Olympiad problems

2005 Federal Competition For Advanced Students, Part 2, 3

Tags: geometry

Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$ extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.

2007 Iran Team Selection Test, 1

Tags: geometric transformation , reflection , analytic geometry , geometry

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position [i]By Sam Nariman[/i]

2015 Macedonia National Olympiad, Problem 3

Tags: combinatorics

All contestants at one contest are sitting in $n$ columns and are forming a "good" configuration. (We define one configuration as "good" when we don't have 2 friends sitting in the same column). It's impossible for all the students to sit in $n-1$ columns in a "good" configuration. Prove that we can always choose contestants $M_1,M_2,...,M_n$ such that $M_i$ is sitting in the $i-th$ column, for each $i=1,2,...,n$ and $M_i$ is friend of $M_{i+1}$ for each $i=1,2,...,n-1$.

2017 Kosovo National Mathematical Olympiad, 4

Tags: combinatorics

Prove the identity $\sum_{k=2}^{n} k(k-1)\binom{n}{k} =\binom{n}{2} 2^{n-1}$ for all $n=2,3,4,...$

2016 Mathematical Talent Reward Programme, MCQ: P 6

Tags: algebra , diophantine equation

Number of solutions of the equation $3^x+4^x=8^x$ in reals is [list=1] [*] 0 [*] 1 [*] 2 [*] $\infty$ [/list]

2021 Moldova EGMO TST, 6

Tags: number theory

How many $3$ digit positive integers are not divided by $5$ neither by $7$?

2025 Balkan MO, 3

Tags: functional equation , algebra

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[f(x+yf(x))+y = xy + f(x+y).\] [i]Proposed by Giannis Galamatis, Greece[/i]

2023 China Team Selection Test, P21

Tags: inequalities

Given integer $n\geq 2$. Find the minimum value of $\lambda {}$, satisfy that for any real numbers $a_1$, $a_2$, $\cdots$, ${a_n}$ and ${b}$, $$\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.$$

1988 National High School Mathematics League, 3

Tags:

$M,N,P$ are three point sets on a plane. $M=\{(x,y)||x|+|y|<1\}$, $N=\{(x,y)|\sqrt{(x-\frac{1}{2})^2+(y+\frac{1}{2})^2}+\sqrt{(x+\frac{1}{2})^2+(y-\frac{1}{2})^2}<2 \sqrt2 \}$, $P=\{(x,y)||x+y|<1,|x|<1,|y|<1\}$.Then $\text{(A)}M\subset P\subset N\qquad\text{(B)}M\subset N\subset P\qquad\text{(C)}P\subset N\subset M\qquad\text{(D)}$ None of$\text{(A)(B)(C)}$

2001 Grosman Memorial Mathematical Olympiad, 6

Tags: diophantine equation , diophantine , number theory

(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$ (b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?

2015 Caucasus Mathematical Olympiad, 1

Tags: digit , number theory , divisible , sum

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

2018 Turkey EGMO TST, 6

Tags: number theory

Let $f:\mathbb{Z}_{+}\rightarrow\mathbb{Z}_{+}$ is one to one and bijective function. Prove that $f(mn)=f (m)f (n)$ if and only if $lcm (f (m),f (n))=f(lcm(m,n)) $

1998 Vietnam National Olympiad, 3

Tags: polynomial , algebra

Find all positive integer $n$ such that there exists a $P\in\mathbb{R}[x]$ satisfying $P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}$.

2006 Estonia Math Open Senior Contests, 6

Tags: rotation , pigeonhole principle , geometry , geometric transformation , combinatorics

Kati cut two equal regular $ n\minus{}gons$ out of paper. To the vertices of both $ n\minus{}gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n\minus{}gons$ so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the $ n\minus{}gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

2017 All-Russian Olympiad, 4

Tags: combinatorics

Every cell of $100\times 100$ table is colored black or white. Every cell on table border is black. It is known, that in every $2\times 2$ square there are cells of two colors. Prove, that exist $2\times 2$ square that is colored in chess order.

2001 Saint Petersburg Mathematical Olympiad, 11.7

Tags: tiling , coloring , rectangle , domino , combinatorics

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder. Proposed by S. Berlov

2015 India PRMO, 7

Tags: number theory , digit

$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$

1974 AMC 12/AHSME, 5

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Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^{\circ}$ and $\measuredangle ADC=68^{\circ}$, find $\measuredangle EBC$. $ \textbf{(A)}\ 66^{\circ} \qquad\textbf{(B)}\ 68^{\circ} \qquad\textbf{(C)}\ 70^{\circ} \qquad\textbf{(D)}\ 88^{\circ} \qquad\textbf{(E)}\ 92^{\circ} $

2011 Singapore Senior Math Olympiad, 5

Tags: inequalities

Given $x_1,x_2,\dots,x_n>0,n\geq 5$, show that \[\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}\]

1967 Spain Mathematical Olympiad, 2

Tags: geometry , rectangle , inversion , collinear

Determine the poles of the inversions that transform four collienar points $A,B, C, D$, aligned in this order, at four points $A' $, $B' $, $C'$ , $D'$ that are vertices of a rectangle, and such that $A'$ and $C'$ are opposite vertices.

2006 Brazil National Olympiad, 3

Tags: function , induction , functional equation , algebra

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that \[f(xf(y)+f(x)) = 2f(x)+xy\] for every reals $x,y$.

2022 MIG, 1

Tags:

In a certain store, all pencils cost the same amount of money. If three pencils can be bought for six dollars, what is the price of two pencils? $\textbf{(A) }\$ 3\qquad\textbf{(B) }\$ 3.5\qquad\textbf{(C) }\$ 4\qquad\textbf{(D) }\$4.5\qquad\textbf{(E) }\$ 5$

2015 Tuymaada Olympiad, 6

Tags: number theory

Is there sequence $(a_n)$ of natural numbers, such that differences $\{a_{n+1}-a_n\}$ take every natural value and only one time and differences $\{a_{n+2}-a_n\}$ take every natural value greater $2015$ and only one time ? [i]A. Golovanov[/i]

2017 Putnam, B2

Tags:

Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[N=a+(a+1)+(a+2)+\cdots+(a+k-1)\] for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?

2015 USAJMO, 4

Tags: function , arithmetic sequence

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)