Olympiad problems

2020 Junior Balkan Team Selection Tests-Serbia, 3#

Tags: inequality

Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.

2008 Germany Team Selection Test, 3

Tags: circumcircle , reflection , cyclic quadrilateral , mixtilinear incircle , geometry

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

Geometry Mathley 2011-12, 4.2

Tags: perpendicular , geometry , circles

Let $ABC$ be a triangle. $(K)$ is an arbitrary circle tangent to the lines $AC,AB$ at $E, F$ respectively. $(K)$ cuts $BC$ at $M,N$ such that $N$ lies between $B$ and $M$. $FM$ intersects $EN$ at $I$. The circumcircles of triangles $IFN$ and $IEM$ meet each other at $J$ distinct from $I$. Prove that $IJ$ passes through $A$ and $KJ$ is perpendicular to $IJ$. Trần Quang Hùng

2010 Contests, 1

Tags: function , vieta , algebra , functional equation

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

2014 Indonesia MO Shortlist, C3

Tags: perimeter , coloring , chessboard , combinatorics

Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.

Kyiv City MO 1984-93 - geometry, 1987.10.1

Tags: geometry , tangential , polygon

Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?

2019 Oral Moscow Geometry Olympiad, 5

Tags: geometry , circumcircle , cyclic quadrilateral , concurrency

On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

2023 BMT, 1

Tags: algebra

Lakshay chooses two numbers, $m$ and $n$, and draws two lines, $y = mx + 3$ and $y = nx + 23$. Given that the two lines intersect at $(20, 23)$, compute $m + n$.

2020 LMT Spring, 27

Tags:

Let $S_n=\sum_{k=1}^n (k^5+k^7).$ Let the prime factorization of $\text{gcd}(S_{2020},S_{6060})$ be $p_1^{k_1}\cdot p_2^{k_2}\cdots p_i^{k_i}$. Compute $p_1+p_2+\cdots +p_i+k_1+k_2+\cdots + k_i $.

2018 Slovenia Team Selection Test, 5

Tags: combinatorics , geometry

Let $n$ be a positive integer. We are given a regular $4n$-gon in the plane. We divide its vertices in $2n$ pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?

2010 LMT, 6

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Al travels for $20$ miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?

2014 Taiwan TST Round 3, 4

Tags: circumcircle , isosceles triangle , angle chasing , inversion , geometry

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

1985 IberoAmerican, 2

Tags: rotation , trigonometry , geometry

Let $ P$ be a point in the interior of the equilateral triangle $ \triangle{}ABC$ such that $ PA \equal{} 5$, $ PB \equal{} 7$, $ PC \equal{} 8$. Find the length of the side of the triangle $ ABC$.

1997 AMC 8, 4

Tags:

Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech? $\textbf{(A)}\ 2250 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 4200 \qquad \textbf{(D)}\ 4350 \qquad \textbf{(E)}\ 5650$

1959 Putnam, B4

Tags: linear algebra , matrix , maximum

Given the following matrix $$\begin{pmatrix} 11& 17 & 25& 19& 16\\ 24 &10 &13 & 15&3\\ 12 &5 &14& 2&18\\ 23 &4 &1 &8 &22 \\ 6&20&7 &21&9 \end{pmatrix},$$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.

2007 Baltic Way, 4

Tags: inequalities

Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]

2011 Romanian Masters In Mathematics, 1

Tags: function , number theory

Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$). Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$; ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$. [i](Romania) Dan Schwarz[/i]

2007 AMC 10, 2

Tags:

Define $ a@b \equal{} ab \minus{} b^{2}$ and $ a\#b \equal{} a \plus{} b \minus{} ab^{2}$. What is $ \frac {6@2}{6\#2}$? $ \textbf{(A)}\ \minus{} \frac {1}{2}\qquad \textbf{(B)}\ \minus{} \frac {1}{4}\qquad \textbf{(C)}\ \frac {1}{8}\qquad \textbf{(D)}\ \frac {1}{4}\qquad \textbf{(E)}\ \frac {1}{2}$

2017 India PRMO, 5

Tags: geometric sequence , geometric progression , algebra , exponential

Let $u, v,w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.

2018 Iran MO (1st Round), 4

Tags: point

There are $5$ points in the plane no three of which are collinear. We draw all the segments whose vertices are these points. What is the minimum number of new points made by the intersection of the drawn segments? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

2024 Romania EGMO TST, P3

Tags: circumcircle , geometry

Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.

2005 National Olympiad First Round, 25

Tags: geometry , rectangle

Let $E$, $F$, $G$ be points on sides $[AB]$, $[BC]$, $[CD]$ of the rectangle $ABCD$, respectively, such that $|BF|=|FQ|$, $m(\widehat{FGE})=90^\circ$, $|BC|=4\sqrt 3 / 5$, and $|EF|=\sqrt 5$. What is $|BF|$? $ \textbf{(A)}\ \dfrac{\sqrt{10} - \sqrt{2}}{2} \qquad\textbf{(B)}\ \sqrt 3 -1 \qquad\textbf{(C)}\ \sqrt 3 \qquad\textbf{(D)}\ \dfrac{\sqrt{11} - \sqrt{3}}{2} \qquad\textbf{(E)}\ 1 $

2012 Indonesia TST, 2

Tags: incenter , circumcircle , geometry

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a,b,c,d$ be positive real numbers with $ a\plus{}b\plus{}c\plus{}d \equal{} 4$. Prove that \[ a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.\]

2016 CHMMC (Fall), 14

Tags: function , geometry

For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.