This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 20

2002 HKIMO Preliminary Selection Contest, 5
Tags: HKIMO , combinatorics
A positive integer is said to be a “palindrome” if it reads the same from left to right as from right to left. For example 2002 is a palindrome. Find the sum of all 4-digit palindromes.

2002 HKIMO Preliminary Selection Contest, 6
Tags: HKIMO , combinatorics
Points $A$ and $B$ lie on a plane. A straight line passing through $A$ will divide the plane into 2 regions. A further straight line through $B$ will altogether divide the plane into 4 regions, and so on. If 1002 and 1000 straight lines are drawn passing through $A$ and $B$ respectively, what is the maximum number of regions formed?

2002 HKIMO Preliminary Selection Contest, 1
Tags: algebra , HKIMO
Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$

2002 HKIMO Preliminary Selection Contest, 14
Tags: HKIMO , geometry
In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$

2002 HKIMO Preliminary Selection Contest, 2
Tags: HKIMO , combinatorics
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.

2002 HKIMO Preliminary Selection Contest, 19
Tags: HKIMO , combinatorics
There are 5 points on the plane. The following steps are used to construct lines. In step 1, connect all possible pairs of the points; it is found that no two lines are parallel, nor any two lines perpendicular to each other, also no three lines are concurrent. In step 2, perpendicular lines are drawn from each of the five given points to straight lines connecting any two of the other four points. What is the maximum number of points of intersection formed by the lines drawn in step 2, including the 5 given points?

2002 HKIMO Preliminary Selection Contest, 18
Tags: HKIMO , combinatorics
Let $A_1A_2\cdots A_{2002}$ be a regular 2002 sided polygon. Each vertex $A_i$ is associated with a positive integer $a_i$ such that the following condition is satisfied: If $j_1,j_2,\cdots, j_k$ are positive integers such that $k<500$ and $A_{j_1}, A_{j_2}, \cdots A_{j_k}$ is a regular $k$ sided polygon, then the values of $a_{j_1},A_{j_2}, \cdots A_{j_k}$ are all different. Find the smallest possible value of $a_1+a_2+\cdots a_{2002}$

2002 HKIMO Preliminary Selection Contest, 4
Tags: HKIMO , combinatorics
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

2002 HKIMO Preliminary Selection Contest, 10
Tags: HKIMO , number theory
How many positive integers less than 500 have exactly 15 positive integer factors?

2002 HKIMO Preliminary Selection Contest, 17
Tags: HKIMO , algebra
Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers

2002 HKIMO Preliminary Selection Contest, 11
Tags: HKIMO , number theory
Find the 2002nd positive integer that is not the difference of two square integers

2002 HKIMO Preliminary Selection Contest, 9
Tags: HKIMO , algebra
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$

2002 HKIMO Preliminary Selection Contest, 16
Tags: HKIMO , combinatorics
Each face and each vertex of a regular tetrahedron is coloured red or blue. How many different ways of colouring are there? (Two tetrahedrons are said to have the same colouring if we can rotate them suitably so that corresponding faces and vertices are of the same colour.

2002 HKIMO Preliminary Selection Contest, 8
Tags: HKIMO , algebra
Given that $0.3010<\log 2<0.3011$ and $0.4771<\log 3<0.4772$. Find the leftmost digit of $12^{37}$

2002 HKIMO Preliminary Selection Contest, 7
Tags: HKIMO , geometry
In $\triangle ABC$, $X, Y$, are points on BC such that $BX=XY=YC$, $M , N$ are points on $AC$ such that $AM=MN=NC$. $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$, find the area of $SMNR$.

2002 HKIMO Preliminary Selection Contest, 12
Tags: HKIMO , geometry , trapezoid
In trapezium $ABCD$, $BC \perp AB$, $BC\perp CD$, and $AC\perp BD$. Given $AB=\sqrt{11}$ and $AD=\sqrt{1001}$. Find $BC$

2002 HKIMO Preliminary Selection Contest, 13
Tags: HKIMO , geometry
Let $ABCD$ be a square of side 5, $E$ a point on $BC$ such that $BE=3, EC= 2$. Let $P$ be a variable point on the diagonal $BD.$ Determine the length of $PB$ if $PE+PC$ is smallest.

2002 HKIMO Preliminary Selection Contest, 20
Tags: HKIMO , geometry
A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper.

2002 HKIMO Preliminary Selection Contest, 3
Tags: HKIMO , combinatorics
Find the sum of all integers from 1 to 1000 which contain at least one “7” in their digits.

2002 HKIMO Preliminary Selection Contest, 15
Tags: HKIMO , geometry
In $\triangle ABC$, $D,E,F$ are respectively the midpoints of $AB, BC, and CA$. Futhermore $AB=10$, $CD=9$, $CD\perp AE$. Find $BF$.