Olympiad problems

2016 Switzerland Team Selection Test, Problem 3

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

1978 Miklós Schweitzer, 3

Tags: number theory , combinatorics , logarithm

Let $ 1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $ \sum_{i\equal{}1}^n 1/a_i \leq 1$. Let $ y$ denote the number of positive integers smaller that $ x$ not divisible by any of the $ a_i$. Prove that \[ y > \frac{cx}{\log x}\] with a suitable positive constant $ c$ (independent of $ x$ and the numbers $ a_i$). [i]I. Z. Ruzsa[/i]

2003 South africa National Olympiad, 5

Tags: algebra , polynomial , number theory

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

Kyiv City MO 1984-93 - geometry, 1993.10.5

Tags: trigonometry , geometry

Prove that for the sides $a, b, c$, the angles $A, B, C$ and the area $S$ of the triangle holds $$\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.$$

2010 Contests, 2

Tags: rectangle , ratio , angle bisector , geometry

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

2009 AMC 12/AHSME, 9

Tags: quadratic

Suppose that $ f(x\plus{}3)\equal{}3x^2\plus{}7x\plus{}4$ and $ f(x)\equal{}ax^2\plus{}bx\plus{}c$. What is $ a\plus{}b\plus{}c$? $ \textbf{(A)}\minus{}\!1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

2012 CHMMC Spring, 4

Tags: algebra

The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}}$$

1967 IMO Shortlist, 3

Tags: 3d geometry , sphere , volume , geometry

Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$

2013 Brazil National Olympiad, 1

Tags: geometry

Let $\Gamma$ be a circle and $A$ a point outside $\Gamma$. The tangent lines to $\Gamma$ through $A$ touch $\Gamma$ at $B$ and $C$. Let $M$ be the midpoint of $AB$. The segment $MC$ meets $\Gamma$ again at $D$ and the line $AD$ meets $\Gamma$ again at $E$. Given that $AB=a$, $BC=b$, compute $CE$ in terms of $a$ and $b$.

2024 CMIMC Team, 8

Tags: team

Compute \[\frac{(1-\tan10^\circ)(1-\tan 20^\circ)(1-\tan30^\circ)(1-\tan40^\circ)}{(1-\tan5^\circ)(1-\tan 15^\circ)(1-\tan25^\circ)(1-\tan35^\circ)}.\] [i]Proposed by Connor Gordon[/i]

2021 Purple Comet Problems, 18

Tags: combinatorics

Three red books, three white books, and three blue books are randomly stacked to form three piles of three books each. The probability that no book is the same color as the book immediately on top of it is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2022 South East Mathematical Olympiad, 3

Tags: combinatorics

There are $n$ people in line, counting $1,2,\cdots, n$ from left to right, those who count odd numbers quit the line, the remaining people press 1,2 from right to left, and count off again, those who count odd numbers quit the line, and then the remaining people count off again from left to right$\cdots$ Keep doing that until only one person is in the line. $f(n)$ is the number of the last person left at the first count. Find the expression for $f(n)$ and find the value of $f(2022)$

1971 AMC 12/AHSME, 16

Tags: ratio

After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was $\textbf{(A) }1:1\qquad\textbf{(B) }35:36\qquad\textbf{(C) }36:35\qquad\textbf{(D) }2:1\qquad \textbf{(E) }\text{None of these}$

2015 Iran Geometry Olympiad, 1

Tags: geometry

let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $ let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $ suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $ now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $

2017 Saudi Arabia Pre-TST + Training Tests, 8

Tags: point , combinatorics , combinatorial geometry

There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

2021 Kyiv Mathematical Festival, 5

Tags: geometry

Let $\omega$ be the circumcircle of a triangle $ABC$ (${AB\ne AC}$), $I$ be the incenter, $P$ be the point on $\omega$ for which $\angle API=90^\circ,$ $S$ be the intersection point of lines $AP$ and $BC,$ $W$ be the intersection point of line $AI$ and $\omega.$ Line which passes through point $W$ orthogonally to $AW$ meets $AP$ and $BC$ at points $D$ and $E$ respectively. Prove that $SD=IE.$ (Ye. Azarov)

2009 Princeton University Math Competition, 4

Tags:

How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?

2000 Tournament Of Towns, 6

Tags: combinatorics , coloring

a) Several black squares of side $1$ cm are nailed to a white plane with a nail of thickness $0 . 1$ cm so that they form a black polygon. Can it happen that the perimeter of this polygon is $1$ km long? (The nail is not allowed to touch the boundary of any of the squares . ) (b) The same problem as in (a) but with a nail of thickness $0$ (a point ) . (c) Several black squares of side $1$ cm lie on a white plane so that they form a black polygon (possibly having more than one piece and/ or having holes) . Can it happen that the ratio of its perimeter (in centimetres) to its area (in square centimetres) is more than $100000$? (Hungarian Folklore)

2016 AIME Problems, 5

Tags:

Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.

2011 Dutch BxMO TST, 3

Tags: absolute value , equation , algebra

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

2006 Sharygin Geometry Olympiad, 9.6

Tags: diagonal , orthocenter , ratio , geometry , convex quadrilateral

A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$, the corresponding diagonals share the intersection points in the same ratio.

2007 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometry , circumcircle , incenter

Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$, respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$, respectively. If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.

PEN K Problems, 27

Tags: function , induction , strong induction , functional equation

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $m,n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

2014 Korea - Final Round, 4

Tags: circumcircle , perpendicular bisector , geometry

Let $ ABC $ be a isosceles triangle with $ AC=BC$. Let $ D $ a point on a line $ BA $ such that $ A $ lies between $ B, D $. Let $O_1 $ be the circumcircle of triangle $ DAC $. $ O_1 $ meets $ BC $ at point $ E $. Let $ F $ be the point on $ BC $ such that $ FD $ is tangent to circle $O_1 $, and let $O_2 $ be the circumcircle of $ DBF$. Two circles $O_1 , O_2 $ meet at point $ G ( \ne D) $. Let $ O $ be the circumcenter of triangle $ BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $ DG \bot FO$.

2011 Laurențiu Duican, 4

Tags: modular arithmetic , arithmetic , base 10 , combinatorics

Let be two natural numbers $ m\ge n $ and a nonnegative integer $ r<2^n. $ How many numbers of $ m $ digits, each digit being either the number $ 1 $ or $ 2, $ are there whose residue modulo $ 2^n $ is $ r? $ [i]Dorel Miheț[/i]