Olympiad problems

1985 Dutch Mathematical Olympiad, 2

Among the numbers $ 11n \plus{} 10^{10}$, where $ 1 \le n \le 10^{10}$ is an integer, how many are squares?

2015 Sharygin Geometry Olympiad, 5

Tags: geometry , midpoint , concurrency

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)

2018 Caucasus Mathematical Olympiad, 1

Tags: combinatorics , parity

A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.

2023 Princeton University Math Competition, A2 / B4

Tags: geometry

Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.

2020 BMT Fall, 17

Tags: algebra , polynomial

Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.

2019 Slovenia Team Selection Test, 3

Tags: combinatorics

Let $n$ be any positive integer and $M$ a set that contains $n$ positive integers. A sequence with $2^n$ elements is christmassy if every element of the sequence is an element of $M$. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.

1990 Tournament Of Towns, (271) 5

Tags: algebra , periodic , sequence , recurrence relation

The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$. (M Kontsevich, Moscow)

1946 Moscow Mathematical Olympiad, 113

Tags: number theory , divisible

Prove that $n^2 + 3n + 5$ is not divisible by $121$ for any positive integer $n$.

1987 Federal Competition For Advanced Students, P2, 2

Tags: combinatorics

Find the number of all sequences $ (x_1,...,x_n)$ of letters $ a,b,c$ that satisfy $ x_1\equal{}x_n\equal{}a$ and $ x_i \not\equal{} x_{i\plus{}1}$ for $ 1 \le i \le n\minus{}1.$

1984 National High School Mathematics League, 3

Tags: geometry

In $\triangle ABC$, $P$ is a point on $BC$. $F\in AB,E\in AC,PF//CA,PE//BA$. If $S_{\triangle ABC}=1$, prove that at least one of $S_{\triangle BPF},S_{\triangle PCE},S_{PEAF}$ is not less than $\frac{4}{9}$.

1985 All Soviet Union Mathematical Olympiad, 400

Tags: trinomial , integer , maximum , polynomial , algebra

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

2002 Korea Junior Math Olympiad, 4

Tags: number theory

For two non-negative integers $i, j$, create a new integer $i \# j$ defined as the following: Express the two numbers in base $2$, and compare each digit. If their $k$th digit is the same, then the $k$th digit of $i \# j$ is $0$. If their $k$th digit is different, then the $k$th digit of $i \# j$ is $1$(of course we are talking in base $2$). For instance, $3 \# 5=6$. Show that for arbitrary positive integer $n$, the number can be expressed with finite operations of $\#$s and integers of the form $2^k-1$.

2005 MOP Homework, 4

Tags: algebra , function , polynomial

Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$, $f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$

2022 Nigerian Senior MO Round 2, Problem 6

Tags: number theory , diophantine equation

Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

1987 Kurschak Competition, 2

Tags: geometry

Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?

2003 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , equal segments , equal angles

In triangle $ABC$, angle $C$ is a right angle. Found on the side $AC$ point $D$, and on the segment $BD$, point $K$ such that $\angle ABC = \angle KAD =\angle AKD$. Prove that $BK = 2DC$.

2012 AMC 12/AHSME, 17

Tags: analytic geometry , graphing lines , slope , trigonometry , geometric transformation , dilation , geometry

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

2009 Kazakhstan National Olympiad, 4

Tags: inequalities

Let $0<a_1 \leq a_2 \leq \cdots\leq a_n $ ($n \geq 3; n \in \mathbb{N}$) be $n$ real numbers. Prove the inequality \[\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n\]

2010 Kyrgyzstan National Olympiad, 7

Tags: number theory

Find all natural triples $(a,b,c)$, such that: $a - )\,a \le b \le c$ $b - )\,(a,b,c) = 1$ $c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.

1994 Hong Kong TST, 1

Tags: trigonometry , geometry

In a $\triangle ABC$, $\angle C=2 \angle B$. $P$ is a point in the interior of $\triangle ABC$ satisfying that $AP=AC$ and $PB=PC$. Show that $AP$ trisects the angle $\angle A$.

2014 Contests, 1

Tags: algebra , system of equations

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

Istek Lyceum Math Olympiad 2016, 2

Tags: inversion , circles , geometry

Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$

2001 APMO, 5

Tags: geometry

Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1,\dots,~X_n$ in the plane with $AB\ne CD$ that satisfy the following condition: for each $i=1,2,\dots,n$ triangles $ABX_i$ and $CDX_i$ are equal.

2001 AMC 10, 15

Tags: geometry , parallelogram , similar triangles

A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes. $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$

2018 Junior Balkan Team Selection Tests - Moldova, 2

Tags: geometry

Let $ABC$ be an acute triangle.Let $OF \| BC$ where $O$ is the circumcenter and $F$ is between $A$ and $B$.Let $H$ be the orthocenter.Let $M$ be the midpoint of $AH$.Prove that $\angle FMC=90$.