Olympiad problems

1956 AMC 12/AHSME, 12

Tags:

If $ x^{ \minus{} 1} \minus{} 1$ is divided by $ x \minus{} 1$ the quotient is: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {1}{x \minus{} 1} \qquad\textbf{(C)}\ \frac { \minus{} 1}{x \minus{} 1} \qquad\textbf{(D)}\ \frac {1}{x} \qquad\textbf{(E)}\ \minus{} \frac {1}{x}$

2011 Akdeniz University MO, 5

Tags: number theory

For all $n \in {\mathbb Z^+}$ we define $$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$ infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$. [b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$ [b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$

2013 USAMTS Problems, 5

Tags: geometry , analytic geometry , ratio , function , calculus , derivative

Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?

1999 French Mathematical Olympiad, Problem 1

Tags: 3d geometry , geometry

What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.

2007 All-Russian Olympiad, 8

Tags: number theory

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

1949-56 Chisinau City MO, 20

Tags: algebra

From point $A$ to point $B$, the car drove at a speed of $50$ km / h, and from $B$ to $A$ , at a speed of $30$ km / h. What was the average vehicle speed?

STEMS 2023 Math Cat A, 6

Tags: combinatorics

Define a positive integer $n$ to be a fake square if either $n = 1$ or $n$ can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer $k \geqslant 2$, there exist distinct positive integers $a_1$, $a_2, \cdots, a_k$ such that the polynomial $(x+a_1)(x+a_2) \cdots (x+a_k)$ takes ‘fake square’ values for all $x = 1,2,\cdots,2023$. [i]Proposed by Prof. Aditya Karnataki[/i]

2019 Vietnam TST, P4

Tags: number theory , diophantine equation

Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.

2025 NCMO, 2

Tags: geometry

In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent. [i]Alan Cheng[/i]

1967 IMO Longlists, 25

Tags: geometry , 3d geometry , sphere , angle

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

1967 IMO Shortlist, 3

Tags: trigonometry , calculus , taylor series , inequality , trigonometric inequality

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

1989 India National Olympiad, 3

Tags: combinatorics

Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.

1991 Putnam, B3

Tags: geometry , rectangle , combinatorics

Can we find $N$ such that all $m\times n$ rectangles with $m,n>N$ can be tiled with $4\times6$ and $5\times7$ rectangles?

2011 Tokio University Entry Examination, 2

Tags: algorithm , number theory , euclidean algorithm

Define real number $y$ as the fractional part of real number $x$ such that $0\leq y<1$ and $x-y$ is integer. Denote this by $<x>$. For real number $a$, define an infinite sequence $\{a_n\}\ (n=1,\ 2,\ 3,\ \cdots)$ inductively as follows. (i) $a_1=<a>$ (ii) If $a\n\neq 0$, then $a_{n+1}=\left<\frac{1}{a_n}\right>$, if $a_n=0$, then $a_{n+1}=0$. (1) For $a=\sqrt{2}$, find $a_n$. (2) For any natural number $n$, find real number $a\geq \frac 13$ such that $a_n=a$. (3) Let $a$ be a rational number. When we express $a=\frac{p}{q}$ with integer $p$, natural number $q$, prove that $a_n=0$ for any natural number $n\geq q$. [i]2011 Tokyo University entrance exam/Science, Problem 2[/i]

2016 Argentina National Olympiad Level 2, 6

Tags: combinatorics

There are $999$ black points marked on a circle, dividing it into $999$ arcs of length $1$. We need to place $d$ arcs of lengths $1, 2, \dots, d$ such that each arc starts and ends at two black points, and none of the $d$ arcs is contained within another. Find the maximum value of $d$ for which this construction is possible. [b]Note:[/b] Two arcs can have one or more black points in common.

1958 AMC 12/AHSME, 33

Tags: quadratic , vieta , algebra , quadratic formula

For one root of $ ax^2 \plus{} bx \plus{} c \equal{} 0$ to be double the other, the coefficients $ a,\,b,\,c$ must be related as follows: $ \textbf{(A)}\ 4b^2 \equal{} 9c\qquad \textbf{(B)}\ 2b^2 \equal{} 9ac\qquad \textbf{(C)}\ 2b^2 \equal{} 9a\qquad \\ \textbf{(D)}\ b^2 \minus{} 8ac \equal{} 0\qquad \textbf{(E)}\ 9b^2 \equal{} 2ac$

2002 China Team Selection Test, 3

Tags: combinatorics

There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.

2001 Saint Petersburg Mathematical Olympiad, 10.5

Tags: geometry , angle , bisector

On the bisector $AL$ of triangle $ABC$ a point $K$ is chosen such that $\angle BKL=\angle KBL=30^{\circ}$. Lines $AB$ and $CK$ intersect at point $M$, lines $AC$ and $BK$ intersect at point $N$. FInd the measure of angle $\angle AMN$ [I]Proposed by D. Shiryaev, S. Berlov[/i]

Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3

Tags: geometry , square , angle chasing , equal angles , angle

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

1964 IMO Shortlist, 5

Tags: geometry , combinatorial geometry , 3d geometry , counting , intersection

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

1950 Miklós Schweitzer, 8

Tags: probability , probability and stats

A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$

IV Soros Olympiad 1997 - 98 (Russia), 11.1

Tags: algebra

Petya digs the garden bed alone for $a$ minutes longer than he does with Vasya. Vasya digs up the same bed for $b$ minutes longer than he would have done with Petya. How many minutes does it take Vasya and Petya to dig up the same bed together? orthogonal).

2024 LMT Fall, 20

Tags: speed

Henry places some rooks and some kings in distinct cells of a $2\times 8$ grid such that no two rooks attack each other and no two kings attack each other. Find the maximum possible number of pieces on the board. (Two rooks [i]attack[/i] each other if they are in the same row or column and no pieces are between them. Two kings attack each other if their cells share a vertex.)

2006 MOP Homework, 1

Tags: function , algebra , functional equation , divisibility

Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

2017 Mathematical Talent Reward Programme, MCQ: P 1

Tags: equation , algebra

The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]