Found problems: 37
1998 Korea Junior Math Olympiad, 4
$n$ lines are on the same plane, no two of them parallel and no three of them collinear(so the plane must be partitioned into some parts). How many parts is the plane partitioned into? Consider only the partitions with finitely large area.
2020 Korea Junior Math Olympiad, 3
The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too.
For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$
Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$
2019 Korea Junior Math Olympiad., 2
In an acute triangle $ABC$, point $D$ is on the segment $AC$ such that $\overline{AD}=\overline{BC}$ and $\overline{AC}^2-\overline{AD}^2=\overline{AC}\cdot\overline{AD}$. The line that is parallel to the bisector of $\angle{ACB}$ and passes the point $D$ meets the segment $AB$ at point $E$. Prove, if $\overline{AE}=\overline{CD}$, $\angle{ADB}=3\angle{BAC}$.
2000 Korea Junior Math Olympiad, 4
Show that for real variables $1 \leq a, b \leq 2$ the following inequality holds.
$$2(a+b)^2 \leq 9ab $$
2003 Korea Junior Math Olympiad, 5
Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.
2019 Korea Junior Math Olympiad., 5
For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.
2000 Korea Junior Math Olympiad, 3
Acute triangle $ABC$ is inscribed in circle $O$. $P$ is the foot of altitude from $A$ to $BC$, and $D$ is the intersection of $O$ and line $AP$. $M, N$ are midpoint of $AB, AC$ respectively. $MP$ and $CD$ intersects at $Q$, and $NP$ and $BD$ intersects at $R$. Show that $AD, BQ, CR$ meet at one point if and only if $AB=AC$.
2017 Korea Junior Math Olympiad, 6
Let triangle $ABC$ be an acute scalene triangle, and denote $D,E,F$ by the midpoints of $BC,CA,AB$, respectively. Let the circumcircle of $DEF$ be $O_1$, and its center be $N$. Let the circumcircle of $BCN$ be $O_2$. $O_1$ and $O_2$ meet at two points $P, Q$. $O_2$ meets $AB$ at point $K(\neq B)$ and meets $AC$ at point $L(\neq C)$. Show that the three lines $EF,PQ,KL$ are concurrent.
2019 Korea Junior Math Olympiad., 8
There are two airlines A and B and finitely many airports. For each pair of airports, there is exactly one airline among A and B whose flights operates in both directions. Each airline plans to develop world travel packages which pass each airport exactly once using only its flights. Let $a$ and $b$ be the number of possible packages which belongs to A and B respectively. Prove that $a-b$ is a multiple of $4$.
The official statement of the problem has been changed. The above is the form which appeared during the contest. Now the condition 'the number of airports is no less than 4'is attached. Cite the following link.
[url]https://artofproblemsolving.com/community/c6h2923697p26140823[/url]
2020 Korea Junior Math Olympiad, 6
for a positive integer $n$, there are positive integers $a_1, a_2, ... a_n$ that satisfy these two.
(1) $a_1=1, a_n=2020$
(2) for all integer $i$, $i$satisfies $2\leq i\leq n, a_i-a_{i-1}=-2$ or $3$.
find the greatest $n$
1998 Korea Junior Math Olympiad, 1
Show that there exist no integer solutions $(x, y, z)$ to the equation
$$x^3+2y^3+4z^3=9$$
1998 Korea Junior Math Olympiad, 8
$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition.
[b]Condition[/b]
The sum of the $1998$ elements is again an element of $T$.