Olympiad problems

2000 Korea Junior Math Olympiad, 1

For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.

1999 Korea Junior Math Olympiad, 8

Tags: set theory , KJMO , combinatorics

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

2019 Korea Junior Math Olympiad., 6

Tags: algebra , functional equation , KJMO , function

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfies the followings. (Note that $\mathbb{R}$ stands for the set of all real numbers) (1) For each real numbers $x$, $y$, the equality $f(x+f(x)+xy) = 2f(x)+xf(y)$ holds. (2) For every real number $z$, there exists $x$ such that $f(x) = z$.

2002 Korea Junior Math Olympiad, 7

Tags: geometry , incenter , KJMO

$I$ is the incenter of $ABC$. $D$ is the intersection of $AI$ and the circumcircle of $ABC$, not $A$. And $P$ is a midpoint of $BI$. If $CI=2AI$, show that $AB=PD$.

2019 Korea Junior Math Olympiad., 3

Tags: number theory , prime numbers , KJMO

Find all pairs of prime numbers $p,\,q(p\le q)$ satisfying the following condition: There exists a natural number $n$ such that $2^{n}+3^{n}+\cdots+(2pq-1)^{n}$ is a multiple of $2pq$.

2017 Korea Junior Math Olympiad, 5

Tags: number theory , KJMO

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

2017 Korea Junior Math Olympiad, 1

Tags: number theory , KJMO

Find all positive integer $n$ and nonnegative integer $a_1,a_2,\dots,a_n$ satisfying: $i$ divides exactly $a_i$ numbers among $a_1,a_2,\dots,a_n$, for each $i=1,2,\dots,n$. ($0$ is divisible by all integers.)

2017 Korea Junior Math Olympiad, 7

Tags: algebra , KJMO

Prove that there is no function $f:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}$ satisfying: $f(x+y^2)\ge f(x)+y$ for all two nonnegative real numbers $x,y$.

1998 Korea Junior Math Olympiad, 7

Tags: geometry , circumcircle , angle bisector , KJMO

$O$ is a circumcircle of non-isosceles triangle $ABC$ and the angle bisector of $A$ meets $BC$ at $D$. If the line perpendicular to $BC$ passing through $D$ meets $AO$ at $E$, show that $ADE$ is an isosceles triangle.

2017 Korea Junior Math Olympiad, 3

Tags: algebra , number theory , KJMO

Find all $n>1$ and integers $a_1,a_2,\dots,a_n$ satisfying the following three conditions: (i) $2<a_1\le a_2\le \cdots\le a_n$ (ii) $a_1,a_2,\dots,a_n$ are divisors of $15^{25}+1$. (iii) $2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)$

2000 Korea Junior Math Olympiad, 2

Tags: combinatorics , KJMO

Along consecutive seven days, from Sunday to Saturday, let us call the days belonging to the same month a MB. For example, if the last day of a month is Sunday, the last MB of that month consists of the last day of that month. If a year is from January first to December $31$, find the maximum and minimum values of MB in one year.

1999 Korea Junior Math Olympiad, 5

Tags: geometry , circumcircle , KJMO

$O$ is a circumcircle of $ABC$ and $CO$ meets $AB$ at $P$, and $BO$ meets $AC$ at $Q$. Show that $BP=PQ=QC$ if and only if $\angle A=60^{\circ}$.

2019 Korea Junior Math Olympiad., 1

Tags: algebraic geometry , combinatorical geometry , combinatorics , KJMO , Coloring

Each integer coordinates are colored with one color and at least 5 colors are used to color every integer coordinates. Two integer coordinates $(x, y)$ and $(z, w)$ are colored in the same color if $x-z$ and $y-w$ are both multiples of 3. Prove that there exists a line that passes through exactly three points when five points with different colors are chosen randomly.

