Olympiad problems

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

Tags:

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

Tags:

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

Tags:

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

Tags:

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.

1997 Croatia National Olympiad, Problem 2

Tags: inequalities

Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum $$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.

VI Soros Olympiad 1999 - 2000 (Russia), 10.10

Tags: number theory

Prove that for every integer $n \ge 1$ there exists a real number $a$ such that for any integer $m \ge 1$ the number $[a^m] + 1$ is divisible by $n$ ($[x]$ denotes the largest integer that does not exceed $x$).

2005 China Team Selection Test, 1

Tags: function , inequalities , floor function

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

2012 AIME Problems, 5

Tags:

Let $B$ be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer 1 is obtained.

2016 Putnam, B3

Tags: putnam geometry , combinatorial geometry

Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$