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@confuest,

Run this command and tell me what the output is:
cat /etc/*-release

This will help in trying to understand your problem

@roboticmind , the output is below

PRETTY_NAME="Raspbian GNU/Linux 9 (stretch)"
NAME="Raspbian GNU/Linux"
VERSION_ID="9"
VERSION="9 (stretch)"
ID=raspbian
ID_LIKE=debian
HOME_URL="http://www.raspbian.org/"
SUPPORT_URL="http://www.raspbian.org/RaspbianForums"
BUG_REPORT_URL="http://www.raspbian.org/RaspbianBugs"

@confuest
Try running these commands:
sudo apt-get install libboost-all-dev -y
sudo apt-get install libssl1.0.2

wget http://ftp.us.debian.org/debian/pool/main/libz/libzip/libzip2_0.11.2-1.2_armhf.deb (all one line)

sudo dpkg -i libzip2_0.11.2-1.2_armhf.deb

Then run the command:
sudo apt-get install gridcoinresearch-qt -y

@roboticmind
For the libssl command it returns the output below:
"Reading package lists... Done
Building dependency tree
Reading state information... Done
libssl1.0.2 is already the newest version (1.0.2l-2).
0 upgraded, 0 newly installed, 0 to remove and 37 not upgraded."

AND at the end after typing "sudo apt-get install gridcoinresearch-qt -y" I got the output below:

"The following packages have unmet dependencies:
gridcoinresearch-qt : Depends: libboost-filesystem1.55.0 but it is not installable
Depends: libboost-program-options1.55.0 but it is not installable
Depends: libboost-system1.55.0 but it is not installable
Depends: libboost-thread1.55.0 but it is not installable
Depends: libssl1.0.0 (>= 1.0.1) but it is not installable
E: Unable to correct problems, you have held broken packages."

@confuest,
Try running these commands:
wget http://ftp.us.debian.org/debian/pool/main/o/openssl/libssl1.0.0_1.0.1t-1+deb8u6_armhf.deb
( one line)

sudo dpkg -i libssl1.0.0_1.0.1t-1+deb8u6_armhf.deb
(one line)

wget http://ftp.us.debian.org/debian/pool/main/b/boost-defaults/libboost-all-dev_1.55.0.2_armhf.deb
(one line)

dpkg -i libboost-all-dev_1.55.0.2_armhf.deb
(one line)

Then run:
sudo apt-get install gridcoinresearch-qt -y
(one line)

@roboticmind ,
libssl solved but others still remaining as you can see the output below:

The following packages have unmet dependencies:
gridcoinresearch-qt : Depends: libboost-filesystem1.55.0 but it is not installable
Depends: libboost-program-options1.55.0 but it is not installable
Depends: libboost-system1.55.0 but it is not installable
Depends: libboost-thread1.55.0 but it is not installable
E: Unable to correct problems, you have held broken packages.

Try running:
wget http://ftp.us.debian.org/debian/pool/main/b/boost1.55/libboost1.55-all-dev_1.55.0+dfsg-3_armhf.deb
(one line)

dpkg - i libboost1.55-all-dev_1.55.0+dfsg-3_armhf.deb

Then run:
sudo apt-get install gridcoinresearch-qt -y

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