Olympiad problems

2007 May Olympiad, 5

Tags: geometry

In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$. Prove that $AM = CM$.

2006 China Second Round Olympiad, 3

Tags: inequalities

Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $

2008 Hanoi Open Mathematics Competitions, 5

Tags: inequalities , algebra

Suppose $x, y, z, t$ are real numbers such that $\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}$ Prove that $x^2 + y^2 + z^2 + t^2 \le 1$.

Today's calculation of integrals, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

1969 IMO Shortlist, 4

Tags: conics , geometry , Locus , IMO Shortlist , IMO Longlist

$(BEL 4)$ Let $O$ be a point on a nondegenerate conic. A right angle with vertex $O$ intersects the conic at points $A$ and $B$. Prove that the line $AB$ passes through a fixed point located on the normal to the conic through the point $O.$

2017 Korea - Final Round, 3

Tags: function , algebra

For a positive integer $n$, denote $c_n=2017^n$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions. 1. For all positive integers $m, n$, $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$. 2. For all positive integer $n$, we have $0<f(c_{n+1})<f(c_n)^{2017}$. Prove that there exists a sequence $a_1, a_2, \cdots $ which satisfies the following. For all $n, k$ which satisfies $a_k<n$, we have $f(n)^{c_k} < f(c_k)^n$.

2018 District Olympiad, 3

Tags: geometry , rectangle , perpendicular

Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.

2002 National Olympiad First Round, 20

Tags:

Which of the following cannot be equal to $x^2+y^2$, if $x^2 + xy + y^2 = 1$ where $x,y$ are real numbers? $ \textbf{a)}\ \dfrac{1}{\sqrt 2} \qquad\textbf{b)}\ \dfrac 12 \qquad\textbf{c)}\ \sqrt 2 \qquad\textbf{d)}\ 3-\sqrt 3 \qquad\textbf{e)}\ \text{None of above} $

2017 IOM, 4

Tags: combinatorics

Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$. Proposed by Mikhail Evdokimov

1987 IMO Longlists, 4

Tags: inequalities , geometry , 3D geometry , inequalities unsolved

Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that \[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\] and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$

2018 Thailand TST, 1

Tags: rectangle , IMO Shortlist , combinatorics , Tiling

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

2015 Thailand TSTST, 1

Tags: algebra

Let $a,b,c$ be a real numbers such that this equations: $a^2x + b^2y + c^2z = 1$ $xy + yz + xz = 1$ have only one solution $(x, y, z)$ in real numbers. Prove that $a, b, c$ are sides of the triangle

2022 HMNT, 3

Tags: combinatorics

Find the number of ordered pairs $(A,B)$ such that the following conditions hold: $\bullet$ $A$ and $B$ are disjoint subsets of $\{1, 2, . . . , 50\}$. $\bullet$ $|A| = |B| = 25$ $\bullet$ The median of $B$ is $1$ more than the median of $A$.

2019 Sharygin Geometry Olympiad, 2

Tags: geometry , Sharygin Geometry Olympiad , perpendicular bisector

Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear.

1964 Poland - Second Round, 4

Tags: algebra , system of equations , System

Find the real numbers $ x, y, z $ satisfying the system of equations $$(z - x)(x - y) = a $$ $$(x - y)(y - z) = b$$ $$(y - z)(z - x) = c$$ where $ a, b, c $ are given real numbers.

2011 CIIM, Problem 2

Tags: CIIM 2011 , undergraduate

Let $k$ be a positive integer, and let $a$ be an integer such that $a-2$ is a multiple of $7$ and $a^6-1$ is a multiple of $7^k$. Prove that $(a + 1)^6-1$ is also a multiple of $7^k$.

2016 Costa Rica - Final Round, G2

Tags: geometry , right triangle

Consider $\vartriangle ABC$ right at $B, F$ a point such that $B - F - C$ and $AF$ bisects $\angle BAC$, $I$ a point such that $A - I - F$ and CI bisect $\angle ACB$, and $E$ a point such that $A- E - C$ and $AF \perp EI$. If $AB = 4$ and $\frac{AI}{IF}={4}{3}$ , determine $AE$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

2010 HMNT, 2

Tags: combinatorics

$16$ progamers are playing in another single elimination tournament. Each round, each of the remaining progamers plays against another and the loser is eliminated. Additionally, each time a progamer wins, he will have a ceremony to celebrate. A player's rst ceremony is ten seconds long, and afterward each ceremony is ten seconds longer than the last. What is the total length in seconds of all the ceremonies over the entire tournament?

1992 IMTS, 3

Tags: algebra unsolved , algebra

For $n$ a positive integer, denote by $P(n)$ the product of all positive integers divisors of $n$. Find the smallest $n$ for which \[ P(P(P(n))) > 10^{12} \]

2009 Indonesia TST, 4

Tags: modular arithmetic , combinatorics proposed , combinatorics

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

2015 Saudi Arabia Pre-TST, 2.2

Tags: algebra , functional

Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$. (Vo Quoc Ba Can)

2002 Moldova Team Selection Test, 2

Tags: combinatorics

Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.

2022 Princeton University Math Competition, A1 / B3

Tags: combinatorics

In the country of PUMaC-land, there are $5$ villages and $3$ cities. Vedant is building roads between the $8$ settlements according to the following rules: a) There is at most one road between any two settlements; b) Any city has exactly three roads connected to it; c) Any village has exactly one road connected to it; d) Any two settlements are connected by a path of roads. In how many ways can Vedant build the roads?

2024 BMT, 3

Tags: geometry

A square with side length $6$ has a circle with radius $2$ inside of it, with the centers of the square and circle vertically aligned. Aarush is standing $4$ units directly above the center of the circle, at point $P.$ What is the area of the region inside the square that he can see? (Assume that Aarush can only see parts of the square along straight lines of sight from $P$ that are unblocked by any other objects.) [center] [img] https://cdn.artofproblemsolving.com/attachments/7/7/ede4444ab82235fc90a27c0b481d320b486cf2.png [/img] [/center]

2020 Baltic Way, 13

Tags: geometry , Baltic Way

Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.