Olympiad problems

1998 Tuymaada Olympiad, 1

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

2012 Hitotsubashi University Entrance Examination, 5

Tags: probability , expected value , probability and stats

At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7. (1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition. (2) Find the expected value of the number of the spot on the top face after the $n$-repetition.

1976 IMO Shortlist, 12

Tags: combinatorics , polynomial , counting , additive combinatorics , algebra

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.

2001 National Olympiad First Round, 6

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How many $5-$digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$? $ \textbf{(A)}\ 3125 \qquad\textbf{(B)}\ 2500 \qquad\textbf{(C)}\ 1845 \qquad\textbf{(D)}\ 1190 \qquad\textbf{(E)}\ \text{None of the preceding} $

2024 LMT Fall, 7

Tags: team

Let $A$, $F$, $L$, $M$, and $T$ be distinct digits such that $\overline{FALL} + \overline{LMT} = 2024$ and $F$, $L > 0$. Find the sum of all possible values of $\overline{FAT}$.

2020 CCA Math Bonanza, T5

Tags: quadratic

Find all pairs of real numbers $(x,y)$ satisfying both equations \[ 3x^2+3xy+2y^2 =2 \] \[ x^2+2xy+2y^2 =1. \] [i]2020 CCA Math Bonanza Team Round #5[/i]

2017 Polish MO Finals, 5

Tags: geometry

Point $M$ is the midpoint of $BC$ of a triangle $ABC$, in which $AB=AC$. Point $D$ is the orthogonal projection of $M$ on $AB$. Circle $\omega$ is inscribed in triangle $ACD$ and tangent to segments $AD$ and $AC$ at $K$ and $L$ respectively. Lines tangent to $\omega$ which pass through $M$ cross line $KL$ at $X$ and $Y$, where points $X$, $K$, $L$ and $Y$ lie on $KL$ in this specific order. Prove that points $M$, $D$, $X$ and $Y$ are concyclic.

2016 CHMMC (Fall), 2

Tags: probability , analytic geometry

Alice and Bob find themselves on a coordinate plane at time $t=0$ at $A(1,0)$ and $B(-1,0)$ respectively. They have no sense of direction, but they want to find each other. They each pick a direction independently and with uniform random probability. Both Alice and Bob travel at a constant speed of $1 \frac{unit}{min}$ in their chosen directions. They continue on their straight line paths forever, each hoping to catch sight of the other. They both have a $1$ unit radius of view; they can see something if and only if its distance from them is at most $1$ unit. What is the probability they never see each other?

1957 Putnam, B7

Tags: combinatorial geometry , polygon , geometry

Let $C$ consist of a regular polygon and its interior. Show that for each positive integer $n$, there exists a set of points $S(n)$ in the plane such that every $n$ points can be covered by $C$, but $S(n)$ cannot be covered by $C.$

2010 IFYM, Sozopol, 2

Tags: inequalities

If $a,b,c>0$ and $abc=3$,find the biggest value of: $\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$

2014 VTRMC, Problem 2

Tags: calculus , integration

Evaluate $\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx$.

2007 AMC 10, 23

Tags: ratio , 3d geometry , pyramid , geometry

A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad \textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 4 \plus{} 2\sqrt{2}$

1964 Polish MO Finals, 1

Tags: trigonometry , inequalities

Prove that the inequality $$ \frac{1}{3} \leq \frac{\tan 3\alpha}{\tan \alpha} \leq 3 $$ is not true for any value of $ \alpha $.

2008 District Olympiad, 2

Tags: algebra , function , domain

Consider the positive reals $ x$, $ y$ and $ z$. Prove that: a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$. b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.

2020 AMC 8 -, 6

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Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car? $\textbf{(A) }\text{Aaron}\qquad \textbf{(B) }\text{Darren}\qquad \textbf{(C) }\text{Karen}\qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}\qquad$

2015 CCA Math Bonanza, I8

Tags: perimeter , algebra

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

2014 Olympic Revenge, 2

Tags: limit , floor function , number theory

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

1997 IMC, 2

Tags: linear algebra , matrix

Let $M \in GL_{2n}(K)$, represented in block form as \[ M = \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] , M^{-1} = \left[ \begin{array}{cc} E & F \\ G & H \end{array} \right] \] Show that $\det M.\det H=\det A$.

2013 Harvard-MIT Mathematics Tournament, 1

Tags: hmmt

Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , angle bisector

Points $H$ and $L$ are, respectively, the feet of the altitude and the angle bisector drawn from the vertex $A$ of the triangle $ABC$, $K$ is the touchpoint of the circle inscribed in the triangle $ABC$ with the side $BC$. Under what conditions will $AK$ be the bisector of the angle $\angle LAH$? (Hryhorii Filippovskyi)

2010 China Second Round Olympiad, 2

Tags: number theory , modular arithmetic

Given a fixed integer $k>0,r=k+0.5$,define $f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$ where $[x]$ denotes the smallest integer not less than $x$. prove that there exists integer $m$ such that $f^m(r)$ is an integer.

2010 All-Russian Olympiad, 1

Tags: number theory

If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.

2007 Purple Comet Problems, 10

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Tom can run to Beth's house in $63$ minutes. Beth can run to Tom's house in $84$ minutes. At noon Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house. When they meet, they both run at Beth's speed back to Beth's house. At how many minutes after noon will they arrive at Beth's house?

2007 Iran MO (2nd Round), 1

Tags: circumcircle , geometry

In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.

2010 Contests, 1

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Mary's top book shelf holds five books with the following widths, in centimeters: $ 6$, $ \frac12$, $ 1$, $ 2.5$, and $ 10$. What is the average book width, in centimeters? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$