Olympiad problems

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

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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

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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]

2024 AMC 12/AHSME, 5

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A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

2008 China Girls Math Olympiad, 1

Tags: combinatorics

[i](a)[/i] Determine if the set $ \{1,2,\ldots,96\}$ can be partitioned into 32 sets of equal size and equal sum. [i](b)[/i] Determine if the set $ \{1,2,\ldots,99\}$ can be partitioned into 33 sets of equal size and equal sum.

2004 India IMO Training Camp, 4

Tags: combinatorics

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

2010 AMC 12/AHSME, 6

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At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$

2016 Math Prize for Girls Olympiad, 2

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Eve picked some apples, each weighing at most $\frac{1}{2}$ pound. Her apples weigh a total of $W$ pounds, where $W > \frac{1}{3}$. Prove that she can place all her apples into $\left\lceil \frac{3W - 1}{2} \right\rceil$ or fewer baskets, each of which holds up to 1 pound of apples. (The apples are not allowed to be cut into pieces.) Note: If $x$ is a real number, then $\lceil x \rceil$ (the ceiling of $x$) is the least integer that is greater than or equal to $x$.

2022 Miklós Schweitzer, 7

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Point-like figures are placed in the vertices of a regular $k$-angle, and then we walk with them. In one step, a piece jumps over another piece, i.e. its new location will be a mirror image of its current location to the current location of another piece. In the case of $k \geq 3$ integers, it is possible to achieve with a series of such steps that the puppets form the vertices of a regular $k$-angle, different in size from the original?

2023 Ecuador NMO (OMEC), 3

Tags: algebra , polynomial

We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$: \[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\] Find the sum of all $a_{2023}$'s possible values.

2014 Serbia National Math Olympiad, 2

Tags: geometry

On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$. [i]Proposed by Milos Milosavljevic[/i]

2010 IFYM, Sozopol, 8

Tags: algebra

Solve this equation with $x \in R$: $x^3-3x=\sqrt{x+2}$

2014 India Regional Mathematical Olympiad, 5

Tags: geometry , circumcircle , asymptote , incenter , perpendicular bisector , angle bisector

Let $ABC$ be a triangle and let $X$ be on $BC$ such that $AX=AB$. let $AX$ meet circumcircle $\omega$ of triangle $ABC$ again at $D$. prove that circumcentre of triangle $BDX$ lies on $\omega$.

2008 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory

Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.

2010 ISI B.Math Entrance Exam, 7

Tags: polynomial , trigonometry , algebra

We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.

2024 Sharygin Geometry Olympiad, 10.6

Tags: geo , geometry

A point $P$ lies on one of medians of triangle $ABC$ in such a way that $\angle PAB =\angle PBC =\angle PCA$. Prove that there exists a point $Q$ on another median such that $\angle QBA=\angle QCB =\angle QAC$.

2008 AIME Problems, 14

Tags: trigonometry , quadratic , geometric transformation , rotation , asymptote , function , geometry

Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.

2015 District Olympiad, 4

Tags: pure geometry , 3d geometry , geometry

Consider the rectangular parallelepiped $ ABCDA'B'C'D' $ and the point $ O $ to be the intersection of $ AB' $ and $ A'B. $ On the edge $ BC, $ pick a point $ N $ such that the plane formed by the triangle $ B'AN $ has to be parallel to the line $ AC', $ and perpendicular to $ DO'. $ Prove, then, that this parallelepiped is a cube.