Olympiad problems

1966 Poland - Second Round, 6

Prove that the sum of the squares of the right-angled projections of the sides of a triangle onto the line $ p $ of the plane of this triangle does not depend on the position of the line $ p $ if and only if it the triangle is equilateral.

2016 Harvard-MIT Mathematics Tournament, 3

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Let $PROBLEMZ$ be a regular octagon inscribed in a circle of unit radius. Diagonals $MR$, $OZ$ meet at $I$. Compute $LI$.

1997 Miklós Schweitzer, 4

Tags: linear algebra , binary matrix

An elementary change in a 0-1 matrix is a change in an element and with it all its horizontal, vertical, and diagonal neighbors (0 to 1 or 1 to 0). Can any 1791 x 1791 0-1 matrix be transformed into a zero matrix with elementary changes?

1955 AMC 12/AHSME, 26

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Mr. A owns a house worth $ \$10000$. He sells it to Mr. B at $ 10 \%$ profit. Mr. B sells the house back to Mr. A at a $ 10 \%$ loss. Then: $ \textbf{(A)}\ \text{Mr. A comes out even} \qquad \textbf{(B)}\ \text{Mr. A makes }\$100 \qquad \textbf{(C)}\ \text{Mr. A makes }\$1000 \\ \textbf{(D)}\ \text{Mr. B loses }\$100 \qquad \textbf{(E)}\ \text{none of the above is correct}$

2024 JHMT HS, 16

Tags: number theory

Let $N_{15}$ be the answer to problem 15. For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.

Kyiv City MO Juniors 2003+ geometry, 2015.9.3

Tags: geometry , trapezoid , construction , square

It is known that a square can be inscribed in a given right trapezoid so that each of its vertices lies on the corresponding side of the trapezoid (none of the vertices of the square coincides with the vertex of the trapezoid). Construct this inscribed square with a compass and a ruler. (Maria Rozhkova)

2020 Azerbaijan National Olympiad, 2

Tags: number theory

$a,b,c$ are positive integer. Solve the equation: $ 2^{a!}+2^{b!}=c^3 $

2008 IMS, 9

Tags: topology , real analysis

Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$ \[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha \] in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , geometric inequality , area , inequalities

Points $A_0,A_1,\ldots,A_{2k}$, in this order, divide a circumference into $2k+1$ equal arcs. Point $A_0$ is connected by chords to all the other points. These $2k$ chords divide the interior of the circle into $2k+1$ parts. These parts are alternately painted red and blue so that there are $k+1$ red and $k$ blue parts. Show that the blue area is larger than the red area.

1998 AMC 12/AHSME, 4

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Define $[a,b,c]$ to mean $\frac{a+b}{c},$ where $c \neq 0$. What is the value of \[[[60,30,90],[2,1,3],[10,5,15]]?\] $\text{(A)} \ 0 \qquad \text{(B)} \ 0.5 \qquad \text{(C)} \ 1 \qquad \text{(D)} \ 1.5 \qquad \text{(E)} \ 2$

2014 Regional Competition For Advanced Students, 1

Tags: algebra , equation

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

2018 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , set

Let $A =\left\{a = q + \frac{1}{q }/ q \in Q^*,q > 0 \right\}$, $A + A = \{a + b |a,b \in A\}$,$A \cdot A =\{a \cdot b | a, b \in A\}$. Prove that: i) $A + A \ne A \cdot A$ ii) $(A + A) \cap N = (A \cdot A) \cap N$. Vasile Pop

2023 Indonesia TST, C

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There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.

2017 Hanoi Open Mathematics Competitions, 11

Tags: geometry , inequalities , geometric inequality , equilateral , angle

Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

2005 USAMTS Problems, 5

Tags: geometry , 3d geometry , sphere

Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.

2022 IFYM, Sozopol, 6

Tags: polynomial , algebra

Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.

2014 Contests, 3

Tags: number theory

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

2016 Brazil Team Selection Test, 4

Tags: 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.

2008 Stanford Mathematics Tournament, 6

Tags: geometry , 3d geometry

A round pencil has length $ 8$ when unsharpened, and diameter $ \frac {1}{4}$. It is sharpened perfectly so that it remains $ 8$ inches long, with a $ 7$ inch section still cylindrical and the remaining $ 1$ inch giving a conical tip. What is its volume?

2010 Romania National Olympiad, 2

Tags: number theory

How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?

2024 LMT Fall, 12

Tags: guts

Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$

2004 Peru MO (ONEM), 4

Tags: geometry , integer , sides of a triangle , area of a triangle , minimum

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

1989 IMO Longlists, 23

Tags: geometry

Let $ ABC$ be a triangle. Prove that there is a unique point $ U$ in the plane of $ ABC$ such that there exist real numbers $ \alpha, \beta, \gamma, \delta$ not all zero, such that \[ \alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \delta UP^2\] is constant for all points $ P$ of the plane, where $ L,M,N$ are the feet of the perpendiculars from $ P$ to $ BC,CA,AB$ respectively. Identify $ U.$

2001 District Olympiad, 1

Tags: matrix , algebra , polynomial , linear algebra

Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$. [i]Daniel Jinga[/i]

2001 Polish MO Finals, 3

Tags: algebra

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.