Olympiad problems

1999 May Olympiad, 1

Two integers between $1$ and $100$ inclusive are chosen such that their difference is $7$ and their product is a multiple of $5$. In how many ways can this choice be made?

PEN H Problems, 53

Tags: diophantine equation , modular arithmetic

Suppose that $a, b$, and $p$ are integers such that $b \equiv 1 \; \pmod{4}$, $p \equiv 3 \; \pmod{4}$, $p$ is prime, and if $q$ is any prime divisor of $a$ such that $q \equiv 3 \; \pmod{4}$, then $q^{p}\vert a^{2}$ and $p$ does not divide $q-1$ (if $q=p$, then also $q \vert b$). Show that the equation \[x^{2}+4a^{2}= y^{p}-b^{p}\] has no solutions in integers.

2016 Regional Competition For Advanced Students, 2

Tags: inequalities , algebra

Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$. Prove that the inequality $$(a+2)(b+2) \ge cd$$ holds and give four numbers $a$, $b$, $c$ and $d$ such that equality holds. (Walther Janous)

1971 All Soviet Union Mathematical Olympiad, 151

Tags: combinatorics , algebra

Some numbers are written along the ring. If inequality $(a-d)(b-c) < 0$ is held for the four arbitrary numbers in sequence $a,b,c,d$, you have to change the numbers $b$ and $c$ places. Prove that you will have to do this operation finite number of times.

2018 CMIMC Combinatorics, 3

Tags: combinatorics

Michelle is at the bottom-left corner of a $6\times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$; the first time Michelle moves to one of these points she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?

2015 India IMO Training Camp, 1

Tags: geometry

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

2015 AMC 12/AHSME, 25

Tags: algebra , polynomial , geometric series , absolute value

A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$? $ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$

2002 Switzerland Team Selection Test, 4

Tags: combinatorial geometry , combinatorics

A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.

2000 Baltic Way, 7

Tags: combinatorics

In a $ 40 \times 50$ array of control buttons, each button has two states: on and off. By touching a button, its state and the states of all buttons in the same row and in the same column are switched. Prove that the array of control buttons may be altered from the all-off state to the all-on state by touching buttons successively, and determine the least number of touches needed to do so.

1994 Spain Mathematical Olympiad, 6

Tags: convex polygon , combinatorics

A convex $n$-gon is dissected into $m$ triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that $m+n$ is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of $m$ and $n$.

2022 District Olympiad, P2

Tags: complex number , inequalities

Let $z_1,z_2$ and $z_3$ be complex numbers of modulus $1,$ such that $|z_i-z_j|\geq\sqrt{2}$ for all $i\neq j\in\{1,2,3\}.$ Prove that \[|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.\][i]Mathematical Gazette[/i]

2006 China Northern MO, 5

Tags: function , inequalities

$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]

1986 AMC 8, 2

Tags:

Which of the following numbers has the largest reciprocal? \[ \textbf{(A)}\ \frac{1}{3} \qquad \textbf{(B)}\ \frac{2}{5} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 1986 \]

2013 NIMO Problems, 8

Tags: euler , number theory , prime factorization

For a finite set $X$ define \[ S(X) = \sum_{x \in X} x \text{ and } P(x) = \prod_{x \in X} x. \] Let $A$ and $B$ be two finite sets of positive integers such that $\left\lvert A \right\rvert = \left\lvert B \right\rvert$, $P(A) = P(B)$ and $S(A) \neq S(B)$. Suppose for any $n \in A \cup B$ and prime $p$ dividing $n$, we have $p^{36} \mid n$ and $p^{37} \nmid n$. Prove that \[ \left\lvert S(A) - S(B) \right\rvert > 1.9 \cdot 10^{6}. \][i]Proposed by Evan Chen[/i]

2014 ASDAN Math Tournament, 7

Tags: team test

Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value?

2009 Princeton University Math Competition, 4

Tags: geometry , rectangle

We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?

2023 CMIMC Algebra/NT, 8

Tags: algebra , number theory

Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$, with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \underline{B} + \underline{C} + \underline{D}$. [i]Proposed by Kyle Lee[/i]

2011 Laurențiu Duican, 3

Tags: function , find all functions , real analysis

Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying $$ \int_0^2 xf(x)dx=f(0)+f(2) . $$ [i]Cristinel Mortici[/i]

2023 India Regional Mathematical Olympiad, 4

Tags: digit , number theory , combinatorics

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition: [i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\ Determine the smallest possible value of $N$.

2005 Today's Calculation Of Integral, 16

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int \sin (\ln x)dx$ [2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$ [3] $\int \frac{x^3}{x^2+1}dx$ [4] $\int \frac{x^2}{\sqrt{2x-1}}dx$ [5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$

2022 JHMT HS, 9

Tags: geometry , 3d geometry , sphere , expected value , probability

Let $B$ and $D$ be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point $A$ (this unit sphere is the set of all points in $\mathbb{R}^3$ a distance of $1$ away from $A$). Compute the expected value of $\sin^2\angle DAB$.

2011 Poland - Second Round, 1

Tags: geometric transformation , rotation , geometry

Points $A,B,C,D,E,F$ lie in that order on semicircle centered at $O$, we assume that $AD=BE=CF$. $G$ is a common point of $BE$ and $AD$, $H$ is a common point of $BE$ and $CD$. Prove that: \[\angle AOC=2\angle GOH.\]

2021/2022 Tournament of Towns, P3

Tags: algebra

Let $n$ be a positive integer. Let us call a sequence $a_1,a_2,\dots,a_n$ interesting if for any $i=1,2,\dots,n$ either $a_i=i$ or $a_i=i+1$. Let us call an interesting sequence even if the sum of its members is even, and odd otherwise. Alice has multiplied all numbers in each odd interesting sequence and has written the result in her notebook. Bob, in his notebook, has done the same for each even interesting sequence. In which notebook is the sum of the numbers greater than by how much? (The answer may depend on $n$.)

1949 Miklós Schweitzer, 1

Tags: limit , real analysis

Let an infinite sequence of measurable sets be given on the interval $ (0,1)$ the measures of which are $ \geq \alpha>0$. Show that there exists a point of $ (0,1)$ which belongs to infinitely many terms of the sequence.

2015 Princeton University Math Competition, B5

Tags:

Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that $p \mid (a^{p-2} - a)$?