Olympiad problems

1995 ITAMO, 4

An acute-angled triangle $ABC$ is inscribed in a circle with center $O$. The bisector of $\angle A$ meets $BC$ at $D$, and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$. Show that $AB = AP$.

1956 AMC 12/AHSME, 17

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The fraction $ \frac {5x \minus{} 11}{2x^2 \plus{} x \minus{} 6}$ was obtained by adding the two fractions $ \frac {A}{x \plus{} 2}$ and $ \frac {B}{2x \minus{} 3}$. The values of $ A$ and $ B$ must be, respectively: $ \textbf{(A)}\ 5x, \minus{} 11 \qquad\textbf{(B)}\ \minus{} 11,5x \qquad\textbf{(C)}\ \minus{} 1,3 \qquad\textbf{(D)}\ 3, \minus{} 1 \qquad\textbf{(E)}\ 5, \minus{} 11$

2011 ELMO Shortlist, 2

Tags: modular arithmetic , number theory

Let $p\ge5$ be a prime. Show that \[\sum_{k=0}^{(p-1)/2}\binom{p}{k}3^k\equiv 2^p - 1\pmod{p^2}.\] [i]Victor Wang.[/i]

2024 Mathematical Talent Reward Programme, 1

Tags: algebra

Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs. $20$, and he sells it at Rs. $24$. One day while riding his cycle with $20$L, Hari trips and loses $5$L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than $10$% impure ($1$L water in $10$L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day? $(A)$ Rs $12$ profit $(B)$ Rs $24$ profit $(C)$ No profit $(D)$ Rs $12$ loss

2003 Gheorghe Vranceanu, 1

Tags: floor function , algebra , equation

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2017 CMIMC Number Theory, 4

Tags: number theory

Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

2009 Baltic Way, 20

Tags: analytic geometry , graphing lines , slope , combinatorics

In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?

1999 Akdeniz University MO, 3

Tags: algebra , inequalities

Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

1977 Chisinau City MO, 138

Tags: geometry , isosceles , angle

In an isosceles triangle $BAC$ ($| AC | = | AB |$) , point $D$ is marked on the side $AC$. Determine the angles of the triangle $BDC$ if $\angle A = 40^o$ and $|BC|: |AD|= \sqrt3$.

2000 IMC, 6

Tags: linear algebra , matrix , algebra , polynomial

Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$ Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$, $P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.

2018 BMT Spring, 6

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Let $x,y,z \in \mathbb{R}$ and $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$. The minimum value of $x^2 + y^2 + z^2$ can be expressed as $\dfrac{a}{b}$ where $a,b \in \mathbb{Z}, \gcd(a,b) = 1$. What is $a + b$?

2011 Tournament of Towns, 3

Tags: weighing , combinatorics

A balance and a set of pairwise different weights are given. It is known that for any pair of weights from this set put on the left pan of the balance, one can counterbalance them by one or several of the remaining weights put on the right pan. Find the least possible number of weights in the set.

1990 AMC 12/AHSME, 21

Tags: geometry , 3d geometry , pyramid , trigonometry , pythagorean theorem

Consider a pyramid $P-ABCD$ whose base $ABCD$ is a square and whose vertex $P$ is equidistant from $A$, $B$, $C$, and $D$. If $AB=1$ and $\angle APD=2\theta$ then the volume of the pyramid is $\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}$

1999 Putnam, 5

Tags: polynomial , integration , calculus , inequalities , real analysis

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

2010 Austria Beginners' Competition, 3

Tags: function

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)

2014 Israel National Olympiad, 1

Tags: modular arithmetic , digit , number theory

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

2001 Brazil Team Selection Test, Problem 2

Tags: number theory

Let $f(n)$ denote the least positive integer $k$ such that $1+2+\cdots+k$ is divisible by $n$. Show that $f(n)=2n-1$ if and only if $n$ is a power of $2$.

2017 AMC 10, 6

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What is the largest number of solid $2\text{-in}\times 2\text{-in}\times 1\text{-in}$ blocks that can fit in a $3\text{-in}\times 2\text{-in}\times 3\text{-in}$ box? $\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$

2004 Manhattan Mathematical Olympiad, 1

Tags: geometry

Suppose two triangles have equal areas and equal perimeters. Prove that, if a side of one triangle is congruent to a side of the other triangle, then the two triangles are congruent.

2008 Thailand Mathematical Olympiad, 4

Tags: inequalities , algebra

Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$ for all real numbers $a, b, c > 0$

2012 Turkey Team Selection Test, 2

Tags: inequalities , pythagorean theorem , geometry

In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that \[ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 \]

2021 Durer Math Competition Finals, 3

Tags: combinatorics

On the evening of Halloween a group of $n$ kids collected $k$ bars of chocolate of the same type. At the end of the evening they wanted to divide the bars so that everybody gets the same amount of chocolate, and none of the bars is broken into more than two pieces. For which $n$ and $k$ is this possible?

2010 Contests, 3

Tags:

Show that , for any positive integer $n$ , the sum of $8n+4$ consecutive positive integers cannot be a perfect square .

2005 AMC 12/AHSME, 18

Tags: geometry , analytic geometry , graphing lines , slope

Let $ A(2,2)$ and $ B(7,7)$ be points in the plane. Define $ R$ as the region in the first quadrant consisting of those points $ C$ such that $ \triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $ R$? $ \textbf{(A)}\ 25 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

2009 Stanford Mathematics Tournament, 5

Tags: algebra , calculus

Compute $\int_{0}^{\infty} t^5e^{-t}dt$