1998 Korea Junior Math Olympiad, 2

Tags: graph theory , combinatorics , KJMO

There are $6$ computers(power off) and $3$ printers. Between a printer and a computer, they are connected with a wire or not. Printer can be only activated if and only if at least one of the connected computer's power is on. Your goal is to connect wires in such a way that, no matter how you choose three computers to turn on among the six, you can activate all $3$ printers. What is the minimum number of wires required to make this possible?

2017 Korea Junior Math Olympiad, 4

Tags: inequalities , KJMO

4. Let $a \geq b \geq c \geq d>0$. Show that \[ \frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}. \] Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad

1999 Korea Junior Math Olympiad, 7

Tags: points , geometry , combinatorics , KJMO

$A_0B, A_0C$ rays that satisfy $\angle BA_0C=14^{\circ}$. You are to place points $A_1, A_2, ...$ by the following rules. [b]Rules[/b] (1) On the first move, place $A_1$ on any point on $A_0B$(except $A_0$). (2) On the $n>1$th move, place $A_n$ on $A_0B$ iff $A_{n-1}$ is on $A_0C$, and place $A_n$ on $A_0C$ iff $A_{n-1}$ is one $A_0B$. $A_n$ must be place on the point that satisfies $A_{n-2}A_n{n-1}=A_{n-1}A_n$. All the points must be placed in different locations. What is the maximum number of points that can be placed?

2000 Korea Junior Math Olympiad, 7

Tags: geometry , computational geometry , KJMO

$ABC$ is a triangle that $2\angle B < \angle A <90^{\circ}$, and $P$ is a point on $AB$ satisfying $\angle A=2\angle APC$. If $BC=a$, $AC=b$, $BP=1$, express $AP$ as a function of $a, b$.

2001 Korea Junior Math Olympiad, 5

Tags: KJMO , Sets , set theory , combinatorics

$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$. [b]Condition[/b] (1) $1 \in A$ (2) If $x \in A$, then $x^2 \in A$. (3) If $(x-3)^2 \in A$, then $x \in A$.

1999 Korea Junior Math Olympiad, 6

Tags: KJMO , Divisors , number theory

For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.

1999 Korea Junior Math Olympiad, 1

Tags: geometry , convex quadrilateral , inscribed quadrilateral , KJMO

There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.

2019 Korea Junior Math Olympiad., 7

Tags: Plane Geometry , geometry , KJMO , circumcircle , Angle Chasing

Let $O$ be the circumcenter of an acute triangle $ABC$. Let $D$ be the intersection of the bisector of the angle $A$ and $BC$. Suppose that $\angle ODC = 2 \angle DAO$. The circumcircle of $ABD$ meets the line segment $OA$ and the line $OD$ at $E (\neq A,O)$, and $F(\neq D)$, respectively. Let $X$ be the intersection of the line $DE$ and the line segment $AC$. Let $Y$ be the intersection of the bisector of the angle $BAF$ and the segment $BE$. Prove that $\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}$.

2001 Korea Junior Math Olympiad, 6

Tags: algebra , KJMO , Inequality

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

1998 Korea Junior Math Olympiad, 6

Tags: inequalities , KJMO

For positive reals $a \geq b \geq c \geq 0$ prove the following inequality: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}$$

2000 Korea Junior Math Olympiad, 8

Tags: counting , derangement , combinatorics , KJMO

$n$ men and one woman are in the meeting room with $n+1$ chairs, each of them having their own seat. Show that the following two number of cases are equal. (1) Number of cases to choose one man to get out of the room, and make the left $n-1$ men to sit to each other's chair. (2) Number of cases to make $n+1$ people to sit to each other's chair.

2009 Korea Junior Math Olympiad, 4

Tags: combinatorics , KJMO

There are $n$ clubs composed of $4$ students out of all $9$ students. For two arbitrary clubs, there are no more than $2$ students who are a member of both clubs. Prove that $n\le 18$. Translator’s Note. We can prove $n\le 12$, and we can prove that the bound is tight. (Credits to rkm0959 for translation and document)