangle ABC
△
A
BC
. Choose two points
P
,
Q
P, Q
P
,
Q
on
A
B
,
A
C
AB, AC
A
B
,
A
C
such that
B
P
=
C
Q
BP = CQ
BP
=
CQ
. Let
M
,
T
M, T
M
,
T
be the midpoints of
B
C
,
P
Q
BC, PQ
BC
,
PQ
. Show that
M
T
MT
MT
is parallel to the angle bisevtor of
∠
B
A
C
\angle BAC
∠
B
A
C
http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7bMBpVRFhbXeYPACK4BGAYYCw/s1600/imoc2017%2Bg1.png
N9
1
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Number Theory 2
Let
a
a
a
be a natural number,
a
>
3
a>3
a
>
3
. Prove there is an infinity of numbers n, for which
a
+
n
∣
a
n
+
1
a+n|a^{n}+1
a
+
n
∣
a
n
+
1