Olympiad problems

2006 Mathematics for Its Sake, 2

Let be a natural number $ n. $ Solve in the set of $ 2\times 2 $ complex matrices the equation $$ \begin{pmatrix} -2& 2007\\ 0&-2 \end{pmatrix} =X^{3n}-3X^n. $$ [i]Petru Vlad[/i]

2013 Singapore Senior Math Olympiad, 1

Tags: geometry , circumcircle , altitude , Isosceles Triangle , Triangles , right triangle , length

In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

2012 Purple Comet Problems, 14

Tags: number theory , relatively prime

A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$.

1965 Vietnam National Olympiad, 3

Tags: algebra , Inequality

1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer. 2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.

1988 Tournament Of Towns, (171) 4

Tags: weighings , combinatorics , power of 2

We have a set of weights with masses $1$ gm, $2$ gm, $4$ gm and so on, all values being powers of $2$ . Some of these weights may have equal mass. Some weights were put on both sides of a balance beam, resulting in equilibrium. It is known that on the left hand side all weights were distinct . Prove that on the right hand side there were no fewer weights than on the left hand side.

2023 ELMO Shortlist, C2

Tags: Elmo , combinatorics

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer $0\le n\le 52$, and cuts the deck into a pile with the top $n$ cards and a pile with the remaining $52-n$. She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size $n$ appear in order, as do the cards that were in the deck of size $52-n$.) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? [i]Proposed by Espen Slettnes[/i]

1993 AMC 8, 22

Tags: AMC

Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits? $\text{(A)}\ 22 \qquad \text{(B)}\ 99 \qquad \text{(C)}\ 112 \qquad \text{(D)}\ 119 \qquad \text{(E)}\ 199$

2025 Macedonian Mathematical Olympiad, Problem 3

Tags: combinatorics

On a horizontally placed number line, a pile of $ t_i > 0 $ tokens is placed on each number $ i \in \{1, 2, \ldots, s\} $. As long as at least one pile contains at least two tokens, we repeat the following procedure: we choose such a pile (say, it consists of $ k \geq 2 $ tokens), and move the top token from the selected pile $ k - 1 $ unit positions to the right along the number line. What is the largest natural number $ N $ on which a token can be placed? (Express $ N $ as a function of $ (t_i;\ i = 1, \ldots, s) $.)

2005 Junior Balkan Team Selection Tests - Moldova, 7

Tags: number theory unsolved , number theory

Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that if $2^p + 3^p = a^n$ then $n=1$

2014 Taiwan TST Round 3, 2

Tags: modular arithmetic , number theory , relatively prime , number theory proposed , primitive root

Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.

2002 Iran MO (3rd Round), 16

Tags: quadratics , Iran , Inequality , algebra

For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$

2011 IFYM, Sozopol, 6

Tags: number theory , Coloring

Let $\sum_{i=1}^n a_i x_i =0$, $a_i\in \mathbb{Z}$. It is known that however we color $\mathbb{N}$ with finite number of colors, then the upper equation has a solution $x_1,x_2,...,x_n$ in one color. Prove that there is some non-empty sum of its coefficients equal to 0.

2011 Morocco National Olympiad, 4

Tags: geometry , ratio , angle bisector , geometry unsolved

Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.

2005 iTest, 40

Tags: number theory

$ITEST + AHSIMC = 6666CS$. Each letter represents a unique digit from $0$ to $9$. How many solutions of the form $(C,A,S,H)$ exist?

1996 Romania National Olympiad, 1

Tags: algebra , inequalities

Let $a$ and $b$ be real numbers such that $a + b = 2$. Show that: $$\min \{|a|,|b|\} < 1 < \max \{|a|,|b|\} \Leftrightarrow a, b \in (-3,1)$$

2020-21 KVS IOQM India, 2

Tags: IOQM , KV , geometry , rectangle

If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

1998 All-Russian Olympiad, 1

Tags: analytic geometry , Vieta , algebra unsolved , algebra

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2023 Harvard-MIT Mathematics Tournament, 4

Tags:

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.

Fractal Edition 1, P3

Tags: Perfect Squares

Can the number $ \overline{abc} + \overline{bca} + \overline{cab} $ be a perfect square?

2005 JHMT, 2

Tags: geometry

Regular hexagon $ABCDEF$ is inscribed in rectangle $PQRS$ with $AB = 1$, A and $B$ on side $PQ$, $C$ on side $QR$, $D$ and $E$ on side $RS$, and $F$ on side $SP$. What is the area of $PQRS$?

2012 Indonesia TST, 1

Tags: induction , algebra proposed , algebra , binary representation , Sequence , recurrence relation

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

2016 Peru IMO TST, 7

Tags: IMO Shortlist , combinatorics , Additive combinatorics

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

2012 Junior Balkan MO, 2

Tags: geometry , JBMO

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

1998 French Mathematical Olympiad, Problem 3

Tags: floor function , algebra , function

Let $k\ge2$ be an integer. The function $f:\mathbb N\to\mathbb N$ is defined by $$f(n)=n+\left\lfloor\sqrt[k]{n+\sqrt[k]n}\right\rfloor.$$Determine the set of values taken by the function $f$.

2020 USOMO, 3

Tags: number theory , Hi

Let $p$ be an odd prime. An integer $x$ is called a [i]quadratic non-residue[/i] if $p$ does not divide $x-t^2$ for any integer $t$. Denote by $A$ the set of all integers $a$ such that $1\le a<p$, and both $a$ and $4-a$ are quadratic non-residues. Calculate the remainder when the product of the elements of $A$ is divided by $p$. [i]Proposed by Richard Stong and Toni Bluher[/i]