Olympiad problems

1984 IMO Longlists, 60

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

1998 National Olympiad First Round, 2

Tags: logarithm , floor function

Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$ $\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$

1990 IMO Longlists, 2

Tags: series summation , summation , combinatorial identity , binomial coefficient , combinatorics

Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

2020 BMT Fall, 10

Tags: geometry

Let $E$ be an ellipse where the length of the major axis is $26$, the length of the minor axis is $24$, and the foci are at points $R$ and $S$. Let $A$ and $B$ be points on the ellipse such that $RASB$ forms a non-degenerate quadrilateral, lines $RA$ and $SB$ intersect at $P$ with segment $PR$ containing $A$, and lines $RB$ and $AS$ intersect at Q with segment $QR$ containing $B$. Given that $RA = AS$, $AP = 26$, the perimeter of the non-degenerate quadrilateral $RP SQ$ is $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.

2014 BMT Spring, 4

Tags: geometry

A cylinder with length $\ell$ has a radius of $6$ meters, and three spheres with radii $3, 4$, and $5$ meters are placed inside the cylinder. If the spheres are packed into the cylinder such that $\ell$ is minimized, determine the length $\ell$.

1972 IMO Shortlist, 8

Tags: recurrence relation , binomial coefficient , factorial , number theory

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

2014 India Regional Mathematical Olympiad, 6

Tags: inequalities , number theory , modular arithmetic

For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$.

2007 Iran MO (3rd Round), 1

Tags: reflection , geometric transformation , parallelogram , geometry

Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.

2007 Iran Team Selection Test, 1

Tags: reflection , geometry , geometric transformation , analytic geometry

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

1960 AMC 12/AHSME, 10

Tags:

Given the following six statements: $\text{(1) All women are good drivers}$ $\text{(2) Some women are good drivers}$ $\text{(3) No men are good drivers}$ $\text{(4) All men are bad drivers}$ $\text{(5) At least one man is a bad driver}$ $\text{(6) All men are good drivers.}$ The statement that negates statement $\text{(6)}$ is: $ \textbf{(A) }(1)\qquad\textbf{(B) }(2)\qquad\textbf{(C) }(3)\qquad\textbf{(D) }(4)\qquad\textbf{(E) }(5) $

PEN N Problems, 5

Tags: more sequences

Prove that there exist two strictly increasing sequences $a_{n}$ and $b_{n}$ such that $a_{n}(a_{n} +1)$ divides $b_{n}^2 +1$ for every natural $n$.

2003 Bulgaria Team Selection Test, 6

Tags: number theory

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

2000 Stanford Mathematics Tournament, 1

Tags:

If $ a\equal{}2b\plus{}c$, $ b\equal{}2c\plus{}d$, $ 2c\equal{}d\plus{}a\minus{}1$, $ d\equal{}a\minus{}c$, what is $ b$?

2011 India IMO Training Camp, 3

Tags: function , number theory

Let $\{a_0,a_1,\ldots\}$ and $\{b_0,b_1,\ldots\}$ be two infinite sequences of integers such that \[(a_{n}-a_{n-1})(a_n-a_{n-2}) +(b_n-b_{n-1})(b_n-b_{n-2})=0\] for all integers $n\geq 2$. Prove that there exists a positive integer $k$ such that \[a_{k+2011}=a_{k+2011^{2011}}.\]

Kyiv City MO Juniors 2003+ geometry, 2018.7.4

Tags: right angle , angle , equal segments , geometry

Inside the triangle $ABC $, the point $P $ is selected so that $BC = AP $ and $\angle APC = 180 {} ^ \circ - \angle ABC $. On the side $AB $ there is a point $K $, for which $AK = KB + PC $. Prove that $\angle AKC = 90 {} ^ \circ $. (Danilo Hilko)

1979 IMO Longlists, 22

Tags: geometry , perpendicular bisector

Consider two quadrilaterals $ABCD$ and $A'B'C'D'$ in an affine Euclidian plane such that $AB = A'B', BC = B'C', CD = C'D'$, and $DA = D'A'$. Prove that the following two statements are true: [b](a)[/b] If the diagonals $BD$ and $AC$ are mutually perpendicular, then the diagonals $B'D'$ and $A'C'$ are also mutually perpendicular. [b](b)[/b] If the perpendicular bisector of $BD$ intersects $AC$ at $M$, and that of $B'D'$ intersects $A'C'$ at $M'$, then $\frac{\overline{MA}}{\overline{MC}}=\frac{\overline{M'A'}}{\overline{M'C'}}$ (if $MC = 0$ then $M'C' = 0$).

2014 Singapore Senior Math Olympiad, 10

Tags: function

If $f(x)=\frac{1}{x}-\frac{4}{\sqrt{x}}+3$ where $\frac{1}{16}\le x\le 1$, find the range of $f(x)$. $ \textbf{(A) }-2\le f(x)\le 4 \qquad\textbf{(B) }-1\le f(x)\le 3\qquad\textbf{(C) }0\le f(x)\le 3\qquad\textbf{(D) }-1\le f(x)\le 4\qquad\textbf{(E) }\text{None of the above} $

2006 Victor Vâlcovici, 2

Tags: derivative , lipschitz continuous , riemann sum , calculus , function , real analysis

Let be a differentiable function $ f:[0,1]\longrightarrow\mathbb{R} $ whose derivative has a positive Lipschitz constant $ L. $ Show that [b]a)[/b] $ x,y\in [0,1]\implies | f(x)-f(y)-f'(y)(x-y) |\le L\cdot (x-y)^2 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n\int_0^1 f(x)dx-\sum_{i=1}^nf\left( \frac{2i-1}{2n} \right) \right) =0. $

2019 Bulgaria EGMO TST, 1

Tags: euler , euler line , centroid , circumcenter , geometry , circumcircle

Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)

2002 APMO, 2

Tags: number theory

Find all positive integers $a$ and $b$ such that \[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \] are both integers.

2013 ELMO Shortlist, 7

Tags: algebra , number theory , polynomial , modular arithmetic

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

1946 Moscow Mathematical Olympiad, 107

Tags: equilateral , geometry , collinear

Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)

2019 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , combinatorial geometry

An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$. Show that there are two dominoes forming a $2 \times 2$ square.

2019 239 Open Mathematical Olympiad, 5

Tags: geometry

Circle $\Gamma$ touches the circumcircle of triangle $ABC$ at point $R$, and it touches the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Rays $PQ$ and $BC$ intersect at point $X$. The tangent line at point $R$ to the circle $\Gamma$ meets the segment $QX$ at point $Y$. The line segment $AX$ intersects the circumcircle of triangle $APQ$ at point $Z$. Prove that the circumscribed circles of triangles $ABC$ and $XY Z$ are tangent.

2008 Princeton University Math Competition, A6/B8

Tags: number theory

What is the largest integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y$, and $z$?