Olympiad problems

VMEO III 2006, 11.2

Tags: geometry

Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)

1974 AMC 12/AHSME, 8

Tags: modular arithmetic

What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$

2022/2023 Tournament of Towns, P1

Tags: combinatorics , geometry

One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).

1997 Moscow Mathematical Olympiad, 4

Tags:

Given real numbers $a_1\leq{a_2}\leq{a_3}$ and $b_1\leq{b_2}\leq{b_3}$ such that $$a_1+a_2+a_3=b_1+b_2+b_3,$$ $$a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.$$ Prove that if $a_1\leq{b_1},$ then $a_3\leq{b_3}$

2005 Today's Calculation Of Integral, 84

Tags: calculus , integration , limit , trigonometry , calculus computations

Evaluate \[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]

2006 AIME Problems, 10

Tags: geometry , analytic geometry , graphing lines , slope , geometric transformation , parallelogram , rectangle

This is the one with the 8 circles? I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...

1983 AIME Problems, 1

Tags: basic maths , logarithm

Let $x$, $y$, and $z$ all exceed 1 and let $w$ be a positive number such that \[\log_x w = 24,\quad \log_y w = 40 \quad\text{and}\quad \log_{xyz} w = 12.\] Find $\log_z w$.

2024 India IMOTC, 8

Tags: number theory

Let $a$ and $n$ be positive integers such that: 1. $a^{2^n}-a$ is divisible by $n$, 2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is [i]not[/i] divisible by $n$. Prove that $n$ has a prime factor [i]smaller[/i] than $2024$. [i]Proposed by Shantanu Nene[/i]

2003 India IMO Training Camp, 3

Tags: function , algebra

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

2009 Dutch IMO TST, 4

Tags: functional equation , functional equation in z , algebra

Find all functions $f : Z \to Z$ satisfying $f(m + n) + f(mn -1) = f(m)f(n) + 2$ for all $m, n \in Z$.

1967 Polish MO Finals, 3

Tags: combinatorics

There are 100 persons in a hall, everyone knowing at least 67 of the others. Prove that there always exist four of them who know each other

1962 All Russian Mathematical Olympiad, 026

Tags: combinatorics , table

Given positive numbers $a_1, a_2, ..., a_m, b_1, b_2, ..., b_n$. Is known that $$a_1+a_2+...+a_m=b_1+b_2+...+b_n.$$ Prove that you can fill an empty table with $m$ rows and $n$ columns with no more than $(m+n-1)$ positive number in such a way, that for all $i,j$ the sum of the numbers in the $i$-th row will equal to $a_i$, and the sum of the numbers in the $j$-th column -- to $b_j$.

2024 AMC 10, 22

Tags: geometry

Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img] $\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$

2012 JBMO ShortLists, 1

Tags: combinatorics

Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee . The cards are arranged in a regular polygon manner . Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name . Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names ?

Champions Tournament Seniors - geometry, 2017.4

Tags: equal segments , angle bisector , circles , geometry

Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$

2004 India IMO Training Camp, 1

Tags: geometry , circumcircle , trigonometry

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4

Tags:

We have a square grid of 4 times 4 points. How many triangles are there with vertices on the points? (The three vertices may not lie on a line.) A. 252 B. 256 C. 360 D. 516 E. 560

2010 Tournament Of Towns, 7

Tags: geometry , rectangle , rotation , induction , combinatorics

A square is divided into congruent rectangles with sides of integer lengths. A rectangle is important if it has at least one point in common with a given diagonal of the square. Prove that this diagonal bisects the total area of the important rectangles

2018 CMIMC Number Theory, 4

Tags: number theory

Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.

2011 Putnam, A2

Tags: ratio , geometric series , summation

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

1991 IberoAmerican, 1

Tags: geometry , 3d geometry , combinatorics

Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.

2012 District Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron , perpendicular

Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$

2012 Online Math Open Problems, 3

Tags:

A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]Lucky numbers are positive. [*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][/hide]

2012 Purple Comet Problems, 20

Tags: vector , analytic geometry , graphing lines , slope , algebra , system of equations

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

1996 Baltic Way, 13

Tags: function , algebra

Consider the functions $f$ defined on the set of integers such that \[f(x)=f(x^2+x+1)\] for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.