Olympiad problems

2018 AMC 10, 1

What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\] $\textbf{(A) } \frac{5}{8} \qquad\textbf{(B) } \frac{11}{7} \qquad\textbf{(C) } \frac{8}{5} \qquad\textbf{(D) } \frac{18}{11} \qquad\textbf{(E) } \frac{15}{8}$

1998 Bosnia and Herzegovina Team Selection Test, 4

Tags: circle , tangent , geometry , perpendicular

Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$

2014 Purple Comet Problems, 14

Tags:

Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$.

2007 Greece JBMO TST, 1

Tags: circumcircle , geometry , angles

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

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 $n$

2015 Belarus Team Selection Test, 3

Tags: geometry , incircle , Concyclic

Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quadrilateral. I.Gorodnin

LMT Team Rounds 2010-20, B16

Tags: algebra

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

1980 Spain Mathematical Olympiad, 4

Tags: analysis , calculus , function , algebra

Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

2009 Polish MO Finals, 6

Tags: algebra unsolved , algebra

Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.

1991 Greece National Olympiad, 3

Tags: number theory , Digits

Find all 2-digit numbers$ n$ having the property: 'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.

May Olympiad L2 - geometry, 1997.2

Tags: geometry

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$

2014-2015 SDML (Middle School), 5

Tags: factorial

In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

2024 IMAR Test, P3

Tags: geometry

Let $ABC$ be a triangle . A circle through $B$ and $C$ crosses sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ on segments $BQ$ and $CP$, respectively, satisfy $\angle ABY=\angle AXP$ and $ACX=\angle AYQ$. Prove that $XY$ and $BC$ are parallel.

1979 Swedish Mathematical Competition, 5

Tags: algebra , trinomial

Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.

2021 All-Russian Olympiad, 8

Tags: combinatorics

Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.

2004 India IMO Training Camp, 1

Tags: ratio , function , geometry unsolved , geometry

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

2005 AMC 10, 16

Tags: modular arithmetic

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $ 6$. How many two-digit numbers have this property? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 19$

1979 Poland - Second Round, 2

Tags: algebra , inequalities

Prove that if $ a, b, c $ are non-negative numbers, then $$ a^3 + b^3 + c^3 + 3abc \geq a^2(b + c) + b^2(a + c) + c^2(a + b).$$

2001 Romania National Olympiad, 1

Tags: algebra , polynomial , algebra unsolved

a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots. b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.

2021 Korea Junior Math Olympiad, 2

Tags: modular arithmetic , Sequence , Integer sequence , number theory

Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. [list] [*] $a_1=2021^{2021}$ [*] $0 \le a_k < k$ for all integers $k \ge 2$ [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. [/list] Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.

1967 IMO Longlists, 16

Tags: number theory , approximation , irrational number

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

1975 Miklós Schweitzer, 1

Tags: advanced fields , advanced fields unsolved , set theory , real analysis

Show that there exists a tournament $ (T,\rightarrow)$ of cardinality $ \aleph_1$ containing no transitive subtournament of size $ \aleph_1$. ( A structure $ (T,\rightarrow)$ is a $ \textit{tournament}$ if $ \rightarrow$ is a binary, irreflexive, asymmetric and trichotomic relation. The tournament $ (T,\rightarrow)$ is transitive if $ \rightarrow$ is transitive, that is, if it orders $ T$.) [i]A. Hajnal[/i]