Olympiad problems

2019 IFYM, Sozopol, 5

The non-decreasing functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are such that $f(r)\leq g(r)$ for $\forall$ rational numbers $r$. Is it true that $f(x)\leq g(x)$ for $\forall$ real numbers $x$?

[b]6.[/b] For which Lebesgue-measurable subsets $E$ of the real line does a positive constant $c$ exist for which $\sup_{-\infty < t<\infty} \left | \int_{E} e^{itx} f(x) dx\right | \leq c \sup_{n=0,\pm 1,\dots } \left | \int_{E} e^{inx} f(x) dx\right |$ for all integrable functions $f$ on $E$? ([b]M.17[/b]) [G. Halász]

2023 Yasinsky Geometry Olympiad, 1

Tags: perpendicular , geometry

Let $BD$ and $CE$ be the altitudes of triangle $ABC$ that intersect at point $H$. Let $F$ be a point on side $AC$ such that $FH\perp CE$. The segment $FE$ intersects the circumcircle of triangle $CDE$ at the point $K$. Prove that $HK\perp EF$ . (Matthew Kurskyi)

2019 Moldova Team Selection Test, 8

Tags: number theory

For any positive integer $k$ denote by $S(k)$ the number of solutions $(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+$ of the system $$\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}$$ where $d$ is the greatest common divisor of positive integers $x$ and $y.$ Determine $S(k)$ as a function of $k$. (Here $\lceil z\rceil$ denotes the smalles integer number which is bigger or equal than $z.$)

1993 Moldova Team Selection Test, 8

Tags: geometry , parallelogram , circumcircle

Inside the parallelogram $ABCD$ points $M, N, K$ and $L{}$ are on sides $AB, BC, CD{}$ and $DA$, respectively. Let $O_1, O_2, O_3$ and $O_4$ be the circumcenters of triangles repesctively $MBN, NCK, KDL$ and $LAM{}$. Prove that the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

BIMO 2022, 6

Tags: geometry

Given a triangle $ABC$ with $AB=AC$ and circumcenter $O$. Let $D$ and $E$ be midpoints of $AC$ and $AB$ respectively, and let $DE$ intersect $AO$ at $F$. Denote $\omega$ to be the circle $(BOE)$. Let $BD$ intersect $\omega$ again at $X$ and let $AX$ intersect $\omega$ again at $Y$. Suppose the line parallel to $AB$ passing through $O$ meets $CY$ at $Z$. Prove that the lines $FX$ and $BZ$ meet at $\omega$. [i]Proposed by Ivan Chan Kai Chin[/i]

2017 ASDAN Math Tournament, 26

Tags: 2017 , Guts Round

A lattice point is a coordinate pair $(a,b)$ where both $a,b$ are integers. What is the number of lattice points $(x,y)$ that satisfy $\tfrac{x^2}{2017}+\tfrac{2y^2}{2017}<1$ and $y\equiv2x\pmod{7}$? Let $C$ be the actual answer, $A$ be the answer you submit, and $D=|A-C|$. Your score will be rounded up from $\max(0,25-e^{D/100})$.

2024 Sharygin Geometry Olympiad, 8.6

Tags: geometry , circumcircle , Sharygin Geometry Olympiad , Sharygin 2024 , reflection

A circle $\omega$ touched lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X'Y'$.

2024 Germany Team Selection Test, 1

Tags: combinatorics

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

2019 ASDAN Math Tournament, 2

Tags: geometry , 2019

A square and a line intersect at a $45^o$ angle. The line bisects the square into two unequal pieces such that the area of one piece is twice that of the other. If the square has side length $6$, compute the length of the cut due to the line. [img]https://cdn.artofproblemsolving.com/attachments/6/4/2eb33fb9766497d25d342001cdbae9a7ffd4b4.png[/img]

2021 HMNT, 5

Tags: algebra

Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.

MathLinks Contest 4th, 2.2

Tags: geometry , 3D geometry , 4th edition

Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.

2016 Taiwan TST Round 2, 1

Tags: geometry , Taiwan , Taiwan TST , geometry solved , radical axis , power of a point , linearity of power of a point

Let $O$ be the circumcenter of triangle $ABC$, and $\omega$ be the circumcircle of triangle $BOC$. Line $AO$ intersects with circle $\omega$ again at the point $G$. Let $M$ be the midpoint of side $BC$, and the perpendicular bisector of $BC$ meets circle $\omega$ at the points $O$ and $N$. Prove that the midpoint of the segment $AN$ lies on the radical axis of the circumcircle of triangle $OMG$, and the circle whose diameter is $AO$.

1970 IMO Longlists, 17

Tags: geometry , 3D geometry , tetrahedron , geometric inequality , IMO , IMO 1970

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

2016 Iran Team Selection Test, 6

Tags:

Suppose that a council consists of five members and that decisions in this council are made according to a method based on the positive or negative vote of its members. The method used by this council has the following two properties: $\bullet$ [b]Ascension:[/b]If the presumptive final decision is favorable and one of the opposing members changes his/her vote, the final decision will still be favorable. $\bullet$ [b]Symmetry:[/b] If all of the members change their vote, the final decision will change too. Prove that the council uses a weighted decision-making method ; that is , nonnegative weights $\omega _1 , \omega _2 , \cdots ,\omega _5$ can be assigned to members of the council such that the final decision is favorable if and only if sum of the weights of those in favor is greater than sum of the weights of the rest. Remark. The statement isn't true at all if you replace $5$ with arbitrary $n$ . In fact , finding a counter example for $n=6$ , was appeared in the same year's [url=https://artofproblemsolving.com/community/c6h1459567p8417532]Iran MO 2nd round P6[/url]

1992 Tournament Of Towns, (328) 5

Tags: combinatorics , weighings

$50$ silver coins ordered by weight and $51$ gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the $51$st place in weight among all $101$ coins) using $7$ weighings? (A. Andjans)

2018 Iran Team Selection Test, 2

Tags: Iran , Iranian TST , maximum value , algebra

Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$, $$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$ [i]Proposed by Morteza Saghafian[/i]

2006 National Olympiad First Round, 23

Tags:

What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 36 $

2012 China Team Selection Test, 3

Tags: algebra , polynomial , inequalities , geometry , geometric transformation , algebra unsolved

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

2003 China Team Selection Test, 3

Tags: number theory unsolved , number theory

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

2019 Romania National Olympiad, 2

Tags: vectorial geometry , vector , geometry

Find all natural numbers which are the cardinal of a set of nonzero Euclidean vectors whose sum is $ 0, $ the sum of any two of them is nonzero, and their magnitudes are equal.

2011 Today's Calculation Of Integral, 730

Tags: calculus , integration , limit , logarithms , calculus computations

Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2008 District Olympiad, 2

Tags: function , real analysis , periodic , analysis , Integral , primitives

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that: [b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic. [b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $

2018 Thailand TSTST, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$

2012 AMC 12/AHSME, 7

Tags: AMC

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $