diff --git a/libcxx/utils/ci/Dockerfile b/libcxx/utils/ci/Dockerfile
index eb3b810c1ef9..aeb1d2f24952 100644
--- a/libcxx/utils/ci/Dockerfile
+++ b/libcxx/utils/ci/Dockerfile
@@ -39,28 +39,36 @@ RUN apt-get update && apt-get install -y bash curl
 RUN apt-get update && apt-get install -y ninja-build python3 python3-sphinx python3-distutils git gdb
 RUN apt-get update && apt-get install -y libc6-dev-i386 # Required to cross-compile to 32 bits
 
-# Install the most recently released LLVM
+# Install Clang <latest>, <latest-1> and ToT, which are the ones we support.
+ENV LLVM_LATEST_VERSION=12
 RUN apt-get update && apt-get install -y lsb-release wget software-properties-common
 RUN wget https://apt.llvm.org/llvm.sh -O /tmp/llvm.sh
-RUN bash /tmp/llvm.sh
-RUN LLVM_VERSION=$(find /usr/bin -regex '^.+/clang-[0-9.]+$') && LLVM_VERSION=${LLVM_VERSION#*clang-} && echo "LLVM_VERSION=$LLVM_VERSION" > /tmp/env.sh
-RUN ln -s $(find /usr/bin -regex '^.+/clang\+\+-[0-9.]+$') /usr/bin/clang++ && [ -e $(readlink /usr/bin/clang++) ]
-RUN ln -s $(find /usr/bin -regex '^.+/clang-[0-9.]+$') /usr/bin/clang && [ -e $(readlink /usr/bin/clang) ]
+RUN bash /tmp/llvm.sh $(($LLVM_LATEST_VERSION - 1)) # previous release
+RUN bash /tmp/llvm.sh $LLVM_LATEST_VERSION          # latest release
+RUN bash /tmp/llvm.sh $(($LLVM_LATEST_VERSION + 1)) # current ToT
 
-# Install the not-yet-released LLVM
-RUN . /tmp/env.sh && echo "LLVM_TOT_VERSION=$(($LLVM_VERSION + 1))" >> /tmp/env.sh
-RUN . /tmp/env.sh && bash /tmp/llvm.sh ${LLVM_TOT_VERSION}
-RUN . /tmp/env.sh && ln -s /usr/bin/clang++-${LLVM_TOT_VERSION} /usr/bin/clang++-tot && [ -e $(readlink /usr/bin/clang++-tot) ]
-RUN . /tmp/env.sh && ln -s /usr/bin/clang-${LLVM_TOT_VERSION} /usr/bin/clang-tot && [ -e $(readlink /usr/bin/clang-tot) ]
+# Make the latest Clang the "default" compiler on the system
+RUN ln -fs /usr/bin/clang++-$LLVM_LATEST_VERSION /usr/bin/c++ && [ -e $(readlink /usr/bin/c++) ]
+RUN ln -fs /usr/bin/clang-$LLVM_LATEST_VERSION /usr/bin/cc && [ -e $(readlink /usr/bin/cc) ]
+
+# Temporarily keep the clang++ and clang++-tot symlinks around for backwards compatibility with builders
+RUN ln -s /usr/bin/clang++-$LLVM_LATEST_VERSION /usr/bin/clang++ && [ -e $(readlink /usr/bin/clang++) ]
+RUN ln -s /usr/bin/clang-$LLVM_LATEST_VERSION /usr/bin/clang && [ -e $(readlink /usr/bin/clang) ]
+RUN ln -s /usr/bin/clang++-$(($LLVM_LATEST_VERSION + 1)) /usr/bin/clang++-tot && [ -e $(readlink /usr/bin/clang++-tot) ]
+RUN ln -s /usr/bin/clang-$(($LLVM_LATEST_VERSION + 1)) /usr/bin/clang-tot && [ -e $(readlink /usr/bin/clang-tot) ]
 
 # Install clang-format
-RUN . /tmp/env.sh && apt-get install -y clang-format-$LLVM_VERSION
-RUN ln -s $(find /usr/bin -regex '^.+/clang-format-[0-9.]+$') /usr/bin/clang-format && [ -e $(readlink /usr/bin/clang-format) ]
-RUN ln -s $(find /usr/bin -regex '^.+/git-clang-format-[0-9.]+$') /usr/bin/git-clang-format && [ -e $(readlink /usr/bin/git-clang-format) ]
+RUN apt-get install -y clang-format-$LLVM_LATEST_VERSION
+RUN ln -s /usr/bin/clang-format-$LLVM_LATEST_VERSION /usr/bin/clang-format && [ -e $(readlink /usr/bin/clang-format) ]
+RUN ln -s /usr/bin/git-clang-format-$LLVM_LATEST_VERSION /usr/bin/git-clang-format && [ -e $(readlink /usr/bin/git-clang-format) ]
 
-# Install the most recent GCC version and the previous one (which are the two we support)
+# Install the most recent GCC
+ENV GCC_LATEST_VERSION=11
 RUN add-apt-repository ppa:ubuntu-toolchain-r/test
-RUN apt-get update && apt install -y gcc-10 g++-10 gcc-11 g++-11
+RUN apt-get update && apt install -y gcc-$GCC_LATEST_VERSION g++-$GCC_LATEST_VERSION
+
+# Temporarily keep installing GCC 10 for backwards compatibility with build bots
+RUN apt-get update && apt install -y gcc-10 g++-10
 
 # Install a recent CMake
 RUN wget https://github.com/Kitware/CMake/releases/download/v3.18.2/cmake-3.18.2-Linux-x86_64.sh -O /tmp/install-cmake.sh