Olympiad problems

1949 Putnam, B4

Tags: expansion

Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion $$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$ are positive integers.

2013 IPhOO, 8

Tags: geometry , 3d geometry , prism , optics , snell's law

A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$. [i](Proposed by Ahaan Rungta)[/i]

2013 National Olympiad First Round, 16

Tags: probability

$16$ white and $4$ red balls that are not identical are distributed randomly into $4$ boxes which contain at most $5$ balls. What is the probability that each box contains exactly $1$ red ball? $ \textbf{(A)}\ \dfrac{5}{64} \qquad\textbf{(B)}\ \dfrac{1}{8} \qquad\textbf{(C)}\ \dfrac{4^4}{\binom{16}{4}} \qquad\textbf{(D)}\ \dfrac{5^4}{\binom{20}{4}} \qquad\textbf{(E)}\ \dfrac{3}{32} $

2016 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , parallelogram , perpendicular

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

2006 Stanford Mathematics Tournament, 14

Tags: geometry

Determine the area of the region defined by [i]x[/i]²+[i]y[/i]²≤[i]π[/i]² and [i]y[/i] ≥ sin [i]x[/i].

2008 Germany Team Selection Test, 2

Tags: geometry , trapezoid , homothety , ddit

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

2009 JBMO TST - Macedonia, 4

Tags: geometry , rectangle , algorithm , induction , combinatorics

In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?

2024 Switzerland Team Selection Test, 12

Tags: integer , functional equation

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

2004 Chile National Olympiad, 4

Tags: sum of digits , digit , number theory

Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

2015 Iran Team Selection Test, 3

Tags: number theory

Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .

1998 AIME Problems, 11

Tags: geometry , 3d geometry , vector , analytic geometry , symmetry , geometric transformation

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

2018 Canadian Open Math Challenge, B1

Tags:

Source: 2018 Canadian Open Math Challenge Part B Problem 1 ----- Let $(1+\sqrt2)^5 = a+b\sqrt2$, where $a$ and $b$ are positive integers. Determine the value of $a+b.$

2011 Vietnam National Olympiad, 1

Tags: modular arithmetic , quadratic , algebra , polynomial , number theory

Define the sequence of integers $\langle a_n\rangle$ as; \[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\] Prove that $a_{2012}-2010$ is divisible by $2011.$

2000 Spain Mathematical Olympiad, 2

Tags: trigonometry , combinatorics

Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$

2019-IMOC, C1

Tags: combinatorics , partition , distributions , parity

Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies $$2\mid x_1,x_2,\ldots,x_k$$ $$x_1+x_2+\ldots+x_k=n$$ then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.

2004 Romania National Olympiad, 4

Tags: superior algebra , group theory , abstract algebra , slope

Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$. (a) Prove that $-1$ is the square of an element from $\mathcal K.$ (b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$. (c) Can $0$ be written in the same way? [i]Marian Andronache[/i]

1992 Balkan MO, 2

Tags: inequalities

Prove that for all positive integers $n$ the following inequality takes place \[ (2n^2+3n+1)^n \geq 6^n (n!)^2 . \] [i]Cyprus[/i]

2014 Taiwan TST Round 1, 1

Tags: function , integration , logarithm

Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.

1997 All-Russian Olympiad Regional Round, 8.6

Tags: combinatorics , number theory

The numbers from 1 to 37 are written in a line so that the sum of any first several numbers is divided by the number following them. What number is worth in third place, if the number 37 is written in the first place, and in the second, 1?

LMT Guts Rounds, 2020 F19

Tags:

Find the second smallest prime factor of $18!+1.$ [i]Proposed by Kaylee Ji[/i]

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3

Tags: modular arithmetic

What is the last digit of $ 17^{1996}$? A. 1 B. 3 C. 5 D. 7 E. 9

2008 ITest, 10

Tags:

Tony has an old sticky toy spider that very slowly "crawls" down a wall after being stuck to the wall. In fact, left untouched, the toy spider crawls down at a rate of one inch for every two hours it's left stuck to the wall. One morning, at around $9$ o' clock, Tony sticks the spider to the wall in the living room three feet above the floor. Over the next few mornings, Tony moves the spider up three feet from the point where he finds it. If the wall in the living room is $18$ feet high, after how many days (days after the first day Tony places the spider on the wall) will Tony run out of room to place the spider three feet higher?

2021 JHMT HS, 1

Tags: general

Walter owns $11$ dumbbells, which have weights, in pounds, $1,$ $2,$ $\ldots,$ $10,$ $11.$ Walter wants to split his dumbbells into three groups of equal total weight. What is the smallest possible product that the dumbbell weights in any one of these groups can have?

2019 Saudi Arabia Pre-TST + Training Tests, 2.3

Tags: equilateral , collinear , equal segments , geometry

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

2018 AMC 10, 21

Tags:

Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$? $\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$