Olympiad problems

KoMaL A Problems 2023/2024, A. 858

Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$

2008 May Olympiad, 3

Tags: number theory , prime

In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.

2020 Philippine MO, 2

Tags: algebra

Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$

2020 Tournament Of Towns, 5

Tags: right angle , trapezoid , equal segments , trapezium , geometry

Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

2012 Dutch IMO TST, 1

Tags: perfect power , number theory , prime , gcd , greatest common divisor

For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2019 Taiwan APMO Preliminary Test, P4

Tags: algebra , floor function , sequence , recursive

We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$ (1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

1975 Dutch Mathematical Olympiad, 3

Tags: inequalities , sum , algebra

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

2023 Assam Mathematics Olympiad, 8

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If $n$ is a positive even number, find the last two digits of $(2^{6n}+26)-(6^{2n}-62)$.

1987 AIME Problems, 11

Tags: calculus , integration , quadratic , inequalities , algorithm , algebra

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

2021 Bangladeshi National Mathematical Olympiad, 3

Tags: number theory , integer part , floor function

Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$?

2006 National Olympiad First Round, 16

Tags: inequalities

How many positive integer tuples $ (x_1,x_2,\dots, x_{13})$ are there satisfying the inequality $x_1+x_2+\dots + x_{13}\leq 2006$? $ \textbf{(A)}\ \frac{2006!}{13!1993!} \qquad\textbf{(B)}\ \frac{2006!}{14!1992!} \qquad\textbf{(C)}\ \frac{1993!}{12!1981!} \qquad\textbf{(D)}\ \frac{1993!}{13!1980!} \qquad\textbf{(E)}\ \text{None of above} $

LMT Theme Rounds, 10

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Let $S=\{1,2,3,4,5,6\}.$ Find the number of bijective functions $f:S\rightarrow S$ for which there exist exactly $6$ bijective functions $g:S\rightarrow S$ such that $f(g(x))=g(f(x))$ for all $x\in S$. [i]Proposed by Nathan Ramesh

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: number theory , divisor

Find the three-digit number that has the greatest number of different divisors.

2024 Belarusian National Olympiad, 10.7

Tags: number theory

Let's call a pair of positive integers $(k,n)$ [i]interesting[/i] if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$ Find the number of interesting pairs $(k,n)$ with $k \leq 100$ [i]M. Karpuk[/i]

2023 India Regional Mathematical Olympiad, 3

Tags: number theory

For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \text { and }(s(m))^2=n . $$

2023 Brazil Team Selection Test, 1

Tags: combinatorics , integer sequence

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

2016 USAJMO, 5

Tags: geometry

Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, respectively. Given that $$AH^2=2\cdot AO^2,$$ prove that the points $O,P,$ and $Q$ are collinear.

2016 Online Math Open Problems, 5

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Jay notices that there are $n$ primes that form an arithmetic sequence with common difference $12$. What is the maximum possible value for $n$? [i]Proposed by James Lin[/i]

2003 AMC 10, 18

Tags: algebra , polynomial , vieta , big big vieta

What is the sum of the reciprocals of the roots of the equation \[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? \] $ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$

2018 Online Math Open Problems, 13

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Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$. [i]Proposed by James Lin

2009 CHKMO, 4

Tags: combinatorics

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

2024 USAJMO, 5

Tags: function , functional equation , pointwise trap

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[ f(x^2-y)+2yf(x)=f(f(x))+f(y) \] for all $x,y\in\mathbb{R}$. [i]Proposed by Carl Schildkraut[/i]

1997 Portugal MO, 5

Tags: line segment , geometry

A square region of side $12$ contains a water source that supplies an irrigation system constituted by several straight channels forming polygonal lines. Considers the source as a point and each channel as a line segment. Knowing that a point is irrigated if it is not more than $1$ distance from any channel and that the system was designed so that the entire region is irrigated, proves that the total length of irrigation channels exceeds $70$.

2025 Harvard-MIT Mathematics Tournament, 1

Tags: algebra , number theory

Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1.$

Brazil L2 Finals (OBM) - geometry, 2018.4

Tags:

a) In $XYZ$ triangle, the incircle touches $XY$ and $XZ$ in $T$ and $W$, respectively. Prove that: $$XT=XW=\frac{XY+XZ-YZ}2$$ Let $ABC$ a triangle and $D$ the foot of the perpendicular of $A$ in $BC$. Let $I$, $J$ be the incenters of $ABD$ and $ACD$, respectively. The incircles of $ABD$ and $ACD$ touch $AD$ in $M$ and $N$, respectively. Let $P$ be where the incircle of $ABC$ touches $AB$. The circle with centre $A$ and radius $AP$ intersects $AD$ in $K$. b) Show that $\triangle IMK \cong \triangle KNJ$. c) Show that $IDJK$ is cyclic.