Olympiad problems

2021 Bangladeshi National Mathematical Olympiad, 4

$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$.

1997 Putnam, 3

Tags: integration , calculus , induction

Evaluate the following : \[ \int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots \right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2}+\cdots \right)\,\mathrm{d}x \]

2000 Argentina National Olympiad, 1

Tags: digit , number theory

The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$. Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.

2022 Benelux, 2

Tags: combinatorics

Let $n$ be a positive integer. There are $n$ ants walking along a line at constant nonzero speeds. Different ants need not walk at the same speed or walk in the same direction. Whenever two or more ants collide, all the ants involved in this collision instantly change directions. (Different ants need not be moving in opposite directions when they collide, since a faster ant may catch up with a slower one that is moving in the same direction.) The ants keep walking indefinitely. Assuming that the total number of collisions is finite, determine the largest possible number of collisions in terms of $n$.

2024 Bulgaria MO Regional Round, 12.1

Tags: incentre , geometry , conditional geometry , right angle , isosceles

Let $ABC$ be an acute triangle with midpoint $M$ of $AB$. The point $D$ lies on the segment $MB$ and $I_1, I_2$ denote the incenters of $\triangle ADC$ and $\triangle BDC$. Given that $\angle I_1MI_2=90^{\circ}$, show that $CA=CB$.

LMT Team Rounds 2010-20, B29

Tags: algebra

Alicia bought some number of disposable masks, of which she uses one per day. After she uses each of her masks, she throws out half of them (rounding up if necessary) and reuses each of the remaining masks, repeating this process until she runs out of masks. If her masks lasted her $222$ days, how many masks did she start out with?

V Soros Olympiad 1998 - 99 (Russia), 9.9

Tags: geometry , geometric inequality , area

What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$ and $c$ from a certain point (where $a$ is the distance to the vertex of the right angle)?

1977 IMO Shortlist, 2

Tags: combinatorics , lattice points , combinatorial geometry , circles

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let $k$ be a circle with radius $r \geq 2$, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle $k$ that has a neighboring point lying outside $k$. Similarly, an exterior boundary point is a lattice point lying outside the circle $k$ that has a neighboring point lying inside $k$. Prove that there are four more exterior boundary points than interior boundary points.

2013 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , circumcircle , antipode

In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$

2019 Bundeswettbewerb Mathematik, 4

Tags: digit , sqrt , number theory

In the decimal expansion of $\sqrt{2}=1.4142\dots$, Isabelle finds a sequence of $k$ successive zeroes where $k$ is a positive integer. Show that the first zero of this sequence can occur no earlier than at the $k$-th position after the decimal point.

2018 Federal Competition For Advanced Students, P2, 4

Tags: geometry , circumcircle , collinear

Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$. [i](Proposed by Walther Janous)[/i]

1950 AMC 12/AHSME, 29

Tags:

A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes alone is: $\textbf{(A)}\ 8-x=2 \qquad \textbf{(B)}\ \dfrac{1}{8}+\dfrac{1}{x}=\dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{500}{8}+\dfrac{500}{x}=500 \qquad \textbf{(D)}\ \dfrac{x}{2}+\dfrac{x}{8}=1 \qquad\\ \textbf{(E)}\ \text{None of these answers}$

2024 Indonesia TST, 1

Tags: geometry

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

1996 Dutch Mathematical Olympiad, 2

Tags: perfect square , number theory

Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.

2005 MOP Homework, 5

Tags: algebra

Let $S$ be a finite set of positive integers such that none of them has a prime factor greater than three. Show that the sum of the reciprocals of the elements in $S$ is smaller than three.

2002 India IMO Training Camp, 21

Tags: number theory

Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.

2002 AMC 12/AHSME, 15

Tags:

How many four-digit numbers $ N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $ N$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

2011 Math Prize For Girls Problems, 4

Tags: logarithm

If $x > 10$, what is the greatest possible value of the expression \[ {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ? \] All the logarithms are base 10.

2005 Thailand Mathematical Olympiad, 17

Tags: algebra

For $a, b \ge 0$ we define $a * b = \frac{a+b+1}{ab+12}$ . Compute $0*(1*(2*(... (2003*(2004*2005))...)))$.

2017 Online Math Open Problems, 7

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Let $S$ be a set of $13$ distinct, pairwise relatively prime, positive integers. What is the smallest possible value of $\max_{s \in S} s- \min_{s \in S}s$? [i]Proposed by James Lin

2003 Turkey Junior National Olympiad, 1

Tags: geometry , trigonometry , quadratic , cyclic quadrilateral , trig identities , law of cosines

Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.

1992 Poland - First Round, 6

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The sequence $(x_n)$ is determined by the conditions: $x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$. Find $\sum_{n=0}^{1992} 2^nx_n$.

2023 239 Open Mathematical Olympiad, 2

Tags: geometry

The excircles of triangle $ABC$ touch its sides $BC$, $CA$, and $AB$ at points $A_1$, $B_1$, and $C_1$, respectively. Let $B_2$ and $C_2$ be the midpoints of segments $BB_1$ and $CC_1$, respectively. Line $B_2C_2$ intersects line $BC$ at point $W$. Prove that $AW = A_1W$.

1976 Chisinau City MO, 127

Tags: combinatorial geometry , combinatorics

The convex $1976$-gon is divided into $1975$ triangles. Prove that there is a straight line separating one of these triangles from the rest.

2011 Indonesia TST, 4

Tags: modular arithmetic , induction , number theory

A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \] If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.