Olympiad problems

2010 AIME Problems, 5

Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

2011 Purple Comet Problems, 14

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The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits

2000 Iran MO (3rd Round), 1

Tags: geometry

Let us denote $\prod = \{(x, y) | y > 0\}$. We call a [i]semicircle[/i] in $\prod$ with center on the $x-\text{axis}$ a [i]semi-line[/i]. Two intersecting [i]semi-lines [/i]determine four [i]semi-angles[/i]. A bisector of a [i]semi-angle [/i]is a [i]semi-line [/i]that bisects the [i]semi-angle[/i]. Prove that in every [i]semi-triangle [/i](determined by three [i]semi-lines[/i]) the bisectors are concurrent.

V Soros Olympiad 1998 - 99 (Russia), 11.1

Tags: algebra , inequalities , trigonometry

Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

2008 National Olympiad First Round, 3

Tags: algebra , polynomial

Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$? $ \textbf{(A)}\ 840 \qquad\textbf{(B)}\ 816 \qquad\textbf{(C)}\ 969 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

2012 Sharygin Geometry Olympiad, 7

Tags: geometry , pentagon , geometric inequality , convex , diagonal

A convex pentagon $P $ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$. (A.Belov)

2020 Dutch BxMO TST, 4

Tags: equal segments , equilateral , geometry

Three different points $A,B$ and $C$ lie on a circle with center $M$ so that $| AB | = | BC |$. Point $D$ is inside the circle in such a way that $\vartriangle BCD$ is equilateral. Let $F$ be the second intersection of $AD$ with the circle . Prove that $| F D | = | FM |$.

2011 Iran MO (3rd Round), 3

Tags: modular arithmetic , number theory

Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$? [i]Proposed by Mahyar Sefidgaran[/i]

2010 Greece Junior Math Olympiad, 2

Tags: geometry , rectangle , equilateral , equal segments , perpendicular

Let $ABCD$ be a rectangle with sides $AB=a$ and $BC=b$. Let $O$ be the intersection point of it's diagonals. Extent side $BA$ towards $A$ at a segment $AE=AO$, and diagonal $DB$ towards $B$ at a segment $BZ=BO$. If the triangle $EZC$ is an equilateral, then prove that: i) $b=a\sqrt3$ ii) $AZ=EO$ iii) $EO \perp ZD$

2014 Germany Team Selection Test, 3

Tags: number theory

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

2020 Dürer Math Competition (First Round), P2

Tags: combinatorics , table , cell

How many ways can you fill a table of size $n\times n$ with integers such that each cell contains the total number of even numbers in its row and column other than itself? Two tables are different if they differ in at least one cell.

2014 District Olympiad, 4

Tags: superior algebra

Let $(G,\cdot)$ be a group with no elements of order 4, and let $f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$ for all $x\in G$.

1988 AIME Problems, 4

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Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that \[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \] What is the smallest possible value of $n$?

2003 Bulgaria Team Selection Test, 4

Tags: number theory

Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

1985 Austrian-Polish Competition, 6

Tags: geometry , 3d geometry , tetrahedron , volume

Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$. Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$.

2022 Mediterranean Mathematics Olympiad, 1

Tags: combinatorics

Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties: $\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side. $\bullet$ For all $x,y \in S$ with $x\ne y$, it is possible to select a card $C_i$ that shows $x$ on one of its sides and another card $C_j$ (with $i \ne j$) that shows $y$ on one of its sides.

2006 Pre-Preparation Course Examination, 6

Tags: limit , algebra

Suppose that $P_c(z)=z^2+c$. You are familiar with the Mandelbrot set: $M=\{c\in \mathbb{C} | \lim_{n\rightarrow \infty}P_c^n(0)\neq \infty\}$. We know that if $c\in M$ then the points of the dynamical system $(\mathbb{C},P_c)$ that don't converge to $\infty$ are connected and otherwise they are completely disconnected. By seeing the properties of periodic points of $P_c$ prove the following ones: a) Prove the existance of the heart like shape in the Mandelbrot set. b) Prove the existance of the large circle next to the heart like shape in the Mandelbrot set. [img]http://astronomy.swin.edu.au/~pbourke/fractals/mandelbrot/mandel1.gif[/img]

2008 Hungary-Israel Binational, 1

Tags: inequalities , triangle inequality , perpendicular bisector , geometry

Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

2009 ITAMO, 1

Tags: combinatorics

A flea is initially at the point $(0, 0)$ in the Cartesian plane. Then it makes $n$ jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length $1$, the second of length $2$, the third of length $4$, and so on. The $n^{th}$-jump is of length $2^{n-1}$. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the $n$ jumps.

Durer Math Competition CD Finals - geometry, 2011.C5

Tags: geometry , equilateral , angle

Given a straight line with points $A, B, C$ and $D$. Construct using $AB$ and $CD$ regular triangles (in the same half-plane). Let $E,F$ be the third vertex of the two triangles (as in the figure) . The circumscribed circles of triangles $AEC$ and $BFD$ intersect in $G$ ($G$ is is in the half plane of triangles). Prove that the angle $AGD$ is $120^o$ [img]https://1.bp.blogspot.com/-66akc83KSs0/X9j2BBOwacI/AAAAAAAAM0M/4Op-hrlZ-VQRCrU8Z3Kc3UCO7iTjv5ZQACLcBGAsYHQ/s0/2011%2BDurer%2BC5.png[/img]

1974 IMO Longlists, 18

Tags: geometry , triangle , congruent triangles , circles

Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.

2009 Today's Calculation Of Integral, 512

Tags: calculus , integration , trigonometry , calculus computations , geometry

Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$

2017 Tuymaada Olympiad, 6

Tags: number theory

Let $\sigma(n) $ denote the sum of positive divisors of a number $n $. A positive integer $N=2^rb $ is given,where $r $ and $b $ are positive integers and $b $ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma (b) $ are coprime. Tuymaada Q6 Juniors

1992 IMO Longlists, 46

Tags: number theory , power of 2 , arithmetic sequence

Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$

2018 AMC 12/AHSME, 9

Tags: gauss

What is \[ \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? \] $ \textbf{(A) }100,100 \qquad \textbf{(B) }500,500\qquad \textbf{(C) }505,000 \qquad \textbf{(D) }1,001,000 \qquad \textbf{(E) }1,010,000 \qquad $