Olympiad problems

2006 ISI B.Math Entrance Exam, 3

Tags: induction , algebra

Find all roots of the equation :- $1-\frac{x}{1}+\frac{x(x-1)}{2!} - \cdots +(-1)^n\frac{x(x-1)(x-2)...(x-n+1)}{n!}=0$.

1982 IMO Shortlist, 12

Tags: geometry , circles , tangency , euclidean distance

Four distinct circles $C,C_1, C_2$, C3 and a line L are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1, C_2, C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be $1$, determine the distance between its center and $L.$

2023 Chile Junior Math Olympiad, 1

Tags: number theory , digit

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

1964 Poland - Second Round, 4

Tags: algebra , system of equations , system

Find the real numbers $ x, y, z $ satisfying the system of equations $$(z - x)(x - y) = a $$ $$(x - y)(y - z) = b$$ $$(y - z)(z - x) = c$$ where $ a, b, c $ are given real numbers.

III Soros Olympiad 1996 - 97 (Russia), 10.2

Tags: geometry

On a side of the triangle, take four points $K$, $P$, $H$ and $M$, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find $KH$ if $KP = a$, $KM =b$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.1

Tags: algebra , arithmetic progression

The equation $x^2 + bx + c = 0$ has two different roots $x_1$ and $x_2$. It is also known that the numbers $b$, $x_1$, $c$, $x_2$ in the indicated order form an arithmetic progression. Find the difference of this progression.

2000 Croatia National Olympiad, Problem 1

Tags: geometry

Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.

2003 South africa National Olympiad, 6

Tags: geometry

In $\Delta ABC$, the sum of the sides is $2s$ and the radius of the incircle is $r$. Three semicircles with diameters $AB$, $BC$ and $CA$ are drawn on the outside of $ABC$. A circle with radius $t$ touches all three semicircles. Prove that \[ \frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r. \]

2009 Today's Calculation Of Integral, 416

Tags: calculus , integration , trigonometry , calculus computations

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

1992 AIME Problems, 11

Tags: geometry , geometric transformation , reflection , rotation , modular arithmetic

Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\left(R^{(n-1)}(l)\right)$. Given that $l$ is the line $y=\frac{19}{92}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.

2007 Putnam, 1

Tags: symmetry , analytic geometry , graphing lines , slope , quadratic , geometry

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

2017 Estonia Team Selection Test, 1

Tags: number theory , power of 5

Do there exist two positive powers of $5$ such that the number obtained by writing one after the other is also a power of $5$?

2008 JBMO Shortlist, 3

Tags: sum of digits , number theory

Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

2013-2014 SDML (High School), 1

Tags: factorial

What is the smallest integer $m$ such that $\frac{10!}{m}$ is a perfect square? $\text{(A) }2\qquad\text{(B) }7\qquad\text{(C) }14\qquad\text{(D) }21\qquad\text{(E) }35$

1981 AMC 12/AHSME, 28

Tags:

Consider the set of all equations $ x^3 \plus{} a_2x^2 \plus{} a_1x \plus{} a_0 \equal{} 0$, where $ a_2$, $ a_1$, $ a_0$ are real constants and $ |a_i| < 2$ for $ i \equal{} 0,1,2$. Let $ r$ be the largest positive real number which satisfies at least one of these equations. Then $ \textbf{(A)}\ 1 < r < \frac{3}{2}\qquad \textbf{(B)}\ \frac{3}{2} < r < 2\qquad \textbf{(C)}\ 2 < r < \frac{5}{2}\qquad \textbf{(D)}\ \frac{5}{2} < r < 3\qquad \\ \textbf{(E)}\ 3 < r < \frac{7}{2}$

2004 South East Mathematical Olympiad, 1

Tags: inequalities

Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.

1991 Iran MO (2nd round), 1

Tags: number theory

Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.

2021 Nigerian Senior MO Round 2, 1

Tags: geometry , science olympiad

If $x$,$y$ and $z$ are the lengths of a side, a shortest diagonal and a longest diagonal respectively, of a regular nonagon. Write a correct equation consisting of the three lengths

2017 Online Math Open Problems, 25

Tags:

A [i]simple hyperplane[/i] in $\mathbb{R}^4$ has the form \[k_1x_1+k_2x_2+k_3x_3+k_4x_4=0\] for some integers $k_1,k_2,k_3,k_4\in \{-1,0,1\}$ that are not all zero. Find the number of regions that the set of all simple hyperplanes divide the unit ball $x_1^2+x_2^2+x_3^2+x_4^2\leq 1$ into. [i]Proposed by Yannick Yao[/i]

2009 Singapore MO Open, 1

Tags: geometry

let $O$ be the center of the circle inscribed in a rhombus ABCD. points E,F,G,H are chosen on sides AB, BC, CD, DA respectively so that EF and GH are tangent to inscribed circle. show that EH and FG are parallel.

Kharkiv City MO Seniors - geometry, 2019.10.5

Tags: excenter , incenter , angle bisector , midpoint , geometry

In triangle $ABC$, point$ I$ is incenter , $I_a$ is the $A$-excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$, and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$. Prove that $K$ is the orthocenter of triangle $II_aE$.

2013 Putnam, 2

Tags: geometry , 3d geometry

Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which: (i) $f(x)\ge 0$ for all real $x,$ and (ii) $a_n=0$ whenever $n$ is a multiple of $3.$ Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.

2006 AMC 10, 15

Tags:

Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other? $ \textbf{(A) } 29 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 47 \qquad \textbf{(E) } 50$

2010 Contests, 1

Tags: induction , geometry , geometric transformation , reflection , symmetry , combinatorics

For all natural $n$, an $n$-staircase is a figure consisting of unit squares, with one square in the first row, two squares in the second row, and so on, up to $n$ squares in the $n^{th}$ row, such that all the left-most squares in each row are aligned vertically. Let $f(n)$ denote the minimum number of square tiles requires to tile the $n$-staircase, where the side lengths of the square tiles can be any natural number. e.g. $f(2)=3$ and $f(4)=7$. (a) Find all $n$ such that $f(n)=n$. (b) Find all $n$ such that $f(n) = n+1$.

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , rational , equation

Let $x$ and $y$ be two rational numbers and $n$ be an odd positive integer. Prove that, if $x^n - 2x = y^n - 2y$, then $x = y$.