Olympiad problems

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

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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

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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

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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

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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 $323$? $\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$

1991 Arnold's Trivium, 34

Tags: calculus , derivative

Investigate the singular points on the curve $y=x^3$ in the projective plane.

2008 Hanoi Open Mathematics Competitions, 9

Tags: geometry , right triangle

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

2010 Today's Calculation Of Integral, 568

Tags: calculus , integration , probability , limit , calculus computations , logarithm

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

2019 Saudi Arabia Pre-TST + Training Tests, 2.2

Tags: combinatorics , square table

A sequence $(a_1, a_2,...,a_k)$ consisting of pairwise different cells of an $n\times n$ board is called a cycle if $k \ge 4$ and cell ai shares a side with cell $a_{i+1}$ for every $i = 1,2,..., k$, where $a_{k+1} = a_1$. We will say that a subset $X$ of the set of cells of a board is [i]malicious [/i] if every cycle on the board contains at least one cell belonging to $X$. Determine all real numbers $C$ with the following property: for every integer $n \ge 2$ on an $n\times n$ board there exists a malicious set containing at most $Cn^2$ cells.

1996 Bundeswettbewerb Mathematik, 3

Tags: combinatorial geometry , geometry , median

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

2024 VJIMC, 2

Tags: linear algebra , matrices , matrix

Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before. A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v\in \mathbb{R}^{2024}$ with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix

2024 Chile Classification NMO Juniors, 3

Tags: number theory

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

2005 Irish Math Olympiad, 2

Tags: combinatorics

Using the digits: $ 1,2,3,4,5,$ players $ A$ and $ B$ compose a $ 2005$-digit number $ N$ by selecting one digit at a time: $ A$ selects the first digit, $ B$ the second, $ A$ the third and so on. Player $ A$ wins if and only if $ N$ is divisible by $ 9$. Who will win if both players play as well as possible?

1968 Miklós Schweitzer, 5

Tags: inequalities , real analysis

Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\] [i]P. Turan[/i]

1980 IMO, 4

Tags: induction , summation , counting , combinatorics

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)